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1. Now we can visualize the transformed iris dataset gt gt gt import pylab as pl gt gt gt pl scatter XL 0 XL 1 c iris target lt matplotiib collections Collection object at PCA is not just useful for visualization of high dimensional datasets It can also be used as a preprocessing step to help speed up supervised methods that are not efficient with high dimensions 20 4 Dimension Reduction with Principal Component Analysis 359 Scipy lecture notes Edition 2015 2 20 5 Putting it all together face recognition An example showcasing face recognition using Principal Component Analysis for dimension reduction and Support Vector Machines for classification Stripped down version of the face recognition example import numpy as np import pylab as pl from sklearn import cross_val datasets decomposition svm load data lfw_people datasets fetch_lfw_people min_faces_per_person 70 resize 0 4 perm np random permutation lfw_people target size lfw_people data 1lfw_people datal perm lfw_people target 1lfw_people target perm faces np reshape lfw_people data lfw_people target shape 0 1 train test iter cross_val StratifiedKFold lfw_people target k 4 next X_train X_test faces train faces test y_train y_test lfw_people target train lfw_people target test eae dimension reduction pca decomposition RandomizedPCA n_components 150 whiten Tr
2. 0 0000 e eee 362 Index 363 Scipy lecture notes Edition 2015 2 Contents 1 Part I Getting started with Python for science Scipy lecture notes Edition 2015 2 This part of the Scipy lecture notes is a self contained introduction to everything that is needed to use Python for science from the language itself to numerical computing or plotting CHAPTER 1 Scientific computing with tools and workflow Authors Fernando Perez Emmanuelle Gouillart Ga l Varoquaux Valentin Haenel 1 1 Why Python 1 1 1 The scientist s needs Get data simulation experiment control Manipulate and process data Visualize results to understand what we are doing Communicate results produce figures for reports or publications write presentations 1 1 2 Specifications Rich collection of already existing bricks corresponding to classical numerical methods or basic actions we don t want to re program the plotting of a curve a Fourier transform or a fitting algorithm Don t reinvent the wheel Easy to learn computer science is neither our job nor our education We want to be able to draw a curve smooth a signal do a Fourier transform in a few minutes Easy communication with collaborators students customers to make the code live within a lab or a company the code should be as readable as a book Thus the language should contain as few syn tax symbols or unneeded routines as possible that would divert the reader from
3. Convert the image into a graph with the value of the gradient on the edges graph image img to_graph img mask mask Take a decreasing function of the gradient we take it weakly dependant from the gradient the segmentation ts close to a voronor graph data np exp graph data graph data std labels spectral_clustering graph n_clusters 4 eigen_solver arpack label_im np ones mask shape label_im mask labels 12 5 Feature extraction 242 Scipy lecture notes Edition 2015 2 12 6 Measuring objects properties ndimage measurements Synthetic data gt gt gt n 10 gt gt gt l1 256 gt gt gt im np zeros 1 1 gt gt gt points 1 np random random 2 n 2 gt gt gt imL points 0 astype np int points 1 astype np int 1 gt gt gt im ndimage gaussian_filter im sigma 1 4 n gt gt gt mask im gt im mean e Analysis of connected components Label connected components ndimage label gt gt gt label_im nb_labels ndimage label mask gt gt gt nb_labels how many regions 16 gt gt gt plt imshow label_im lt matplotlib image AxesImage object at Ox gt Compute size mean_value etc of each region gt gt gt sizes ndimage sum mask label_im range nb_labels 1 gt gt gt mean_vals ndimage sum im label_im range 1 nb_labels 1 Clean up small connect components gt gt gt mask_size sizes lt 1000 gt gt
4. You can also download each of the examples and run it using regular python but you will loose interactive data manipulation python exercice_1 py You can get source for each step by clicking on the corresponding figure 4 2 1 Plotting with default settings Documentation e plot tutorial e plot command 4 2 Simple plot 33 Scipy lecture notes Edition 2015 2 Matplotlib comes with a set of default settings that allow customizing all kinds of properties You can control the defaults of almost every property in matplotlib figure size and dpi line width color and style axes axis and grid properties text and font properties and so on import numpy as np import matplotlib pyplot as plt X np linspace np pi np pi 256 endpoint True C S np cos X np sin X plt plot X C plt plot X S plt show 4 2 2 Instantiating defaults Documentation e Customizing matplotlib In the script below we ve instantiated and commented all the figure settings that influence the appearance of the plot The settings have been explicitly set to their default values but now you can interactively play with the values to explore their affect see Line properties and Line styles below import numpy as np import matplotlib pyplot as plt Create a figure of size 826 inches 80 dots per inch plt figure figsize 8 6 dpi 80 Create a new subplot from a grid of izi plt
5. cimport the Cython declarations for numpy cimport numpy as np if you want to use the Numpy C API from Cython not strictly necessary for this example but good practice np import_array cdefine the signature of our c function cdef extern from cos_doubles h void cos_doubles double in_array double out_array int size create the wrapper code with numpy type annotations def cos_doubles_func np ndarray double ndim 1 mode c in_array not None np ndarray double ndim 1 mode c out_array not None cos_doubles lt double gt np PyArray_DATA in_array lt double gt np PyArray_DATA out_array in_array shape 0 And can be compiled using distutils from distutils core import setup Extension import numpy from Cython Distutils import build_ext 14 5 Cython 278 Scipy lecture notes Edition 2015 2 setup cmdclass build_ext build_ext ext_modules Extension cos_doubles sources _cos_doubles pyx cos_doubles c include_dirs numpy get_include e As with the previous compiled Numpy examples we need the include_dirs option 1s cos_doubles c cos_doubles h _cos_doubles pyx setup py test_cos_doubles py python setup py build_ext i running build_ext cythoning _cos_doubles pyx to _cos_doubles c building cos_doubles extension creating build creating build temp linux x86_64 2 7 gcc pthread fno strict aliasing g 02 DNDEBUG g fwrapv 03 Wall Wstric
6. 15 3 Linear models multiple factors and analysis of variance 296 Scipy lecture notes Edition 2015 2 15 3 3 Post hoc hypothesis testing analysis of variance ANOVA In the above iris example we wish to test if the petal length is different between versicolor and virginica after removing the effect of sepal width This can be formulated as testing the difference between the coefficient as sociated to versicolor and virginica in the linear model estimated above it is an Analysis of Variance ANOVA For this we write a vector of contrast on the parameters estimated we want to test name T versicolor name T virginica with an F test gt gt gt print Mmodeld Gest ClO 1 1 01 lt F test F array LL 3 24533535 p LL 0 07369059 df_denom 146 df_num 1 gt Is this difference significant Exercice Going back to the brain size IQ data test if the VIQ of male and female are different after removing the effect of brain size height and weight 15 4 More visualization seaborn for statistical exploration Seaborn combines simple statistical fits with plotting on pandas dataframes Let us consider a data giving wages and many other personal information on 500 individuals Berndt ER The Practice of Econometrics 1991 NY Addison Wesley The full code loading and plotting of the wages data is found in corresponding example gt gt gt print data EDUCATION SOUTH SEX EXPERIENCE UNION WAGE AGE RACE
7. And visualize the solution 5 11 Summary exercises on scientific computing 130 Scipy lecture notes Edition 2015 2 gt gt gt plt plot t waveform_1 t model t x lt matplotlib lines Line2D object at gt lt matplotlib lines Line2D object at gt gt gt gt plt legend waveform model lt matplotlib legend Legend object at gt gt gt gt plt show Remark from scipy v0 8 and above you should rather use scipy optimize curve_fit which takes the model and the data as arguments so you don t need to define the residuals any more Going further e Try with a more complex waveform for instance data waveform_2 npy that contains three significant peaks You must adapt the model which is now a sum of Gaussian functions instead of only one Gaussian peak Intensity bins 0 10 20 30 AQ 50 60 70 80 Time ns e In some cases writing an explicit function to compute the Jacobian is faster than letting leastsgq esti mate it numerically Create a function to compute the Jacobian of the residuals and use it as an input for leastsq e When we want to detect very small peaks in the signal or when the initial guess is too far from a good solution the result given by the algorithm is often not satisfying Adding constraints to the parameters of the model enables to overcome such limitations An example of a priori knowledge we can add is the sign of our variables which are all positive With the fo
8. gt gt gt from scipy import ndimage Image processing routines may be sorted according to the category of processing they perform 5 10 1 Geometrical transformations on images Changing orientation resolution gt gt gt from scipy import misc gt gt gt face misc face gray True gt gt gt shifted_face ndimage shift face 50 50 gt gt gt shifted_face2 ndimage shift face 50 50 mode nearest gt gt gt rotated_face ndimage rotate face 30 gt gt gt cropped_face face 50 50 50 50 gt gt gt zoomed_face ndimage zoom face 2 gt gt gt zoomed_face shape 1536 2048 gt gt gt plt subplot 151 lt matplotlib axes _subplots AxesSubplot object at Ox gt gt gt gt plt imshow shifted_face cmap plt cm gray lt matplotlib image AxesImage object at Ox gt gt gt gt plt axis off Osa ULES Ere 065 gt gt gt etc 5 10 2 Image filtering from scipy import misc face misc face gray True face face 512 512 crop out square on right import numpy as np noisy_face np copy face astype np float noisy_face face std 0 5 np random standard_normal face shape blurred_face ndimage gaussian_filter noisy_face sigma 3 median_face ndimage median_filter noisy_face size 5 from scipy import signal wiener_face signal wiener noisy_face 5 5 Many other filters in scipy ndimage filters and scipy signal can be applied to i
9. O 8 O 1 21 O 0 707570 35 2 1 9 O 1 42 O 0 694605 57 3 2 12 O O 1 O 0 824126 19 3 3 12 O O O 0 602060 22 3 15 4 1 Pairplot scatter matrices We can easily have an intuition on the interactions between continuous variables using seaborn pairplot to display a scatter matrix gt gt gt import seaborn gt gt gt seaborn pairplot data vars WAGE AGE EDUCATION kind reg 15 4 More visualization seaborn for statistical exploration 297 Scipy lecture notes Edition 2015 2 Lu z 0 5 Ww l Z O q U D Q u 0 0 50 0 0 5 1 0 15 2 0 10 20 30 405060 70 0 5 10 15 20 WAGE AGE EDUCATION Categorical variables can be plotted as the hue gt gt gt seaborn pairplot data vars WAGE AGE EDUCATION kind reg hue SEX 2 0 Effect of gender 1 Female 0O Male EDUCATION 0 0 5 0 0 0 5 1 0 1 5 2 0 10 20 30 40 50 60 70 O 5 10 15 20 WAGE AGE EDUCATION Look and feel and matplotlib settings Seaborn changes the default of matplotlib figures to achieve a more modern excel like look It does that upon import You can reset the default using gt gt gt from matplotlib import pyplot as plt gt gt gt plt rcdefaults To switch back to seaborn settings or understand better styling in seaborn see the relevent section of the seaborn documentation 15 4 More visualization seaborn for statistical exploration 298 15 4 2 Implot plotting a univariate regression
10. dtype uint8 gt gt gt x view np int16 array 513 1027 dtype int16 gt gt gt 0x0201 0x0403 sis 1027 gt gt gt y view np int16 array 769 1026 dtype int16 e What happened e we need to look into what x LO 1 actually means gt gt gt 0x0301 0x0402 769 1026 8 1 4 Indexing scheme strides Main point The question gt gt gt x np array 1 2 3 ia 5 6l Be 7 8 9 dtype np ints8 gt gt gt str x data x01 x02 x03 x04 x05 x06 x07 x08 t At which byte in x data does the item x 1 2 begin The answer in Numpy e strides the number of bytes to jump to find the next element e 1 stride per dimension gt gt gt x strides 3 1 gt gt gt byte_offset 3 1 1 2 to find x 1 2 gt gt gt x flat byte_offset 6 gt gt gt xill 2 6 simple flexible C and Fortran order gt gt x np array lIi 2 3 4 5 6 dtype np int16 order C gt gt gt x strides 6 2 8 1 Life of ndarray 167 Scipy lecture notes Edition 2015 2 gt gt gt str x data x01 x00 x02 x00 x03 x00 x04 x00 x05 x00 x06 x00 e Need to jump 6 bytes to find the next row e Need to jump 2 bytes to find the next column gt gt gt y np array x order F gt gt gt y strides 2 4 gt gt gt str y data x01 x00 x04 x00 x02 x00 x05 x00 x03 x00 x06 x00 e Need to jump 2 bytes to find the next row e Need to jump 4
11. e manually view dtype Casting e Casting in arithmetic in nutshell only type not value of operands matters largest safe type able to represent both is picked scalars can lose to arrays in some situations e Casting in general copies data 8 1 Life of ndarray sss 6A Scipy lecture notes Edition 2015 2 gt gt gt x np array 1 2 3 4 dtype np float 22 gt X array lrs 2 5 Sig 4al gt gt gt y x astype np int8 gt gt gt y array 1 2 3 4 dtype int8 gt gt y 1 array 2 3 4 5 dtype int8 gt gt gt y 256 array 257 258 259 260 dtype int16 gt gt gt y 256 0 array 2572 2505 259 260 gt gt gt y np array 256 dtype np int32 array 257 258 259 260 dtype int32 e Casting on setitem dtype of the array is not changed on item assignment gt gt gt yll y 1 5 gt gt gt y array 2 3 4 5 dtype int8 Fxact rules see numpy documentation Re interpretation viewing e Data block in memory 4 bytes x01 0x02 I 0x08 I 0x04 4of uint8 OR 4of int8 OR 2 of int16 OR 1 of int32 OR 1 of float32 OR How to switch from one to another 1 Switch the dtype gt gt gt x np array 1 2 3 4 dtype np uint8 yo X2GL ype ZY 22 gt X array 513 1027 dtype int16 gt gt gt 0x0201 0x0403 E13 1027 0x01 0x02 I 0x03 0x04 little endian least significant byte
12. gt gt gt from skimage import restoration gt gt gt filtered_camera restoration denoise_bilateral camera gt gt gt type filtered_camera lt type numpy ndarray gt Other Python packages are available for image processing and work with NumPy arrays e scipy ndimage for nd arrays Basic filtering mathematical morphology regions properties e Mahotas Also powerful image processing libraries have Python bindings e OpenCV computer vision e ITK 3D images and registration 17 1 Introduction and concepts 309 Scipy lecture notes Edition 2015 2 e and many others but they are less Pythonic and NumPy friendly to a variable extent 17 1 2 What s to be found in scikit image e Website http scikit image org e Gallery of examples as in matplotlib or scikit learn http scikit image org docs stable auto_examples Different kinds of functions from boilerplate utility functions to high level recent algorithms e Filters functions transforming images into other images NumPy machinery Common filtering algorithms e Data reduction functions computation of image histogram position of local maxima of corners etc e Other actions I O visualization etc 17 2 Input output data types and colorspaces I O skimage io gt gt gt from skimage import io Reading from files skimage io imread gt gt gt import os gt gt gt filename os path join skimage data_dir camera png
13. gt gt gt morphology binary_opening a morphology diamond 1 astype np uint8 array l0 0 0 0 Ol O 0 Ie 00l On e 1 oe 0 One Os 0 LO 0 0 O OJ dtype uint8 Opening removes small objects and smoothes corners Grayscale mathematical morphology Mathematical morphology operations are also available for non binary grayscale images int or float type Erosion and dilation correspond to minimum resp maximum filters Higher level mathematical morphology are available tophat skeletonization etc See also 17 3 Image preprocessing enhancement 314 Scipy lecture notes Edition 2015 2 Basic mathematical morphology is also implemented in scipy ndimage morphology The scipy ndimage implementation works on arbitrary dimensional arrays Example of filters comparison image denoising gt gt gt from skimage morphology import disk gt gt gt coins data coins gt gt gt coins_zoom coins 10 80 300 370 gt gt gt median_coins filters rank median coins_ zoom disk 1 gt gt gt from skimage import restoration gt gt gt tv_coins restoration denoise_tv_chambolle coins_zoom weight 0 1 gt gt gt gaussian_coins filters gaussian_filter coins sigma 2 Gaussian filter Median filter TV filter 17 4 Image segmentation Image segmentation is the attribution of different labels to different regions of the image for example in order to extract the pixels of an object of
14. types 0 types 1 types 2 PyObject NPY_BYTE type of first input arg NPY_BYTE type of second input arg NPY_BYTE type of third input arg python_ufunc PyUFunc_FromFuncAndData ufunc_loop NULL type S 1 ntypes 2 num_inputs 1 num outputs identity_element name docs unus tring ed e Aufunc can also support multiple different input output type combinations Making it eas ier 3 ufunc_loop is of very generic form and Numpy provides pre made ones PyUfunc_f_f float elementwise_func float input_1 PyUfunc_ff_f float elementwise_func float input_1 float input_2 PyUfunc_d_d double elementwise_func double input_1 PyUfunc_dd_d double elementwise_func double input_1 double input_2 PyUfunc_D_D elementwise_func npy_cdouble input npy_cdouble output PyUfunc_DD_D elementwise_func npy_cdouble ini npy_cdouble in2 npy_cdoublex out e Only elementwise_func needs to be supplied e except when your elementwise function is not in one of the above forms 8 2 Universal functions 174 Scipy lecture notes Edition 2015 2 8 2 2 Exercise building an ufunc from scratch The Mandelbrot fractal is defined by the iteration z z C where c x iy is a complex number This iteration is repeated if z stays finite no matter how long the iteration runs c belongs to the Mandelbrot set e Make ufunc called mandel z0 c that computes Zz
15. w In 71 fp close In 72 a os path abspath junk txt In 73 a Out 73 Users cburns src scipy2009 scipy_2009_tutorial source junk txt In 74 os path split a Out 74 Users cburns src scipy2009 scipy_2009_tutorial source Junk txt In 78 os path dirname a Out 78 Users cburns src scipy2009 scipy_2009_tutorial source In 79 os path basename a Out 79 jumk trt In 80 os path splitext os path basename a 2 7 Standard Library 36 Scipy lecture notes Edition 2015 2 Out 20 Junk txt In 84 os path exists junk txt Out 84 True In 86 os path isfile junk txt Out 86 True In 87 os path isdir junk txt Out 87 False In 88 os path expanduser local Out 88 Users cburns local In 92 os path join os path expanduser local bin Out 92 Users cburns local bin Running an external command In 8 os system ls basic_types rst demo py functions rst python_language rst standard_library rst control_flow rst exceptions rst io rst python logo png demo2 py first_steps rst oop rst reusing _code rst Alternative to os system A noteworthy alternative to os system is the sh module Which provides much more convenient ways to obtain the output error stream and exit code of the external command In 20 import sh In 20 com sh 1s In 21 print com basic_types rst exceptions rst oop rst standard_libr
16. 1 41421356 2 23606798 3 16227766 84 12310563 as 2 23606798 2582649712 360555128 942472135951 lac 3 16227766 3 60555128 4 24264069 B5 T 2 4 12310563 4247213595 5 5 65685425 Or in color gt gt gt plt pcolor distance gt gt gt plt colorbar Ul UJ N _ Remark the numpy ogrid function allows to directly create vectors x and y of the previous example with two significant dimensions gt gt gt x y np ogrid 0 5 0 5 Poo ky array 0 i 2 3 4 array llO 1 2 3 4 gt gt gt x shape y shape CER E gt gt gt distance np sqrt x 2 y 2 3 2 Numerical operations on arrays 64 Scipy lecture notes Edition 2015 2 So np ogridis very useful as soon as we have to handle computations on a grid On the other hand np mgrid directly provides matrices full of indices for cases where we can t or don t want to benefit from broadcasting gt gt gt x y np mgrid l0 4 0 4 gt gt gt X array 110 0 0 OJ a gt See acs lel Eon 62 Eee ea 8s 85 EIN gt gt gt y array l0 L 2 3r OE Oa a OPE Te R 3 2 4 Array shape manipulation Flattening gt gt gt a np array 1 2 3 4 5 6 gt gt gt a ravel array Gil 2 3 4 5 6 gt gt gt a T array 1 4 PS 3 61 gt gt gt a T ravel array Clin 4 2 by 3 6J Higher dimensions last dimensions ravel out first
17. 7 2 gt gt gt x_min optimize brent f It actually converges in 9 iterations gt gt gt x _min 0 699999999 gt gt gt xX min 7 216005 6 10 Brents method can also be used for optimization constrained to an interval using scipy optimize fminbound In scipy 0 11 scipy optimize minimize_scalar gives a generic interface to 1D scalar minimization 13 2 2 Gradient based methods Some intuitions about gradient descent Here we focus on intuitions not code Code will follow Gradient descent basically consists in taking small steps in the direction of the gradient that is the direction of the steepest descent Table 13 1 Fixed step gradient descent 20 40 60 80 100 A well conditionned quadratic function 50 T00 150 200 An ill conditionned quadratic function The core problem of gradient methods on ill conditioned problems is that the gradient tends not to point in the direction of the minimum We can see that very anisotropic ill conditionned functions are harder to optimize 13 2 A review of the different optimizers 252 Scipy lecture notes Edition 2015 2 Take home message conditioning number and preconditioning If you know natural scaling for your variables prescale them so that they behave similarly This is related to preconditioning Also it clearly can be advantageous to take bigger steps This is done in gradient descent code using a line search
18. S str Hello world 1 3 The interactive workflow IPython and a text editor T Scipy lecture notes Edition 2015 2 From a script to functions While it is tempting to work only with scripts that is a file full of instructions following each other do plan to progressively evolve the script to a set of functions e A script is not reusable functions are e Thinking in terms of functions helps breaking the problem in small blocks 1 3 3 IPython Tips and Tricks The IPython user manual contains a wealth of information about using IPython but to get you started we want to give you a quick introduction to four useful features history magic functions aliases and tab completion Like a UNIX shell Python supports command history Type up and down to navigate previously typed com mands in t 10 In 2 lt UP gt in lI 10 Python supports so called magic functions by prefixing a command with the character For example the run and whos functions from the previous section are magic functions Note that the setting automagic which is enabled by default allows you to omit the preceding sign Thus you can just type the magic function and it will work Other useful magic functions are e cd to change the current directory In 2 cd tmp tmp e Ztimeit allows you to time the execution of short snippets using the timeit module from the standard library In 3 timeit x 10 10000000 loops best of 3 39 ns per lo
19. e Style spaces after commas around etc A certain number of rules for writing beautiful code and more importantly using the same con ventions as everybody else are given in the Style Guide for Python Code and the Docstring Con ventions page to manage help strings e Except some rare cases variable names and comments in English 3 4 Advanced operations Section contents e Polynomials e Loading data files 3 4 1 Polynomials Numpy also contains polynomials in different bases For example 3x 2x 1 gt gt gt p np polyid 3 2 1 gt gt gt p 0 l gt gt gt p roots array I i 0 33333333 gt gt gt p order 2 gt gt gt x np linspace 0 1 20 gt gt gt y np cos x 0 3 np random rand 20 gt gt gt p np polyid np polyfit x y 3 gt gt gt t np linspace 0 1 200 gt gt gt plit plot z ye o t PU lt matplotlib lines Line2D object at gt lt matplotlib lines Line2D object at 3 4 Advanced operations gt 12 Scipy lecture notes Edition 2015 2 1 3 1 2 1 1 1 0 0 9 0 8 0 7 O 0 2 0 4 0 6 0 8 1 0 See http docs scipy org doc numpy reference routines polynomials polyld html for more More polynomials with more bases Numpy also has a more sophisticated polynomial interface which supports e g the Chebyshev basis 3x 2x 1 gt gt gt p np polynomial Polynomial 1 2 3 coefs in differen
20. nnm Plot Mandelbrot Plot the Mandelbrot ensemble nnnm import numpy as np import mandel x np linspace 1 7 0 6 1000 I y np linspace 1 4 4 1000 8 2 Universal functions 179 Scipy lecture notes Edition 2015 2 x None 1j yL None mandel mandel c c N Q import matplotlib pyplot as plt plt imshow abs z 2 lt 1000 extent 1 7 0 6 1 4 1 4 plt gray plt show Most of the boilerplate could be automated by these Cython modules http wiki cython org MarkLodato CreatingUfuncs Several accepted input types E g supporting both single and double precision versions cdef void mandel_single _point double complex z_in double complex c_in double complex z_out nogil cdef void mandel_single _point_singleprec float complex z_in float complex c_in float complex z_out nogil cdef PyUFuncGenericFunction loop_funcs 2 cdef char input_output_types 3 2 cdef void elementwise_funcs 1 2 loop_funcs 0 PyUFunc_DD_D input_output_types 0 NPY_CDOUBLE input _output_types 1 NPY_CDOUBLE input _output_types 2 NPY_CDOUBLE elementwise_funcs 0 lt void gt mandel_single_point loop_funcs 1 PyUFunc_FF_F input_output_types 3 NPY_CFLOAT input_output_types 4 NPY_CFLOAT input_output_types 5 NPY_CFLOAT elementwise_funcs 1 lt void gt mandel_single_point_singleprec mandel PyUFunc_FromFuncAndData loop_func elementwise_funcs input_output_t
21. or cylinder or point or sphere Default sphere name the name of the vtk object created representation the representation type used for the surface Must be surface or wireframe or points or mesh or fancymesh Default surface resolution The resolution of the glyph created For spheres for instance this is the number of divisions along theta and phi Must be an integer Default 8 scalars optional scalar data scale_factor scale factor of the glyphs used to represent the vertices in fancy_mesh mode Must be a float Default 0 05 scale_mode the scaling mode for the glyphs vector scalar or none transparent make the opacity of the actor depend on the scalar tube_radius radius of the tubes used to represent the lines in mesh mode If None simple lines are used tube_sides number of sides of the tubes used to represent the lines Must be an integer Default 6 vmax vmax is used to scale the colormap If None the max of the data will be used vmin vmin is used to scale the colormap If None the min of the data will be used 19 1 Mlab the scripting interface 344 Scipy lecture notes Edition 2015 2 Example In 1 import numpy as np In 2 r theta np mgrid 0 10 np pi np pi 10j In 3 r np cos theta In 4 r np sin theta FSi np sin r r In 6 from mayavi import mlab In 7 mlab mesh x y z colormap gist_earth extent 0 1 0 1 0 1 Outl
22. s to be found in scikit image e Input output data types and colorspaces Data types Colorspaces e Image preprocessing enhancement Local filters Non local filters Mathematical morphology e Image segmentation Binary segmentation foreground background Marker based methods e Measuring regions properties e Data visualization and interaction e Feature extraction for computer vision 17 1 Introduction and concepts Images are NumPy s arrays np ndarray image np ndarray pixels array values a 2 3 channels array dimensions image encoding dtype np uint8 np uint16 np float 308 Scipy lecture notes Edition 2015 2 filters functions numpy skimage scipy gt gt gt import numpy as np gt gt gt check np zeros 9 9 gt gt gt check 712 1722 1 gt gt gt check 1 2 21 1 gt gt gt import matplotlib pyplot as plt gt gt gt plt imshow check cmap gray interpolation nearest 17 1 1 scikit image and the SciPy ecosystem Recent versions of scikit image is packaged in most Scientific Python distributions such as Anaconda or Enthought Canopy It is also packaged for Ubuntu Debian gt gt gt import skimage gt gt gt from skimage import data most functions are in subpackages Most scikit image functions take NumPy ndarrays as arguments gt gt gt camera data camera gt gt gt camera dtype dtype uint8 gt gt gt camera shape C212
23. self _storage storage_value def _get_spillage self new_storage self _storage self release self inflows overflow new_storage self max_storage return max overflow 0 def print_state self print Storage tRelease tInflows tSpillage str_format t join 7 2f for i in range 4 print str_format format self storage self release self inflows self spillage pring gt 79 if name _main __ projectA Reservoir name Project A max_storage 30 18 3 What are Traits 330 Scipy lecture notes Edition 2015 2 max release 5 hydraulic_head 60 efficiency 0 8 state ReservoirState reservoir projectA storage 25 state release 4 O state inflows state print_state Caching property Heavy computation or long running computation might be a problem when accessing a property where the inputs have not changed The cached_property decorator can be used to cache the value and only recompute them once invalidated Let s extend the TraitsUI introduction with the ReservoirState example from traits api import HasTraits Instance DelegatesTo Float Range Property from traitsui api import View Item Group VGroup from reservoir import Reservoir class ReservoirState HasTraits Keeps track of the reservoir state given the initial storage For the simplicity of the example the release is considered in hm3 timestep and not in m3 s reservoir Instance Rese
24. spe has data array specialized cialized 11 2 Storage Schemes 223 Scipy lecture notes Edition 2015 2 11 3 Linear System Solvers e sparse matrix eigenvalue problem solvers live in scipy sparse linalg e the submodules dsolve direct factorization methods for solving linear systems isolve iterative methods for solving linear systems eigen sparse eigenvalue problem solvers e all solvers are accessible from gt gt gt import scipy sparse linalg as spla gt gt gt spla __all__ LinearOperator Tester arpack aslinearoperator bicg iblegstab cg ees cs matrix csr matrik dsollve eigen eigen_symmetric factorized gmres interface isolve iterative lgmres linsolve lobpcg lsqr minres np qmr speigs spilu splu spsolve svd test umfpack use_solver utils warnings 11 3 1 Sparse Direct Solvers e default solver SuperLU 4 0 included in SciPy real and complex systems both single and double precision e optional umfpack real and complex systems double precision only recommended for performance wrappers now live in scikits umfpack check out the new scikits suitesparse by Nathaniel Smith Examples e import the whole module and see its docstring gt gt gt from scipy sparse linalg import dsolve gt gt gt help dsolve e both superlu and umfpack can be used if t
25. timeit a copy 10 loops best of 3 124 ms per loop In 3 timeit a 1 10 loops best of 3 112 ms per loop e Beware of cache effects Memory access is cheaper when it is grouped accessing a big array in a continuous way is much faster than random access This implies amongst other things that smaller strides are faster see CPU cache effects In 1 c np zeros 1e4 1e4 order C In 2 timeit c sum axis 0 1 loops best of 3 3 89 s per Toop In 3 timeit c sum axis 1 1 loops best of 3 188 ms per loop In 4 c strides Out 4 80000 8 This is the reason why Fortran ordering or C ordering may make a big difference on operations In 5 a np random rand 20 2 18 In 6 b np random rand 20 2 18 In 7 timeit np dot b a T 1 loops best of 3 194 ms per loop In 8 c np ascontiguousarray a T In 9 timeit np dot b c 10 loops best of 3 84 2 ms per loop Note that copying the data to work around this effect may not be worth it In 10 timeit c np ascontiguousarray a T 10 loops best of 3 106 ms per loop Using numexpr can be useful to automatically optimize code for such effects e Use compiled code The last resort once you are sure that all the high level optimizations have been explored is to transfer the hot spots i e the few lines or functions in which most of the time is spent to compiled code For compiled code the preferred option is to use Cy
26. 1 4 x i 12 x 0 2 4 x 0 4 x 0 2 gt gt gt optimize fmin_ncg f 2 2 fprime fprime fhess hessian Optimization terminated successfully Current function value 0 000000 Iterations 9 Function evaluations 11 Gradient evaluations 19 Hessian evaluations 9 array 1 1 At very high dimension the inversion of the Hessian can be costly and unstable large scale gt 250 Newton optimizers should not to be confused with Newton s root finding method based on the same princi ples scipy optimize newton 13 2 A review of the different optimizers 255 Scipy lecture notes Edition 2015 2 Quasi Newton methods approximating the Hessian on the fly BFGS BFGS Broyden Fletcher Goldfarb Shanno algorithm refines at each step an approximation of the Hes sian An ill conditionned quadratic function On a exactly quadratic function BFGS is not as fast as Newton s method but still very fast An ill conditionned non quadratic function Here BFGS does better than Newton as its empirical estimate of the curvature is better than that given by the Hessian An ill conditionned very non quadratic function CIEL gt gt gt def f x The rosenbrock function return 5 1 x 0 2 x 1 x 0 2 2 gt gt gt def fprime x return np array 2 5 1 x 0 4 xlO x 1 x 0 2 2 x 1 x 0 2 gt gt gt optimize fmin_bfgs f 2 2 fprime fprime Optimiza
27. 1 3 1 1 rotation 1 shear 0 7 translation 210 50 image warp data checkerboard tform inverse output_shape 350 350 coords corner_peaks corner_harris image min_distance 5 coords_subpix corner_subpix image coords window_size 13 this ex ample is taken from the plot_corner example in scikit image Points of interest such as corners can then be used to match objects in different images as described in the plot_matching example of scikit image 17 7 Feature extraction for computer vision 321 CHAPTER 18 Traits building interactive dialogs Author Didrik Pinte The Traits project allows you to simply add validation initialization delegation notification and a graphical user interface to Python object attributes In this tutorial we will explore the Traits toolset and learn how to dramatically reduce the amount of boilerplate code you write do rapid GUI application development and understand the ideas which underly other parts of the Enthought Tool Suite Traits and the Enthought Tool Suite are open source projects licensed under a BSD style license Intended Audience Intermediate to advanced Python programmers Requirements e Either wxPython PyQt or PySide e Numpy and Scipy e Enthought Tool Suite e All required software can be obtained by installing the EPD Free Tutorial content e Introduction e Example e What are Traits Initialisation Validation Documentation Visual
28. 15 1 Data representation and interaction 286 Scipy lecture notes Edition 2015 2 gt gt gt data columns It has columns Index u Unnamed 0 u Gender u FSIQ u VIQ u PIQ u Weight u Height u MRI_Count dtype object gt gt gt print datal Gender Columns can be addressed by name O Female 1 Male 2 Male 3 Male 4 Female gt gt gt Simpler selector gt gt gt data datal Gender Female VIQ mean 109 45 For a quick view on a large dataframe use its describe method pandas DataFrame describe groupby splitting a dataframe on values of categorical variables gt gt gt groupby_gender data groupby Gender gt gt gt for gender value in groupby_gender VIQ print gender value mean Female 109 45 Male 115 25 groupby_gender is a powerful object that exposes many operations on the resulting group of dataframes gt gt gt groupby_gender mean Unnamed O FSIQ VIQ PIQ Weight Height MRI_Count Gender Female 19 65 111 9 109 45 110 45 137 200000 65 765000 862654 6 Male 21 35 115 0 115 25 111 60 166 444444 71 431579 954855 4 Use tab completion on groupby_gender to find more Other common grouping functions are median count useful for checking to see the amount of missing values in different subsets or sum Groupby evaluation is lazy no work is done until an aggregation function is applied Female FSIQ VIQ PIQ FSIQ VIQ PIQ 15 1 Data repres
29. Basic visualization Indexing and slicing Copies and views Fancy indexing 3 1 1 What are Numpy and Numpy arrays Numpy arrays Python objects e high level number objects integers floating point e containers lists costless insertion and append dictionaries fast lookup Numpy provides e extension package to Python for multi dimensional arrays e closer to hardware efficiency e designed for scientific computation convenience e Also known as array oriented computing 43 Scipy lecture notes Edition 2015 2 gt gt gt import numpy as np gt gt gt a np array 0 1 2 3 gt gt gt a aria ClO 1 2 3J For example An array containing e values of an experiment simulation at discrete time steps e signal recorded by a measurement device e g sound wave e pixels of an image grey level or colour e 3 D data measured at different X Y Z positions e g MRI scan J Why it is useful Memory efficient container that provides fast numerical operations In 1 L range 1000 In 2 timeit i 2 for i in L 1000 loops best of 3 403 us per loop In 3 a np arange 1000 In 4 timeit a 2 100000 loops best of 3 12 7 us per loop Numpy Reference documentation e On the web http docs scipy org e Interactive help in S np array String Form lt built in function array Docstring array object dtype None copy True order None subok False ndmin 0 e Looking for
30. First load the data into a Numpy array gt gt gt data np loadtxt data populations txt gt gt gt year hares lynxes carrots data T trick columns to variables Then plot it gt gt gt from matplotlib import pyplot as plt ee plt axes l0 2 0 17 0 5 0 8 gt gt gt plt plot year hares year lynxes year carrots gt gt gt plt legend Hare Lynx Carrot loc 1 05 0 5 80000 70000 60000 Hare 50000 Lynx Carrot 40000 30000 20000 10000 O 1900 1905 1910 1915 1920 The mean populations over time gt gt gt populations datal 1 gt gt gt populations mean axis 0 array 34080 95238095 20166 66666667 42400 The sample standard deviations gt gt gt populations std axis 0 array 20897 90645809 16254 59153691 3322 50622558 Which species has the highest population each year gt gt gt np argmax populations axis 1 array Ghee 2a 00 dis p2 2 a SO 2O On O 2 2 2 2a 3 2 Numerical operations on arrays 60 Scipy lecture notes Edition 2015 2 Worked Example diffusion using a random walk algorithm 1 2 1 2 KNIN 1 O 1 d t Let us consider a simple 1D random walk process at each time step a walker jumps right or left with equal probability We are interested in finding the typical distance from the origin of a random walker after t left or right jumps We are going to simulate many walkers to find this law and we are going to
31. In 6 1 4 AttributeError list object has no attribute foobar As you can see there are different types of exceptions for different errors 2 8 2 Catching exceptions try except In 10 while True try x int raw_input Please enter a number break except ValueError print That was no valid number Try again Please enter a number a That was no valid number Try again Please enter a number 1 In 9 x Out 9 1 try finally Ty x int raw_input Please enter a number gt finally print Thank you for your input Please enter a number a Thank you for your input ValueError invalid literal for int wath base 10 2 8 Exception handling in Python 40 Scipy lecture notes Edition 2015 2 Important for resource management e g closing a file Easier to ask for forgiveness than for permission In 11 def print_sorted collection try collection sort except AttributeError pass print collection In 12 print_sorted 1 3 21 i 2 Si In 13 print_sorted set 1 3 2 seu i 2 31 In 14 print_sorted 132 152 2 8 3 Raising exceptions e Capturing and reraising an exception In 15 def filter_name name try name name encode ascii except UnicodeError e if name Ga l print OK Ga l else raise e return name In 16 filter_name Ga l OK Ga l Out 16 Ga xc3 xabl In 17 filter_
32. Reshaping The inverse operation to flattening gt gt gt a shape 2 3 gt gt gt b a ravel gt gt gt b b reshape 2 3 gt gt gt b array Iil 2 3 4 5 6 Or gt gt gt a reshape 2 1 unspecified 1 value is inferred array I1 2 3l 4 5 6 A ndarray reshape may return a view cf help np reshape or copy 3 2 Numerical operations on arrays 65 Scipy lecture notes Edition 2015 2 gt gt gt bLO O 99 gt gt gt a array LI99 2 3 La 5 6J Beware reshape may also return a copy gt gt gt a np zeros 3 2 gt gt gt b a T reshape 3 2 gt gt gt b 0 9 gt gt a array To understand this you need to learn more about the memory layout of a numpy array Adding a dimension Indexing with the np newaxis object allows us to add an axis to an array you have seen this already above in the broadcasting section gt gt gt z np array 1 2 3 PP Z array id 2 310 gt gt gt z np newaxis array i Els 3 gt gt gt gt z np newaxis array Cre 2 311 Dimension shuffling a np arange 4 3 2 reshape 4 3 2 a shape 3 2 alo 2 a a transpose 1 2 0 b shape 2 4 bi2 15 0 Also creates a view gt gt gt b 2 1 0 1 gt gt gt a0 2 1 i Resizing Size of an array can be changed with ndarray resize gt gt gt a np arange 4 gt gt gt a
33. Y ly 2 2 gt 1x ly 4 Masks face mask 0 Fancy indexing face range 400 range 400 255 12 3 1 Statistical information gt gt gt face misc face gray True gt gt gt face mean 113 48026784261067 gt gt gt face max face min 250 0 12 3 Basic manipulations Scipy lecture notes Edition 2015 2 234 Scipy lecture notes Edition 2015 2 np histogram Exercise e Open as an array the scikit image logo http scikit image org _static img logo png or an im age that you have on your computer e Crop a meaningful part of the image for example the python circle in the logo e Display the image array using matplotlib Change the interpolation method and zoom to see the difference e Transform your image to greyscale e Increase the contrast of the image by changing its minimum and maximum values Optional use scipy stats scoreatpercentile read the docstring to saturate 5 of the darkest pixels and 5 of the lightest pixels e Save the array to two different file formats png jpg tiff scikits image image processing in python 12 3 2 Geometrical transformations face misc face gray True lx ly face shape Cropping crop_face face lx 4 lx 4 ly 4 ly 4 up lt gt down flip flip_ud_face np flipud face rotation rotate_face ndimage rotate face 45 rotate_face_noreshape ndimage rotate face 45 reshape False 12 4 Image
34. and L pop is our first example of object oriented pro gramming OOP Being a list the object r owns the method function that is called using the notation No further knowledge of OOP than understanding the notation is necessary for going through this tutorial Discovering methods Reminder in Ipython tab completion press tab In 28 r lt TAB gt r __add__ r iadd r __class__ r __imul__ r __contains__ Geo nae e r __delattr__ rr _iter__ r __delitem__ r le _ r __delslice__ r __len__ eel OG am r lt r __eq__ nomu r __format__ r __ne__ r __ge__ r __new__ r __getattribute__ r __reduce__ r __getitem__ r __reduce_ex__ r __getslice__ r repr r gt r __reversed__ r __hash__ ee Strings em Wi gee ad Le Oe Oe em BR oo Piel R RT __setattr__ __setitem__ __setslice__ __sizeof__ SET __subclasshook__ append count extend index insert pop remove reverse SOrt Different string syntaxes simple double or triple quotes 2 2 Basic types 15 Scipy lecture notes Edition 2015 2 Hello how are you Hi what s up Hello tripling the quotes allows the how are you g un Hi 2 what 1 s up we the string to span more than one line File lt ipython console gt line 1 Hi what s lps a SyntaxError invalid syntax The newline character is n and the tab character is t Strings are collections like lists Hence they can be indexed and sl
35. buffer interface 2 the array interface 3 the new buffer interface PEP 3118 8 3 2 The old buffer protocol Only 1 D buffers No data type information C level interface PyBufferProcs tp_as_buffer in the type object e But it s integrated into Python e g strings support it 8 3 Interoperability features 182 Scipy lecture notes Edition 2015 2 Mini exercise using Pillow Python Imaging Library See also pilbuffer py from PIL import Image data np zeros 200 200 4 dtype np int8 datal 255 0 0 255 Red In PIL RGBA images consist of 32 bit integers whose bytes are RR GG BB AA data data view np int32 squeeze img Image frombuffer RGBA 200 200 data raw RGBA O 1 img save test png Check what happens if data is now modified and img saved again 8 3 3 The old buffer protocol nnn From buffer Show how to exchange data between numpy and a library that only knows the buffer interface nnm import numpy as np import Image Let s make a sample image RGBA format x np zeros 200 200 4 dtype np int8 x 0 254 red x 3 255 opaque data x view np int32 Check that you understand why this is OK img Image frombuffer RGBA 200 200 data img save test png Modify the original data and save again It turns out that PIL which knows next to nothing about Numpy happily shares the same data
36. c x 1j y for j in xrange N_max Zz z 2 c Point x y belongs to the Mandelbrot set if c lt some_threshold Do this computation by 1 Construct a grid of c x 1j y values in range 2 1 x 1 5 1 5 2 Do the iteration 3 Form the 2 d boolean mask indicating which points are in the set 4 Save the result to an image with gt gt gt import matplotlib pyplot as plt gt gt gt plt imshow mask T extent 2 1 1 5 1 5 lt matplotlib image AxesImage object at gt 3 5 Some exercises 80 Scipy lecture notes Edition 2015 2 gt gt gt plt gray gt gt gt plt savefig mandelbrot png Solution Python source file 3 5 6 Markov chain 2 Pj 1 J Markov chain transition matrix P and probability distribution on the states p 1 O lt P i j lt 1 probability to go from state i to state j 2 Transition rule Pnew P pold 3 all sum P axis 1 1 p sum 1 normalization Write a script that works with 5 states and e Constructs a random matrix and normalizes each row so that it is a transition matrix e Starts from a random normalized probability distribution p and takes 50 steps gt p_50 e Computes the stationary distribution the eigenvector of P T with eigenvalue 1 numerically closest to 1 gt p_stationary Remember to normalize the eigenvector I didn t e Checks if p_50 and p_stationary are equal to tolerance le 5 Toolbox np random rand dot np li
37. gen f gt gt gt next gen Stane 3 gt gt gt next gen middle 4 gt gt gt next gen finished Traceback most recent call last StopIteration Traceback most recent call last StopIteration Contrary to a normal function where executing f would immediately cause the first print to be executed gen is assigned without executing any statements in the function body Only when gen next is invoked by next the statements up to the first yield are executed The second next prints middle and execution halts on the second yield The third next prints finished and falls of the end of the function Since no yield was reached an exception is raised What happens with the function after a yield when the control passes to the caller The state of each generator is stored in the generator object From the point of view of the generator function is looks almost as if it was running in a separate thread but this is just an illusion execution is strictly single threaded but the interpreter keeps and restores the state in between the requests for the next value Why are generators useful As noted in the parts about iterators a generator function is just a different way to create an iterator object Everything that can be done with yield statements could also be done with next methods Nevertheless using a function and having the interpreter perform its magic to create an iterator ha
38. gt gt gt misc imsave face png face First we need to create the PNG file gt gt gt face misc imread face png gt gt gt type face lt numpy ndarray gt gt gt gt face shape face dtype 768 1024 3 dtype uint8 dtype is uint8 for 8 bit images 0 255 12 1 Opening and writing to image files 231 Scipy lecture notes Edition 2015 2 Opening raw files camera 3 D images gt gt gt face tofile face raw Create raw file gt gt gt face_from_raw np fromfile face raw dtype np uint8 gt gt gt face_from_raw shape 2359296 gt gt gt face_from_raw shape 768 1024 3 Need to know the shape and dtype of the image how to separate data bytes For large data use np memmap for memory mapping gt gt gt face_memmap np memmap face raw dtype np uint8 shape 768 1024 3 data are read from the file and not loaded into memory Working on a list of image files gt gt gt for i in range 10 im np random random_integers 0 255 10000 reshape 100 100 za misc imsave random_ 02d png i im gt gt gt from glob import glob gt gt gt filelist glob random png gt gt gt filelist sort 12 2 Displaying images Use matplotlib and imshow to display an image inside amatplotlib figure gt gt gt f misc face gray True retrieve a grayscale image gt gt gt import matplotlib pyplot as plt gt gt gt plt imshow f cmap pl
39. gt gt t 12345 54321 hello gt gt gt t 0 12345 gt gt t 12345 54321 2 hele gt gt gt u 0 2 Sets unordered unique items gt gt gt s set a gt gt gt S set la cle Ube gt gt gt s difference a b set c 2 2 Basic types 17 Scipy lecture notes Edition 2015 2 2 2 3 Assignment operator Python library reference says Assignment statements are used to re bind names to values and to modify attributes or items of mutable objects In short it works as follows simple assignment 1 an expression on the right hand side is evaluated the corresponding object is created obtained 2 aname on the left hand side is assigned or bound to the r h s object Things to note e asingle object can have several names bound to it In fil a 1 O38 In 2 b a ibaa Fed a ue oie ie 2 yy sail In 4 b Outl4aie io 2 3 In 5 a is b Out 5 True In 6 bli hi In 7 a pueli bis hit T3 e to change a list in place use indexing slices in bis a 2 3 in fs a Duttok Ei 23 In 4 a a b c Creates another object In 5 a outil frar Tbs Tel In 6 id a Out 6 138641676 in 7 ale 1 2 3 Modifies object in place In 8 a Out 6 12 3 In 9 id a Out 9 138641676 Same as in Outl6 yours will differ e the key concept here is mutable vs immutable mutable objects can be changed in
40. gt remove_pixel mask_size label_im 12 6 Measuring objects properties ndimage measurements 243 Scipy lecture notes Edition 2015 2 gt gt gt remove_pixel shape 256 256 gt gt gt label_imlLremove_pixel 0 gt gt gt plt imshow label_im lt matplotlib image AxesImage object at Ox gt Now reassign labels with np searchsorted gt gt gt labels np unique label_im gt gt gt label_im np searchsorted labels label_im Find region of interest enclosing object gt gt gt slice_x slice_y ndimage find_objects label_im 4 0 gt gt gt roi im slice_x slice_y gt gt gt plt imshow roi lt matplotlib image AxesImage object at Ox gt De Other spatial measures ndimage center_of_mass ndimage maximum_position etc Can be used outside the limited scope of segmentation applications Example block mean gt gt gt from scipy import misc gt gt gt f misc face gray True 12 6 Measuring objects properties ndimage measurements 244 Scipy lecture notes Edition 2015 2 sx sy f shape X Y np ogrid 0 sx O sy regions sy 6 X 4 Y 6 note that we use broadcasting block_mean ndimage mean f labels regions index np arange 1 regions max 1 block_mean shape sx 4 sy 6 When regions are regular blocks it is more efficient to use stride tricks Example fake dimensions with strides Non regularly spaced blocks radial mean
41. lt iterator object at gt gt gt gt nums _iter__ lt iterator object at gt gt gt gt nums __reversed__ lt reverseiterator object at gt gt gt gt it iter nums gt gt gt next it 1 gt gt gt next it 2 gt gt gt next it 3 gt gt gt next it Traceback most recent call last File lt stdin gt line 1 in lt module gt StopIteration Traceback most recent call last File lt stdin gt line 1 in lt module gt StopIteration When used in a loop StopIteration is swallowed and causes the loop to finish But with explicit invocation we can see that once the iterator is exhausted accessing it raises an exception Using the for in loop also uses the __iter__ method This allows us to transparently start the iteration over a sequence But if we already have the iterator we want to be able to use it in an for loop in the same way In order to achieve this iterators in addition to next are also required to have a method called __iter__ which returns the iterator self Support for iteration is pervasive in Python all sequences and unordered containers in the standard library allow this The concept is also stretched to other things e g file objects support iteration over lines gt gt gt f open etc fstab gt gt gt f is f __iter True The file is an iterator itself and it s __iter__ method doesn t create a separate object only a single th
42. new When the release is higher than zero warn all the inhabitants of the valley nnm if new gt 0 print Warning we are releasing hm3 of water format new if oname _main __ projectA Reservoir name Project A max_storage 30 max release 100 0 hydraulic_head 60 efficiency 0 8 state ReservoirState reservoir projectA storage 10 state release 90 state inflows 0 state print_state The static trait notification signatures can be def _release_changed self pass def _release_changed self new pass def _release_changed self old new pass def _release_changed self name old new pass Listening to all the changes To listen to all the changes on a HasTraits class the magic _any_trait_changed method can be imple mented In many situations you do not know in advance what type of listeners need to be activated Traits offers the ability to register listeners on the fly with the dynamic listeners from reservoir import Reservoir from reservoir_state_property import ReservoirState def wake_up_watchman_if_spillage new_value if new_value gt 0 print Wake up watchman Spilling hm3 format new_value if oname _main __ projectA Reservoir name Project A max_storage 30 max release 100 0 hydraulic_head 60 efficiency 0 8 18 3 What are Traits 334 Scipy lecture notes Edition 2015 2 state ReservoirState rese
43. pandas read_csv examples brain_size csv sep na_values gt gt gt data Unnamed O Gender FSIQ VIQ PIQ Weight Height MRI_Count O 1 Female 133 132 124 118 64 5 816932 1 2 Male 140 150 124 NaN 12 5 1001121 2 3 Male 139 123 150 143 T33 1038437 3 4 Male 133 7129123 172 68 8 965353 4 5 Female 137 132 134 147 65 0 951545 Missing values The weight of the second individual is missing in the CSV file If we don t specify the missing value NA not available marker we will not be able to do statistical analysis Creating from arrays A pandas DataFrame can also be seen as a dictionary of 1D series eg arrays or lists If we have 3 numpy arrays gt gt gt import numpy as np gt gt gt t np linspace 6 6 20 gt gt gt sin_t np sin t gt gt gt cos_t np cos t We can expose them as a pandas DataFrame gt gt gt pandas DataFrame t t sin sin_t cos cos_t CONaOoOFWN KF O Other inputs pandas can input data from SQL excel files or other formats See the pandas documentation COS 960170 609977 024451 570509 945363 955488 596979 008151 583822 sin 279415 192419 999701 821291 326021 295030 802257 999967 3811882 Manipulating data t 000000 368421 736842 105263 473684 842105 210526 578947 947368 data is a pandas DataFrame that resembles R s dataframe gt gt gt data shape 40 8 40 rows and 8 columns
44. scipy io mmwrite e IDL scipy io readsav lf somebody uses it there s probably also a Python library for it Exercise Text data files Write a Python script that loads data from populations txt and drop the last column and the first 5 rows Save the smaller dataset to pop2 txt 3 4 Advanced operations 76 Scipy lecture notes Edition 2015 2 Numpy internals If you are interested in the Numpy internals there is a good discussion in Advanced Numpy 3 5 Some exercises 3 5 1 Array manipulations 1 Form the 2 D array without typing it in explicitly Pi 265 1i Be ite 12 3 8 13 ag TA 5 10 15 and generate a new array containing its 2nd and 4th rows 2 Divide each column of the array gt gt gt import numpy as np gt gt gt a np arange 25 reshape 5 5 elementwise with the array b np array 1 5 10 15 20 Hint np newaxis 3 Harder one Generate a 10 x 3 array of random numbers in range 0 1 For each row pick the number closest to 0 5 e Use abs and argsort to find the column j closest for each row e Use fancy indexing to extract the numbers Hint ali j the array i must contain the row num bers corresponding to stuff in j 3 5 2 Picture manipulation Framing a Face Let s do some manipulations on numpy arrays by starting with an image of a racoon scipy provides a 2D array of this image with the scipy misc face function gt
45. the g ufunc operates elementwise e matrix multiplication this way could be useful for operating on many small matrices at once 8 2 Universal functions 181 Scipy lecture notes Edition 2015 2 Generalized ufunc loop Matrix multiplication m n n p gt m p void gufunc_loop void xargs int dimensions int steps void data char input_1 char args 0 these are as previously char input_2 char args 1 char xoutput char args 2 int input_i_stride_m steps 3 strides for the core dimensions int input_i_stride_n steps 4 are added after the non core int input_2_strides_n steps 5 steps int input_2_strides_p steps 6 int output_strides_n steps 7 int output_strides_p steps 8 int m dimension 1 core dimensions are added after int n dimension 2 the main dimension order as in int p dimension 3 signature int i for i 0 i lt dimensions 0 i matmul_for_strided_matrices input_1 input_2 output strides for each array input_1 steps 0 input_2 steps 1 output steps 2 Bi 8 3 Interoperability features 8 3 1 Sharing multidimensional typed data Suppose you 1 Write a library than handles multidimensional binary data 2 Want to make it easy to manipulate the data with Numpy or whatever other library 3 but would not like to have Numpy as a dependency Currently 3 solutions 1 the old
46. try do_something var finally manager __exit__ In other words the context manager protocol defined in PEP 343 permits the extraction of the boring part of a try except finally structure into a separate class leaving only the interesting do_something block 1 The __enter__ method is called first It can return a value which will be assigned to var The as part is optional if it isn t present the value returned by __enter__ is simply ignored 2 The block of code underneath with is executed Just like with try clauses it can either execute success fully to the end or it can break continue or return or it can throw an exception Either way after the block is finished the __exit__ method is called If an exception was thrown the information about the 7 3 Context managers 155 Scipy lecture notes Edition 2015 2 exception is passed to __exit__ which is described below in the next subsection In the normal case exceptions can be ignored just like in a finally clause and will be rethrown after ___exit__ is finished Let s say we want to make sure that a file is closed immediately after we are done writing to it gt gt gt class closing object def _init__ self obj self obj obj def _enter__ self return self obj def _exit__ self args self obj close gt gt gt with closing open tmp file w as f f write the contents n Here we have made sure that the f close is called when the
47. 0O 1 2 3 4 5 6 7 The digits dataset consists of 1797 images where each one is an 8x8 pixel image representing a hand written digit gt gt gt digits datasets load_digits gt gt gt digits images shape Gon B8 gt gt gt import pylab as pl gt gt gt pl imshow digits images 0 cmap pl cm gray_r lt matplotlib image AxesImage object at gt To use this dataset with the scikit we transform each 8x8 image into a vector of length 64 gt gt gt data digits images reshape digits images shape 0 1 20 1 1 Learning and Predicting Now that we ve got some data we would like to learn from it and predict on new one In scikit learn we learn from existing data by creating an estimator and calling its fit X Y method gt gt gt from sklearn import svm gt gt gt clf svm LinearSVC gt gt gt clf fit iris data iris target learn from the data Linear s C Once we have learned from the data we can use our model to predict the most likely outcome on unseen data Pee Cli predict IE G 0 se 64 le a 5 60 251 array 0 We can access the parameters of the model via its attributes ending with an underscore gt gt gt clf coef_ array CLL Gan T11 20 2 Classification 20 2 1 k Nearest neighbors classifier The simplest possible classifier is the nearest neighbor given a new observation take the label of the training samples closest to it in n dimensional space where n is the
48. 1 2147483647 2147483647 Unsigned integers gt gt gt np iinfo np uint32 max 2 32 1 4294967295 4294967295 Long integers Scipy lecture notes Edition 2015 2 Python 2 has a specific type for long integers that cannot overflow represented with an L at the end In Python 3 all integers are long and thus cannot overflow gt gt gt np iinfo np int64 max 2 63 1 9223372036854775807 9223372036854775807L Floating point numbers 3 3 More elaborate arrays 69 Scipy lecture notes Edition 2015 2 floati float32 float64 64 bits same as float float96 96 bits platform dependent same as np longdoub1le floati28 128 bits platform dependent same as np longdoub1le gt gt gt np finfo np float32 eps 1 1920929e 07 gt gt gt np finfo np float64 eps 2 2204460492503131e 16 gt gt gt np float32 1e 8 np float32 1 1 True gt gt gt np float64 1e 8 np float64 1 1 False Complex floating point numbers complex64 two 32 bit floats complex128 two 64 bit floats complex192 two 96 bit floats platform dependent complex256 two 128 bit floats platform dependent Smaller data types If you don t know you need special data types then you probably don t Comparison on using float32 instead of float64 e Half the size in memory and on disk e Half the memory bandwidth required may be a bit faster in some operations In 1 a np zeros
49. 2015 2 gt gt gt message Hello how are you gt gt gt message split returns a list Hello how are you gt gt gt for word in message split print word Hello how are you Few languages in particular languages for scientific computing allow to loop over anything but inte gers indices With Python it is possible to loop exactly over the objects of interest without bothering with indices you often don t care about This feature can often be used to make code more readable Not safe to modify the sequence you are iterating over Keeping track of enumeration number Common task is to iterate over a sequence while keeping track of the item number e Could use while loop with a counter as above Or a for loop gt gt gt words cool powerful readable gt gt gt for i in range 0 len words oe print i words i O cool 1 powerful 2 readable e But Python provides a built in function enumerate for this gt gt gt for index item in enumerate words pee print index item CO cool 1 powerful 2 readable Looping over a dictionary Use items gt gt gt d a 1 b 1 2 c 1j gt gt gt for key val in sorted d items print Key Zs has value s key val Key a has value 1 Key b has value 1 2 Key c has value 1j The ordering of a dictionary in random thus we use sorted which will sort on the
50. 3 4 ene oa 415 1 2 3 4 dtype np ints8 using only as_strided Hint byte_offset stride 0 index 0 stride 1 index 1 Spoiler Stride can also be 0 gt gt gt x np array 1 2 3 4 dtype np int8 gt gt gt y as_strided x strides 0 1 shape 3 4 gt gt gt y array 1 2 3 4 a ees ees 1 2 3 4 dtype ints8 gt gt gt y base base is x True Broadcasting e Doing something useful with it outer product of 1 2 3 4 and 5 6 7 gt gt gt x np array 1 2 3 4 dtype np inti6 gt gt gt x2 as_strided x strides 0 1 2 shape 3 4 gt gt gt x2 array 1 2 3 4 Ee eel 1 2 3 4 dtype int16 gt gt gt y np array 5 6 7 dtype np int16 gt gt gt y2 as_strided y strides 1 2 0 shape 3 4 gt gt gt y2 array l5 5 5 5 6 6 6 6 7 7 7 7 dtype int16 gt gt gt x2 y2 array iis 10 15 20 e D els 2A 7 14 21 28 dtype int16 seems somehow familiar gt gt gt x np array 1 2 3 4 dtype np inti6 gt gt gt y np array 5 6 7 dtype np inti6 gt gt gt x np newaxis yl np newaxis array il 5 10 15 20 Leci 1e 24 7 14 21 28 dtype int16 e Internally array broadcasting is indeed implemented using 0 strides More tricks diagonals See also 8 1 Life of ndarray 170 Scipy lecture notes Edition 2015 2 stride diagonals py Challenge
51. 3 What is the difficulty 13 5 Optimization with constraints 13 5 1 Box bounds Box bounds correspond to limiting each of the individual parameters of the optimization Note that some problems that are not originally written as box bounds can be rewritten as such via change of variables al E T i e scipy optimize fminbound for 1D optimization e scipy optimize fmin_l_bfgs_b a quasi Newton method with bound constraints 13 5 Optimization with constraints 261 Scipy lecture notes Edition 2015 2 gt gt gt def f x return np sqrt xLl0 3 2 x 1 2 2 gt gt gt optimize fmin_1_bfgs_b f np array 0 0 approx_grad 1 bounds 1 5 1 5 1 5 1 5 array 1 5 1 5 1 5811338300341898 1 13 5 2 General constraints Equality and inequality constraints specified as functions f x 0 and g x lt 0 e scipy optimize fmin_slsqp Sequential least square programming equality and inequality con k straints gt gt gt def f x return np sqrt x 0 3 2 x 1 2 2 gt gt gt def constraint x return np atleast_1d 1 5 np sum np abs x gt gt gt optimize fmin_slsqp f np array 0 0 ieqcons constraint Optimization terminated successfully Exit mode 0 Current function value 2 47487373504 Iterations 5 Function evaluations 20 Gradient evaluations 5 array 1 25004696 0 24995304 e scipy optimize fmin_cobyla Constraints optimization by lin
52. Esc gt 15 make M autocmd FileType tex mp rst python set autowrite e In emacs In your emacs binds F5 to pyf lakes 9 1 Avoiding bugs 193 Scipy lecture notes Edition 2015 2 defun pyflakes thisfile interactive compile format pyflakes 4s buffer file name define minor mode pyflakes mode Toggle pyflakes mode With no argument this command toggles the mode Non null prefix argument turns on the mode Null prefix argument turns off the mode The initial value nil The indicator for the mode line Pyflakes The minor mode bindings 5 pyflakes thisfile add hook python mode hook lambda pyflakes mode t A type as go spell checker like integration e Invim Use the pyflakes vim plugin 1 download the zip file from http www vim org scripts script php script_id 2441 2 extract the files in vim ftplugin python 3 make sure your vimrc has filetype plugin indent on F i anc Il B L r Er aF ja el nprob obs Alternatively use the syntastic plugin This can be configured to use flake8 too and also handles on the fly checking for many other languages data load_dat data data debug_file py at least two spaces before inline comment e In emacs Use the flymake mode with pyflakes documented on http www plope com Members chrism flymake mode add the following to your emacs file when load flymake t defun flymake pyfl
53. Findings in dissection data type ndarray e memory block may be shared base data oe 0O e data type descriptor structured data sub arrays byte order casting viewing astype view e strided indexing strides C F order slicing w integers as_strided broadcasting stride tricks diag CPU cache coherence 8 2 Universal functions 8 2 1 What they are e Ufunc performs and elementwise operation on all elements of an array Examples np add np subtract scipy special e Automatically support broadcasting casting e The author of an ufunc only has to supply the elementwise operation Numpy takes care of the rest e The elementwise operation needs to be implemented in C or e g Cython 8 2 Universal functions 173 Scipy lecture notes Edition 2015 2 Parts of an Ufunc 1 Provided by user void ufunc_loop void xargs int dimensions int steps void data Ve 4 x nt8 output elementwise_function int8 input_1 int8 input_2 This function must compute the ufunc for many values at once 4 char char char int for n the way shown below input_1 char args 0 input_2 char args 1 xoutput char args 2 as i 0 i lt dimensions 0 i output elementwise_function input_1 input_2 input_1 steps 0 input_2 steps 1 output steps 2 2 The Numpy part built by char types 3
54. LUT Vector LUT ModuleManager F vV Mayavi Scene 1 LUT Look Up Table Manager v GridSource v PolyDataNormals Lut mode v amp Colors and legends E m Surface Number of colors m Outline Reverse lut C Mayavi Scene gt in pman m Axes Edit LUT properties k GridSource v PolyDataNormals v m Surface m Title Number of labels 0 1 64 Shadow O Use default name in Launch LUT editor Show legend 4 Data name scalars Edit bar Title Edit bar Text Edit bar Actor Edit bar Widget e Set the background of the figure in the Mayavi Scene node e Set the colormap in the Colors and legends node e Right click on the node to add modules or filters 19 2 Interactive work 346 Scipy lecture notes Edition 2015 2 19 2 2 The script recording button To find out what code can be used to program these changes click on the red button as you modify those properties and it will generate the corresponding lines of code 19 3 Slicing and dicing data sources modules and filters 19 3 1 An example inspecting magnetic fields Suppose we are simulating the magnetic field generated by Helmholtz coils The examples compute_field py script does this computation and gives you a B array that is 3 x n where the first axis is the direction of the field Bx By Bz and the second axis the index number
55. Login scipy ndimage morphology binary_dilation View Log DifftoSVN Discussion Source Review status Being written SciPy Multi dimensional image processing mod scipy ndimage binary_dilation input structure None iterations 1 mask None output None border_value 0 origin 0 e_force False Multi dimensional binary dilation with the given structuring element Parameters input array_like Binary array_like to be dilated Non zero True elements form the subset to be dilated structure array_like optional Structuring element used for the dilation Non zero elements are considered True If no structuring element is provided an element is generated with a square connectivity equal to one at optional repeated iterations times one by default If iterations is less than 1 the dilation is repeated until ore optional Array of the same shape as input into which the output is placed By default a new array is created in int or tuple of ints optional Placement of the filter by default 0 border_value int cast to 0 or 1 Value at the border in the output array Returns out ndarray of bool Dilation of the input by the structuring element See Also Find l help Previous amp Next Highlight all Match case Reached end of page continued from top e Numpy s and Scipy s documentation is enriched and updated on a regular basis by users on a wiki http docs scipy org
56. Plot the scatter matrix for males only and for females only Do you think that the 2 sub populations correspond to gender 15 2 Hypothesis testing comparing two groups For simple statistical tests we will use the scipy stats sub module of scipy gt gt gt from scipy import stats See also Scipy is a vast library For a quick summary to the whole library see the scipy chapter 15 2 Hypothesis testing comparing two groups 289 Scipy lecture notes Edition 2015 2 15 2 1 Student s t test the simplest statistical test l sample t test testing the value of a population mean scipy stats ttest_1samp tests if the population mean of data is likely to be equal to a given value tech nically if observations are drawn from a Gaussian distributions of given population mean It returns the T statistic and the p value see the function s help gt gt gt stats ttest_1samp data VIQ 0 30 088099970 1 32891964 e 28 With a p value of 104 28 we can claim that the population mean for the IQ VIQ measure is not 0 2 sample t test testing for difference across populations We have seen above that the mean VIQ in the male and female populations were different To test if this is significant we do a 2 sample t test with scipy stats ttest_ind gt gt gt female_viq dataldata Gender Female VIQ gt gt gt male_viq data data Gender Male VIQ gt gt gt stats tt
57. Program received signal SIGSEGV Segmentation fault _strided_byte_copy dst 0x8537478 360 343G outstrides 4 src Ox86c0690 lt Address Ox86c0690 out of bounds gt instrides 32 N 3 elsize 4 at numpy core src multiarray ctors c 365 365 _FAST_MOVE Int32 gdb We get a segfault and gdb captures it for post mortem debugging in the C level stack not the Python call stack We can debug the C call stack using gdb s commands gdb up 1 Ox004af4f5 in _copy_from_same_shape dest lt value optimized out gt src lt value optimized out gt myfunc 0x496780 lt _strided_byte_copy gt swap 0 at numpy core src multiarray ctors c 748 748 myfunc dit gt dataptr dest gt strides maxaxis As you can see right now we are in the C code of numpy We would like to know what is the Python code that triggers this segfault so we go up the stack until we hit the Python execution loop gdb up 8 Ox080ddd23 in call_function f Frame 0x85371ec for file home varoquau usr lib python2 6 site packages numpy core arrayprint py line 156 at Python ceval c 3750 3750 Python ceval c No such file or directory in Python ceval c gdb up 9 PyEval_EvalFrameEx f Frame 0x85371ec for file home varoquau usr lib python2 6 site packages numpy core arrayprint py line 156 at Python ceval c 2412 2412 in Python ceval c gdb Once we are in the Python execution loop we can use our special Python helper function For instance w
58. Range on_trait_change class Visualization HasTraits n_turns Range 0O 30 11 scene Instance MlabSceneModel def _init_ self HasTraits _init__ self X y Z curve self n_turns self plot self scene mlab plot3d x y z on_trait_change n_turns def update_plot self X y Z curve self n_turns self plot mlab_source set x x y y z z view View Item scene height 300 show_label False editor SceneEditor HGroup n_turns resizable True Fire up the dialog Visualization configure_traits Full code of the example examples mlab_dialog py 19 6 Putting it together 19 6 Putting it together 351 Scipy lecture notes Edition 2015 2 Exercise Using the code from the magnetic field simulation create a dialog that enable to move the 2 coils change their parameters Hint to define a dialog entry for a vector of dimension 3 direction Array float value 0 0 1 cols 3 shape 3 You can look at the example_coil_application py to see a full blown application for coil design in 270 lines of code 19 6 Putting it together 352 CHAPTER 20 scikit learn machine learning in Python Authors Fabian Pedregosa Gael Varoquaux machine learning in Python Prerequisites Numpy Scipy Python matplotlib scikit learn http scikit learn org Chapters contents Loading an example dataset Learning and Predicting Classification k Nearest neighbors classifie
59. Scipy lecture notes Edition 2015 2 A regression capturing the relation between one variable and another eg wage and eduction can be plotted using seaborn 1lmplot gt gt gt seaborn Implot y WAGE x EDUCATION data data 2 0 1 5 1 0 LLJ lt 0 5 0 0 0 5 0 5 10 15 20 EDUCATION Robust regression Given that in the above plot there seems to be a couple of data points that are outside of the main cloud to the right they might be outliers not representative of the population but driving the regression To compute a regression that is less sentive to outliers one must use a robust model This is done in seaborn using robust True in the plotting functions or in statsmodels by replacing the use of the OLS by a Robust Linear Model statsmodels formula api rlm 15 5 Testing for interactions 2 0 gender e female e male 0 0 e 0 5 10 15 20 education males than females 15 5 Testing for interactions Do wages increase more with education for 299 Scipy lecture notes Edition 2015 2 The plot above is made of two different fits We need to formulate a single model that tests for a variance of slope across the to population This is done via an interaction gt gt gt result sm ols formula wage education gender education gender os data data fit gt gt gt print result summary coef std err iE Pell o gt 0 Comm lint Intercept 0 29985 0 07
60. Table 13 2 Adaptive step gradient descent iterations function calls Error on f x eae e 60 80 100 120 140 A well conditionned quadratic function iterations function calls Error on f x 5 109 200 400 600 800 100012001400 An ill conditionned quadratic function iterations function calls Error on f x 200 400 600 800 1000 1200 An ill conditionned non quadratic function Error on f x An ill conditionned very non quadratic function The more a function looks like a quadratic function elliptic iso curves the easier it is to optimize Conjugate gradient descent The gradient descent algorithms above are toys not to be used on real problems As can be seen from the above experiments one of the problems of the simple gradient descent algorithms is that it tends to oscillate across a valley each time following the direction of the gradient that makes it cross 13 2 A review of the different optimizers 253 Scipy lecture notes Edition 2015 2 the valley The conjugate gradient solves this problem by adding a friction term each step depends on the two last values of the gradient and sharp turns are reduced Table 13 3 Conjugate gradient descent iterations function calls Error on f x 60 80 100 120 An ill conditionned non quadratic function iterations function calls Error on f x ae 60 80 100 120 An ill condition
61. Types of Plots examples and exercises 95 ax plot_surface X Y Z rstride 1 cstride 1 cmap hot Click on figure for solution See also 3D plotting with Mayavi 4 4 12 Text Scipy lecture notes Edition 2015 2 a Have a look at the matplotlib logo Try to do the same from scratch Click on figure for solution Quick read If you want to do a first quick pass through the Scipy lectures to learn the ecosystem you can directly skip to the next chapter Scipy high level scientific computing The remainder of this chapter is not necessary to follow the rest of the intro part But be sure to come back and finish this chapter later 4 5 Beyond this tutorial Matplotlib benefits from extensive documentation as well as a large community of users and developers Here are some links of interest 4 5 1 Tutorials 4 5 Pyplot tutorial Introduction Controlling line properties Working with multiple figures and axes Working with text Image tutorial Startup commands Importing image data into Numpy arrays Plotting numpy arrays as images Text tutorial Text introduction Basic text commands Text properties and layout Writing mathematical expressions Text rendering With LaTeX Beyond this tutorial 96 Scipy lecture notes Edition 2015 2 e Annotating text e Artist tutorial e Introduction e Customizing your objects e Object containers e Figure container e Axes container e Axis
62. When discussing generators it was said that we prefer generators to iterators implemented as classes because they are shorter sweeter and the state is stored as local not instance variables On the other hand as de scribed in Bidirectional communication the flow of data between the generator and its caller can be bidirec tional This includes exceptions which can be thrown into the generator We would like to implement context managers as special generator functions In fact the generator protocol was designed to support this use case contextlib contextmanager def some_generator lt arguments gt lt setup gt try yield lt value gt finally lt cleanup gt The contextlib contextmanager helper takes a generator and turns it into a context manager The gener ator has to obey some rules which are enforced by the wrapper function most importantly it must yield exactly once The part before the yield is executed from __enter__ the block of code protected by the con text manager is executed when the generator is suspended in yield and the rest is executed in __exit__ If an exception is thrown the interpreter hands it to the wrapper through __exit__ arguments and the wrap per function then throws it at the point of the yield statement Through the use of generators the context manager is shorter and simpler Let s rewrite the closing example as a generator contextlib contextmanager def closing obj try yield obj
63. X max 1 1 plt ylim C min 1 1 C max 1 1 4 2 Simple plot 85 Scipy lecture notes Edition 2015 2 4 2 5 Setting ticks Documentation e xticks command e yticks command e Tick container e Tick locating and formatting L 4 L L 3 142 1 571 0 000 1 571 3 142 Current ticks are not ideal because they do not show the interesting values 7 7 2 for sine and cosine We ll change them such that they show only these values plt xticks np pi np pi 2 0 np pi 2 np pil l plt yticks 1 0 11 4 2 0 Setting tick labels 1 Documentation e Working with text i a e xticks command e yticks command e set_xticklabels e set_yticklabels Ticks are now properly placed but their label is not very explicit We could guess that 3 142 is x but it would be better to make it explicit When we set tick values we can also provide a corresponding label in the second argument list Note that we ll use latex to allow for nice rendering of the label plt xticks np pi np pi 2 0 np pi 2 np Pil r Apis s r b pa72e i POR fot pa 25 I Spio pit yticks i 0 1 r 1 r 0 r i 4 2 Simple plot 86 Scipy lecture notes Edition 2015 2 4 2 7 Moving spines Documentation e Spines e Axis container e Transformations tutorial Spines are the lines connecting the axis tick marks and noting the boundaries of the data area They can be p
64. advanced users Not free Other scripting languages Scilab Octave Igor R IDL etc e Advantages Open source free or at least cheaper than Matlab Some features can be very advanced statistics in R figures in Igor etc e Drawbacks Fewer available algorithms than in Matlab and the language is not more advanced Some software are dedicated to one domain Ex Gnuplot or xmgrace to draw curves These pro grams are very powerful but they are restricted to a single type of usage such as plotting What about Python e Advantages Very rich scientific computing libraries a bit less than Matlab though Well thought out language allowing to write very readable and well structured code we code what we think Many libraries for other tasks than scientific computing web server management serial port ac cess etc Free and open source software widely spread with a vibrant community e Drawbacks less pleasant development environment than for example Matlab More geek oriented Not all the algorithms that can be found in more specialized software or toolboxes 1 2 Scientific Python building blocks Unlike Matlab Scilab or R Python does not come with a pre bundled set of modules for scientific computing Below are the basic building blocks that can be combined to obtain a scientific computing environment e Python a generic and modern computing language 1 2 Scientific Pytho
65. api import HasTraits Str Float class Reservoir HasTraits name Str max_storage Float 100 desc Maximal storage hm3 Let s now define the complete reservoir class from traits api import HasTraits Str Float Range class Reservoir HasTraits name Str max_storage Float 1e6 desc Maximal storage hm3 max_release Float 10 desc Maximal release m3 s head Float 10 desc Hydraulic head m efficiency Range 0 1 def energy_production self release Returns the energy production Wh for the given release m3 s power 1000 9 81 self head release self efficiency return power 3600 if coname _ main __ reservoir Reservoir name Project A max_storage 30 max release 100 0 head 60 efficiency 0 8 release 80 print Releasing m3 s produces kWh format release reservoir energy_production release 18 3 What are Traits 326 Scipy lecture notes Edition 2015 2 18 3 4 Visualization opening a dialog The Traits library is also aware of user interfaces and can pop up a default view for the Reservoir class reservoiri Reservoir reservoirl edit_traits Poe Paes Pe Edit properties Efficiency 0 000 NA 1008 Head 10 Hydraulic head 60 Max release 100 0 Max storage 30 0 Name Proj ect A OK Cancel TraitsUI simplifies the way user interfaces are created Every trait on a HasTraits class has a default editor that wi
66. argmin index of minimum O gt gt gt x argmax index of maximum 1 Logical operations gt gt gt np all True True False False gt gt gt np any True True False True 3 2 Numerical operations on arrays 58 Scipy lecture notes Edition 2015 2 Can be used for array comparisons gt gt gt a np zeros 100 100 gt gt gt np any a 0 False gt gt gt np all a a True gt gt gt a np array 1 2 3 2 gt gt gt b np array 2 2 3 2 gt gt gt c np array 6 4 4 5 gt gt gt MUG lt b amp ie et True Statistics gt gt gt x np array 1 2 3 1 gt gt gt y np array 1 2 3 5 6 1 gt gt gt x mean 1 75 gt gt gt np median x 1 5 gt gt gt np median y axis 1 last aris array Gl 265 7521 gt gt gt x stdQ full population standard dev 0 82915619758884995 and many more best to learn as you go Exercise Reductions e Given there is a sum what other function might you expect to see e What is the difference between sum and cumsum 3 2 Numerical operations on arrays 59 Scipy lecture notes Edition 2015 2 Worked Example data statistics Data in populations txt describes the populations of hares and lynxes and carrots in northern Canada during 20 years You can view the data in an editor or alternatively in Python both shell and notebook In 1 icat data populations txt
67. array 4 3 1 2 gt gt gt j_max np argmax a gt gt gt j_min np argmin a gt gt gt j_max j_min 0 2 3 2 Numerical operations on arrays 67 Scipy lecture notes Edition 2015 2 Exercise Sorting e Try both in place and out of place sorting e Try creating arrays with different dtypes and sorting them e Use all or array_equal to check the results e Look at np random shuffle for a way to create sortable input quicker e Combine ravel sort and reshape e Look at the axis keyword for sort and rewrite the previous exercise 3 2 6 Summary What do you need to know to get started e Know how to create arrays array arange ones zeros e Know the shape of the array with array shape then use slicing to obtain different views of the array array L 2 etc Adjust the shape of the array using reshape or flatten it with ravel e Obtain a subset of the elements of an array and or modify their values with masks gt gt gt ala lt 0 0 e Know miscellaneous operations on arrays such as finding the mean or max array max array mean No need to retain everything but have the reflex to search in the documentation on line docs help lookfor e For advanced use master the indexing with arrays of integers as well as broadcasting Know more Numpy functions to handle various array operations Quick read If you want to do a first quick pass through the Scipy lectures to learn the eco
68. as filters gt gt gt camera_sobel filters sobel camera gt gt gt camera_sobel max 0 591502 A In the example above we use the filters submodule of scikit image that has been renamed from filter to filters between versions 0 10 and 0 11 in order to avoid a collision with Python s built in name filter Utility functions are provided in skimage to convert both the dtype and the data range following skimage s conventions util img_as_float util img_as_ubyte etc See the user guide for more details 17 2 2 Colorspaces Color images are of shape N M 3 or N M 4 when an alpha channel encodes transparency gt gt gt face scipy misc face kes face shape 768 1024 3 17 2 Input output data types and colorspaces 311 Scipy lecture notes Edition 2015 2 Routines converting between different colorspaces RGB HSV LAB etc are available in skimage color color rgb2hsv color lab2rgb etc Check the docstring for the expected dtype and data range of input images 3D images Most functions of skimage can take 3D images as input arguments Check the docstring to know if a function can be used on 3D images for example MRI or CT images Exercise Open a color image on your disk as a NumPy array Find a skimage function computing the histogram of an image and plot the histogram of each color channel Convert the image to grayscale and plot its histogram 17 3 Image
69. bugs 3 Complain on the mailing list 8 6 4 Contributing features 0 Ask on mailing list if unsure where it should go 1 Write a patch add an enhancement ticket on the bug tracket 2 OR create a Git branch implementing the feature add enhancement ticket e Especially for big invasive additions e http projects scipy org numpy wiki GitMirror e http www spheredev org wiki Git_for_the_lazy Clone numpy repository git clone origin svn http projects scipy org git numpy git numpy cd numpy Create a feature branch git checkout b name of my feature branch svn trunk lt edit stuff gt git commit a e Create account on https github com or anywhere e Create a new repository Github e Push your work to github git remote add github git github USERNAME REPOSITORYNAME git git push github name of my feature branch 8 6 Contributing to Numpy Scipy 190 Scipy lecture notes Edition 2015 2 8 6 5 How to help in general e Bug fixes always welcome What irks you most Browse the tracker e Documentation work API docs improvements to docstrings Know some Scipy module well User guide Needs to be done eventually x Want to think Come up with a Table of Contents http scipy org Developer_Zone UG_Toc e Ask on communication channels numpy discussion list scipy dev list 8 6 Contributing to Numpy Scipy 191 CHAPTER 9 Debugging code Author Ga l Varoquaux This section explor
70. cos_func 3 14159265359 Out 6 1 0 Again we test for robustness and we see that we get a better error message although strictly speaking in 14 4 SWIG 273 Scipy lecture notes Edition 2015 2 Python there is no double type In 7 cos_module cos_func foo TypeError Traceback most recent call last lt ipython input 7 11bee483665d gt in lt module gt gt 1 cos_module cos_func foo TypeError in method cos_func argument 1 of type double 14 4 2 Numpy Support Numpy provides support for SWIG with the numpy i file This interface file defines various so called typemaps which support conversion between Numpy arrays and C Arrays In the following example we will take a quick look at how such typemaps work in practice We have the same cos_doubles function as in the ctypes example void cos_doubles double in_array double out_array int size include lt math h gt Compute the cosine of each element in in_array storing the result in out_array void cos_doubles double in_array double out_array int size inte a for i 0 i lt size i out_array i cos in_arrayLi This is wrapped as cos_doubles_func using the following SWIG interface file Example of wrapping a C function that takes a C double array as input using numpy typemaps for SWIG module cos_doubles ht the resulting C file should be built as a python extension define SWIG_FILE_WIT
71. cos_module h cos_module i setup py python setup py build_ext inplace running build_ext building _cos_module extension SWigging cos_module i to cos_module_wrap c swig python o cos_module_wrap c cos_module i creating build creating build temp linux x86_64 2 7 gcc pthread fno strict aliasing g 02 DNDEBUG g fwrapv 03 Wall Wstrict prototypes fPIC I home esc an gcc pthread fno strict aliasing g 02 DNDEBUG g fwrapv 03 Wall Wstrict prototypes fPIC I home esc an gcc pthread shared build temp linux x86_64 2 7 cos_module o build temp linux x86_64 2 7 cos_module_wrap o L 1s build cos_module c cos_module h cos_module i cos_module py _cos_module so cos_module_wrap c setup py We can now load and execute the cos_module as we have done in the previous examples In 1 import cos_module In 2 cos_module Type module String Form lt module cos_module from cos_module py gt File home esc git working scipy lecture notes advanced interfacing with_c swig cos_module py Docstring lt no docstring gt In 3 dir cos_module putissi ms builtins A SENEC OC _file__ name_ __package__ cos_module newclass object _swig_getattr _swig_property _Swig_repr _swig_setattr _swig_setattr_nondynamic cos fu unc In 4 cos_module cos_func 1 0 Out 4 0 5403023058681398 In 5 cos_module cos_func 0 0 aye Bed pagel 6 In 6 cos_module
72. doc numpy As a result some docstrings are clearer or more detailed on the wiki 137 Scipy lecture notes Edition 2015 2 and you may want to read directly the documentation on the wiki instead of the official documentation website Note that anyone can create an account on the wiki and write better documentation this is an easy way to contribute to an open source project and improve the tools you are using e Scipy central http central scipy org gives recipes on many common problems frequently encoun amp smatplotlib home search examples gallery docs modules index Click on any image to see full size image and source code Ww A e3 m Go gt a wy g Enter search terms or a module te class or function name A i As amp matplotlib Bg NENN KEK DE NNN tered such as fitting data points solving ODE etc aa Ta e Matplotlib s website http matplotlib org features a very nice gallery with a large number of plots each of them shows both the source code and the resulting plot This is very useful for learning by example More standard documentation is also available Finally two more technical possibilities are useful as well e In Ipython the magical function psearch search for objects matching patterns This is useful if for example one does not know the exact name of a function In 3 import numpy as np In 4 psearch np diag np diag np diagflat np diagonal e numpy
73. documentation for numpy ndarray ctypes and numpy ctypeslib For the following example let s consider a C function in a library that takes an input and an output array computes the cosine of the input array and stores the result in the output array The library consists of the following header file although this is not strictly needed for this example we list it for completeness void cos_doubles double in_array double out_array int size The function implementation resides in the following C source file include lt math h gt Compute the cosine of each element in in_array storing the result in out_array void cos_doubles double in_array double out_array int size int ij for i 0 i lt size i out_array i cos in_arrayLi And since the library is pure C we can t use distutils to compile it but must use a combination of make and gcc m PHONY clean libcos_doubles so cos_doubles o gcc shared Wl soname libcos_doubles so o libcos_doubles so cos_doubles o cos_doubles o cos_doubles c gcc c fPIC cos_doubles c o cos_doubles o clean rm vf libcos_doubles so cos_doubles o cos_doubles pyc 14 3 Ctypes 270 Scipy lecture notes Edition 2015 2 We can then compile this on Linux into the shared library 1ibcos_doubles so ls cos_doubles c cos_doubles h cos_doubles py makefile test_cos_doubles py make gcc c fPIC cos doubles ic o cos doubles o gcc
74. exercise by using the wind speeds measured over 21 years The measurement period is around 90 minutes the original period was around 10 minutes but the file size has been reduced for making the exercise setup easier The data are stored in numpy format inside the file examples sprog windspeeds npy Do not look at the source code for the plots until you have completed the exercise 5 11 Summary exercises on scientific computing 125 Scipy lecture notes Edition 2015 2 Cumulative probability Vso 32 98 m s 0 06 22 24 26 28 30 32 34 Annual wind speed maxima m s Figure 5 1 Solution Python source file 5 11 Summary exercises on scientific computing 126 Scipy lecture notes Edition 2015 2 e The first step will be to find the annual maxima by using numpy and plot them as a matplotlib bar figure Annual wind speed maxima m s Year Figure 5 2 Solution Python source file e The second step will be to use the Gumbell distribution on cumulative probabilities p_i defined as log log p_i for fitting a linear quantile function remember that you can define the degree of the UnivariateSpline Plotting the annual maxima versus the Gumbell distribution should give you the following figure e The last step will be to find 34 23 m s for the maximum wind speed occurring every 50 years 5 11 2 Non linear least squares curve fitting application to point extraction in topographical lidar data The goal of this exercise is
75. fast Fourier transform gt gt gt from scipy import fftpack gt gt gt sample_freq fftpack fftfreq sig size d time_step gt gt gt sig _fft fftpack fft sig Because the resulting power is symmetric only the positive part of the spectrum needs to be used for finding the frequency gt gt gt pidxs np where sample_freq gt 0 gt gt gt freqs sample_freq pidxs gt gt gt power np abs sig_fft pidxs 5 4 Fast Fourier transforms scipy fftpack 104 Scipy lecture notes Edition 2015 2 Peak frequency __ 0 050 100 150 200 250 300 350 400 45 0 5 10 15 20 25 Frequency Hz The signal frequency can be found by gt gt gt freq freqs power argmax gt gt gt np allclose freq 1 period check that correct freq is found True Now the high frequency noise will be removed from the Fourier transformed signal gt gt gt sig _fft np abs sample_freq gt freq 0 The resulting filtered signal can be computed by the scipy fftpack ifft function gt gt gt main_sig fftpack ifft sig_fft The result can be viewed with gt gt gt import pylab as plt gt gt gt plt figure lt matplotlib figure Figure object at 0x gt gt gt gt plt plot time_vec sig lt matplotlib lines Line2D object at Ox gt gt gt gt plt plot time_vec main_sig linewidth 3 lt matplotlib lines Line2D object at Ox gt gt gt gt plt xlabel Time s lt matplotiib te
76. foobar gt 5 print lst len 1st 6 ipdb gt list i gt Small snippet to raise an IndexError ts 2 3 def index_error 4 lst list foobar gt 5 print lst len 1st 6 T if __name__ __main__ 8 index_error 9 ipdb gt len 1st 6 ipdb gt print lst len 1st 1 Y ipdb gt quit In 3I 9 3 Using the Python debugger 196 Scipy lecture notes Edition 2015 2 Post mortem debugging without IPython In some situations you cannot use Python for instance to debug a script that wants to be called from the command line In this case you can call the script with python m pdb script py python m pdb index_error py gt home varoquau dev scipy lecture notes advanced optimizing index_error py 1 lt module gt gt Small snippet to raise an IndexError Pdb continue Traceback most recent call last File usr lib python2 6 pdb py line 1296 in main pdb _runscript mainpyfile File usr lib python2 6 pdb py line 1215 in _runscript self run statement File usr lib python2 6 bdb py line 372 in run exec cmd in globals locals File lt string gt line 1 in lt module gt File index_error py line 8 in lt module gt index error File index_error py line 5 in index_error print lst len 1st IndexError list index out of range Uncaught exception Entering post mortem debugging Running cont or step will restart the program gt home v
77. for different classes of global optimization problems exist but this is out of the scope of scipy Some useful packages for global optimization are OpenOpt IPOPT PyGMO and PyEvolve scipy used to contain the routine anneal it has been deprecated since SciPy 0 14 0 and removed in SciPy 0 16 0 To find the local minimum lets constraint the variable to the interval 0 10 using scipy optimize fminbound gt gt gt xmin_local optimize fminbound f 0 10 gt gt gt xmin_local 3 8374671 Finding minima of function is discussed in more details in the advanced chapter Mathematical optimization finding minima of functions Finding the roots of a scalar function To find a root ie a point where f x 0 of the function f above we can use for example scipy optimize fsolve gt gt gt root optimize fsolve f 1 our initial guess is 1 gt gt gt root array F 0 1 Note that only one root is found Inspecting the plot of f reveals that there is a second root around 2 5 We find the exact value of it by adjusting our initial guess gt gt gt root2 optimize fsolve f 2 5 gt gt gt root2 array 2 47948183 Curve fitting Suppose we have data sampled from f with some noise gt gt gt xdata np linspace 10 10 num 20 gt gt gt ydata f xdata np random randn xdata size Now if we know the functional form of the function from which the samples were drawn x72 sin x in this case but
78. from numpy random import rand gt gt gt data np round rand 2 3 gt gt gt data array lL boe ies el EO 11 2 Storage Schemes 215 Scipy lecture notes Edition 2015 2 e assign the data using fancy indexing gt gt gt mtx 2 1 2 3 data gt gt gt mtx lt 4x5 sparse matrix of type lt numpy float64 gt with 5 stored elements in LInked List format gt gt gt gt print mtx Of 1 0 OTD OS 71 0 Fee 10 Oe A gt gt gt mtx todense matriz Il O24 Glee te tes 20 7 Oe sey Os lite Oe Ons One BOs One Oae Ore 106 Oey sO O 1d gt gt gt mtx toarray array ll Oe aa RiGee d E Osy O O 0 2 2 O O O re O O re re 2 e more slicing and indexing gt gt gt mtx sparse lil matrix 0 1 2 0 3 0 1 0l 1 0 0 Hid gt gt gt mtx todense Matrix LPO 1 2 Ol 3 0 1s 0l EER NA o T EE gt gt gt print mtx Cond 1 Om 2 1 0 1 2 2 0 257 3 gt gt gt mex 2 lt lt 2x4 sparse matrix of type lt numpy int64 gt with 4 stored elements in LInked List format gt gt gt gt mtx 2 todense matrix l0 1 2 0 3 0 aT ol gt gt gt mtx 1 2 0 2 todense matrix LS 11 3 gt gt gt mtx todense matrix 0 1 2 0 B0 les Ol tes BOs OO ig A eae eee WwW DY Dictionary of Keys Format DOK e subclass of Python dict keys are row column index tuples no duplica
79. gt include lt numpy arrayobject h gt include lt math h gt wrapped cosine function static PyObject cos_func_np PyObject self PyObject args PyArrayObject in_array PyObject out_array NpyIter in_iter NpyIter out_iter NpyIter_IterNextFunc in_iternext NpyIter_IterNextFunc out_iternext parse single numpy array argument if PyArg ParseTuple args 0 amp PyArray_Type in_array return NULL construct the output array like the input array out_array PyArray_NewLikeArray in_array NPY_ANYORDER NULL 0 if out_array NULL return NULL create the iterators in_iter NpyIter_New in_array NPY_ITER_READONLY NPY_KEEPORDER NPY_NO_CASTING NULL if in_iter NULL goto fail out_iter NpyIter_New PyArrayObject out_array NPY_ITER_READWRITE NPY_KEEPORDER NPY_NO_CASTING NULL if out_iter NULL NpyIter_Deallocate in_iter goto fail in_iternext NpyIter_GetIterNext in_iter NULL out_iternext NpyIter_GetIterNext out_iter NULL if in_iternext NULL out_iternext NULL NpyIter_Deallocate in_iter NpyIter_Deallocate out_iter goto fail double in_dataptr double NpyIter_GetDataPtrArray in_iter double out_dataptr double NpyIter_GetDataPtrArray out_iter iterate over the arrays do xout_dataptr cos in_dataptr while in_iternext in_iter amp amp out_iternext out_iter clean up and re
80. gt gt gt 8 4 3 2 Type conversion casting gt gt gt float 1 1 0 2 2 Basic types 12 Scipy lecture notes Edition 2015 2 A Integer division In Python 2 22 gt 3 2 1 In Python 3 gt 22 32 T5 To be safe use floats gt gt gt 3 2 25 gt gt gt a gt gt gt gt gt gt a 1 gt gt gt a float b 5 oO Il 3 2 b S In Python 2 Future behavior to always get the behavior of Python3 gt gt gt from __future__ import division a e S If you explicitly want integer division use gt gt gt 3 0 2 16 The behaviour of the division operator has changed in Python 3 2 2 2 Containers Python provides many efficient types of containers in which collections of objects can be stored Lists A list is an ordered collection of objects that may have different types For example gt gt gt 1 red blue green black white gt gt gt type 1 lt type list gt Indexing accessing individual objects contained in the list gt gt gt 1 2 green Counting from the end with negative indices gt gt gt 1 1 white gt gt gt 1 2 black A Indexing starts at 0 as in C not at 1 as in Fortran or Matlab Slicing obtaining sublists of regularly spaced elements 2 2 Basic types 13 Scipy lecture notes Edition 2015 2 gt gt gt 1 red blue green black white gt gt gt 1 2 4 gr
81. gt gt gt camera io imread filename Works with all data formats supported by the Python Imaging Library or any other I O plugin provided to imread with the plugin keyword argument Also works with URL image paths gt gt gt logo io imread http scikit image org _static img logo png Saving to files 17 2 Input output data types and colorspaces 310 Scipy lecture notes Edition 2015 2 gt gt gt io imsave local_logo png logo imsave also uses an external plugin such as PIL 17 2 1 Data types Image ndarrays can be represented either by integers signed or unsigned or floats Careful with overflows with integer data types gt gt gt camera data camera gt gt gt camera dtype dtype uint8 gt gt gt camera_multiply 3 camera Different integer sizes are possible 8 16 or 32 bytes signed or unsigned A Animportant if questionable skimage convention float images are supposed to lie in 1 1 in order to have comparable contrast for all float images gt gt gt from skimage import img _as_float gt gt gt camera_float img_as_float camera gt gt gt camera max camera_float max 255 1 0 Some image processing routines need to work with float arrays and may hence output an array with a different type and the data range from the input array gt gt gt try from skimage import filters except ImportError from skimage import filter
82. gt gt gt diff sin 2 x x 3 8 cos 2 x 16 3 3 Series expansion SymPy also knows how to compute the Taylor series of an expression at a point Use series expr var gt gt gt series cos x x 2 4 x x 7 6X eo 2 24 16 3 Calculus 8 Scipy lecture notes Edition 2015 2 gt gt gt series 1 cos x x 2 4 X 5 x 6 1 O x 2 24 Exercises 1 Calculate lim _ 9 sin x x 2 Calculate the derivative of log x for x 16 3 4 Integration SymPy has support for indefinite and definite integration of transcendental elementary and special functions via integrate facility which uses powerful extended Risch Norman algorithm and some heuristics and pattern matching You can integrate elementary functions gt gt gt integrate 6 x 5 x 6 Z gt gt gt integrate sin x x Cos x gt gt gt integrate log x x log Ga ie ce gt gt gt integrate 2 x sinh x x 2 x cosh x Also special functions are handled easily gt gt gt integrate exp x 2 erf x x EEN 2 pi erf x It is possible to compute definite integral gt gt gt integrate x 3 x 1 1 O gt gt gt integrate sin x x 0 pi 2 1 gt gt gt integrate cos x x pi 2 pi 2 2 Also improper integrals are supported as well gt gt gt integrate exp x x 0 00 il gt gt gt integrate exp x 2 x 00 00 pi 16 3 5 Exercises 16 4 Equation solving S
83. gt gt gt print x 10600 pr 21 07 0 4 6 60 tes sO 1 0 0 j gt gt gt print Error 7s 4 mtx2 x rhs Error TOOS Oe tO Or Oe ees Ong Oo nnnm Solve a linear system Construct a 1000x1000 lil_matrix and add some values to it convert it to CSR format and solve A x b for s and solve a linear system with a direct solver import numpy as np import scipy sparse as sps from matplotlib import pyplot as plt from scipy sparse linalg dsolve import linsolve rand np random rand mtx sps lil_matrix 1000 1000 dtype np float64 mtx 0 100 rand 100 mtx 1 100 200 mtx 0 100 mtx setdiag rand 1000 plt clf plt spy mtx marker markersize 2 plt show mtx mtx tocsr rand 1000 linsolve spsolve mtx rhs 11 3 Linear System Solvers 225 Scipy lecture notes Edition 2015 2 print rezidual r np linalg norm mtx x rhs e examples direct_solve py 11 3 2 Iterative Solvers e the isolve module contains the following solvers bicg BIConjugate Gradient bicgstab BIConjugate Gradient STABilized cg Conjugate Gradient symmetric positive definite matrices only cgs Conjugate Gradient Squared gmres Generalized Minimal RESidual minres MINimum RESidual qmr Quasi Minimal Residual Common Parameters e mandatory A sparse matrix dense matrix LinearOperator The N by N matrix of the linear system b array matrix Right hand side
84. gt gt populations bad_years 1 np ma masked gt gt gt populations mean axis 0 masked_array data 40472 72727272727 18627 272727272728 42400 0 mask False False False fill value 1e 20 gt gt gt populations std axis 0 masked_array data 21087 656489006717 15625 799814240254 3322 5062255844787 mask False False False fill_value 1e 20 Note that Matplotlib knows about masked arrays gt gt gt plt plot year populations o lt matplotlib lines Line2D object at gt 80000 70000 60000 50000 40000 A 30000 20000 10000 O 1900 1905 1910 1915 1920 8 4 Array siblings chararray maskedarray matrix 187 Scipy lecture notes Edition 2015 2 8 4 3 recarray purely convenience gt gt gt arr np array C a 1 Cb 2 dtype Cx Si 4 Cy ant gt gt gt arr2 arr view np recarray gt gt gt arr2 x chararray Da le dtype S1 gt gt gt arr2 y array Li 21 8 4 4 matrix convenience e always 2 D e is the matrix product not the elementwise one gt gt gt np matrix 1 0 0 1 np matrix 1 2 3 4 matrix 1 21 3 4 8 5 Summary e Anatomy of the ndarray data dtype strides e Universal functions elementwise operations how to make new ones e Ndarray subclasses e Various buffer interfaces for integration with other tools e Recent additions PEP 3118 generalized ufuncs 8 6 Contributing to Numpy Sci
85. gt gt y as_strided x shape 5 5 strides 5 5 5 5 x itemsize 5 5 1 x itemsize gt gt gt s2 y sum CPU cache effects Memory layout can affect performance In 1 x np zeros 20000 In 2 y np zeros 20000 67 67 In 3 x shape y shape 20000 20000 In 4 timeit x sum 100000 loops best of 3 0 180 ms per loop In 5 timeit y sum 100000 loops best of 3 2 34 ms per loop In 6 x ctrides y strides COT E e Smaller strides are faster cache block size e CPU pulls data from main memory to its cache in blocks e If many array items consecutively operated on fit in a single block small stride gt fewer transfers needed gt faster See also numexpr is designed to mitigate cache effects in array computing Example inplace operations caveat emptor e Sometimes gt gt gt a b is not the same as gt gt gt a b copy 8 1 Life of ndarray 172 gt gt gt x np array 1 2 3 4 gt gt gt x x transpose gt gt gt X array IE On SII 4 0 Scipy lecture notes Edition 2015 2 gt gt gt y np array 1 2 3 4 gt gt gt y y T copy gt gt gt y array L OAI i aie O11 e xandx transpose share data e x x transpose modifies the data element by element e because x and x transpose have different striding modified data re appears on the RHS 8 1 5
86. gt mtx 2 1 1 3 todense matrix I 1 0 EO Coordinate Format COO e also known as the ijv or triplet format three NumPy arrays row col data datali is value at rowLi colli position permits duplicate entries subclass of _data_matrix sparse matrix classes with data attribute e fast format for constructing sparse matrices e constructor accepts dense matrix array 11 2 Storage Schemes 217 Scipy lecture notes Edition 2015 2 sparse matrix shape tuple create empty matrix data ij tuple e very fast conversion to and from CSR CSC formats e fast matrix vector sparsetools fast and easy item wise operations manipulate data array directly fast NumPy machinery e no slicing no arithmetics directly e use facilitates fast conversion among sparse formats when converting to other format usually CSR or CSC duplicate entries are summed together facilitates efficient construction of finite element matrices Examples e create empty COO matrix gt gt gt mtx sparse coo_matrix 3 4 dtype np int8 gt gt gt mtx todense matrix 0 0 0 0 10540 0 OE 0 0 0 OJ dtype int8 e create using data ij tuple gt gt gt row np array 0 3 1 0 gt gt gt col np array 0 3 1 2 gt gt gt data np array 4 5 7 9 gt gt gt mtx sparse coo_matrix data row col shape 4 4 gt gt gt mtx lt 4
87. is a function containing the keyword yield It must be noted that the mere presence of this keyword completely changes the nature of the function this yield statement doesn t have to be invoked or even reachable but causes the function to be marked as a generator When a normal function is called the instructions contained in the body start to be executed When a generator is called the execution stops before the first instruction in the body An invocation of a generator function creates a generator object adhering to the iterator protocol As with normal function invocations concurrent and recursive invocations are allowed When next is called the function is executed until the first yield Each encountered yield statement gives a value becomes the return value of next After executing the yield statement the execution of this function is suspended gt gt gt def f yield 1 yield 2 gt gt gt f lt generator object f at 0x gt gt gt gt gen f gt gt gt next gen 1 gt gt gt next gen 2 gt gt gt next gen Traceback most recent call last File lt stdin gt line 1 in lt module gt StopIteration 7 1 Iterators generator expressions and generators 144 Scipy lecture notes Edition 2015 2 Let s go over the life of the single invocation of the generator function gt gt gt def f princ start yield 3 print middle yield 4 zes print finished gt gt gt
88. keys 2 3 6 List Comprehensions gt gt gt i 2 for i in range 4 o 1 4 9 2 3 Control Flow 21 Scipy lecture notes Edition 2015 2 Exercise Compute the decimals of Pi using the Wallis formula oe 4i m 2 i s Izo 2 4 Defining functions 2 4 1 Function definition In 56 def test Eee print in test function In 57 test in test function A Function blocks must be indented as other control flow blocks 2 4 2 Return statement Functions can optionally return values In 6 def disk_area radius return 3 14 radius radius In 8 disk_area 1 5 Out 8 7 0649999999999995 By default functions return None Note the syntax to define a function e the def keyword e is followed by the function s name then e the arguments of the function are given between parentheses followed by a colon e the function body e andreturn object for optionally returning values 2 4 3 Parameters Mandatory parameters positional arguments In 81 def double_it x judas return x 2 In 82 double _it 3 Out 82 6 In 83 double_it 2 4 Defining functions 22 Scipy lecture notes Edition 2015 2 Traceback most recent call last File lt stdin gt line 1 in lt module gt TypeError double_it takes exactly 1 argument 0 given Optional parameters keyword or named arguments In 84 def double_it x 2 re return x 2 In 85 double_it Out 85 4 In 8
89. mask sand_labels ravel reshape sand_labels shape 8 Compute the mean size of bubbles gt gt gt bubbles_labels bubbles_nb ndimage label void gt gt gt bubbles_areas np bincount bubbles_labels ravel i gt gt gt mean_bubble_ size bubbles_areas mean gt gt gt median_bubble_size np median bubbles_areas gt gt gt mean_bubble_size median _bubble_size 5 11 Summary exercises on scientific computing 135 Scipy lecture notes Edition 2015 2 16997 G15 65 0 5 11 Summary exercises on scientific computing 136 CHAPTER 6 Getting help and finding documentation Author Emmanuelle Gouillart Rather than knowing all functions in Numpy and Scipy it is important to find rapidly information throughout the documentation and the available help Here are some ways to get information e In Ipython help function opens the docstring of the function Only type the beginning of the func tion s name and use tab completion to display the matching functions In 204 help np v np vander np vdot np version np void0O np vstack np var np vectorize np void np vsplit In 204 help np vander In Ipython it is not possible to open a separated window for help and documentation how ever one can always open a second Ipython shell just to display help and docstrings Numpy and Scipy Documentation SciPy v0 8 dev Reference Guide DRAFT next modules index SciPy Release 0 8 dev
90. matplotlib means the whole window in the user interface Within this figure there can be sub plots So far we have used implicit figure and axes creation This is handy for fast plots We can have more control over the display using figure subplot and axes explicitly While subplot positions the plots in a regular grid axes allows free placement within the figure Both can be useful depending on your intention We ve already worked with figures and subplots without explicitly calling them When we call plot matplotlib calls gca to get the current axes and gca in turn calls gcf to get the current figure If there is none it calls figure to make one strictly speaking to make a subplot 111 Let s look at the details 4 3 1 Figures A figure is the windows in the GUI that has Figure as title Figures are numbered starting from 1 as opposed to the normal Python way starting from 0 This is clearly MATLAB style There are several parameters that determine what the figure looks like wm ft _ umber of figure fi Bue fi igs ize figure s size in in inches width height apt T resolution in dots perinel The defaults can be specified in the resource file and will be used most of the time Only the number of the figure is frequently changed As with other objects you can set figure properties also setp or with the set_something methods When you work with the GUI you can close a figure by clicking on the x in the upp
91. nested functions gt gt gt def replacing decorator_with_args arg print defining the decorator def _decorator function in this inner function arg ts avatlable too print doing decoration r arg def _wrapper args kwargs print inside wrapper r Zr args kwargs return function args xkwargs return _wrapper return _decorator gt gt gt replacing decorator_with_args abc def function args xkwargs print inside function r sr args kwargs return 14 defining the decorator doing decoration abc gt gt gt function 11 12 inside wrapper 11 12 inside function 11 12 4 14 The _wrapper function is defined to accept all positional and keyword arguments In general we cannot know what arguments the decorated function is supposed to accept so the wrapper function just passes everything to the wrapped function One unfortunate consequence is that the apparent argument list is misleading Compared to decorators defined as functions complex decorators defined as classes are simpler When an object is created the __init__ method is only allowed to return None and the type of the created object cannot be changed This means that when a decorator is defined as a class it doesn t make much sense to use the argument less form the final decorated object would just be an instance of the decorating class returned by the constructor call which is not very useful Therefore it s eno
92. number of features in each sample 20 2 Classification 355 Scipy lecture notes Edition 2015 2 Sepal width Sepal length The k nearest neighbors classifier internally uses an algorithm based on ball trees to represent the samples it is trained on KNN k nearest neighbors classification example gt gt gt Create and fit a nearest neighbor classifier gt gt gt from sklearn import neighbors gt gt gt knn neighbors KNeighborsClassifier gt gt gt knn fit iris data iris target KNeighborsClassifier gt gt gt kun predict 0 1 0 2 0 3 0 4 array 0 Training set and testing set When experimenting with learning algorithms it is important not to test the prediction of an estima tor on the data used to fit the estimator Indeed with the kNN estimator we would always get perfect prediction on the training set gt gt gt perm np random permutation iris target size gt gt gt iris data iris datal perm gt gt gt iris target iris target perm gt gt gt knn fit iris data 100 iris target 100 KNeighborsClassifier gt gt gt knn score iris data 100 iris target 100 07 95999 Bonus question why did we use a random permutation 20 2 2 Support vector machines SVMs for classification Linear Support Vector Machines SVMs try to construct a hyperplane maximizing the margin between the two classes It selects a subset of the input called the suppo
93. of _data_matrix sparse matrix classes with data attribute offset for each diagonal Ois the main diagonal negative offset below positive offset above e fast matrix vector sparsetools fast and easy item wise operations manipulate data array directly fast NumPy machinery e constructor accepts dense matrix array sparse matrix 11 2 Storage Schemes 213 Scipy lecture notes Edition 2015 2 shape tuple create empty matrix data offsets tuple e no slicing no individual item access e use rather specialized solving PDEs by finite differences with an iterative solver Examples e create some DIA matrices gt gt gt data np array 1 2 3 4 repeat 3 axis 0 gt gt gt data array CUI 2 3 741 lets 2 Shh 4 1 2 a gt gt gt offsets np array LO 1 2 gt gt gt mtx sparse dia_matrix data offsets shape 4 4 gt gt gt mtx lt 4x4 sparse matrix of type lt numpy int64 gt with 9 stored elements 3 diagonals in DIAgonal format gt gt gt gt mtx todense Matrix 1 0 5 3 0J ts 2 eral OT 3 Ol 0 0 3 41 gt gt gt data np arange 12 reshape 3 4 1 gt gt gt data array l ae 25 83 A tS SNS m 2 I torai 20 gt gt gt mtx sparse dia_matrix data offsets shape 4 4 gt gt gt mtx data array L 1 2 3 Al IS G a eile oor TAa gt gt gt mtx offsets array 0 1 2 dt
94. of the point Arrays X Y and Z give the positions of these data points Excercise Visualize this field Your goal is to make sure that the simulation code is correct Suggestions e If you compute the norm of the vector field you can apply an isosurface to it e using mayavi mlab quiver3d you can plot vectors You can also use the masking options in the GUI to make the plot a bit less dense 19 3 2 Different views on data sources and modules As we see above it may be desirable to look at the same data in different ways Mayavi visualization are created by loading the datain a data source and then displayed on the screen using modules 19 3 Slicing and dicing data sources modules and filters 347 Scipy lecture notes Edition 2015 2 This can be seen by looking at the pipeline view By right clicking on the nodes of the pipeline you can add new modules Quiz Why is it not possible to add a VectorCutPlane to the vectors created by mayavi mlab quiver3d Different sources scatters and fields Data comes in different descriptions e A 3D block of regularly spaced value is structured it is easy to know how one measurement is related to another neighboring and how to continuously interpolate between these We can call such data a field borrowing from terminology used in physics as it is continuously defined in space e A set of data points measured at random positions in a random order gives rise to mu
95. or SfePy 5 9 Signal processing scipy signal gt gt gt from scipy import signal e scipy signal detrend remove linear trend from signal gt gt gt t np linspace 0 5 100 gt gt gt x t np random normal size 100 gt gt gt plt plot t x linewidth 3 lt matplotlib lines Line2D object at gt gt gt gt plt plot t signal detrend x linewidth 3 lt matplotlib lines Line2D object at gt 5 9 Signal processing scipy signal 118 Scipy lecture notes Edition 2015 2 e scipy signal resample resample a signal to n points using FFT gt gt gt t np linspace 0 5 100 gt gt gt x np sin t gt gt gt plt plot t x linewidth 3 lt matplotlib lines Line2D object at gt gt gt gt plt plot t 2 signal resample x 50 ko lt matplotlib lines Line2D object at gt l 5 1 0 0 5 0 0 0 5 1 0 a Notice how on the side of the window the resampling is less accurate and has a rippling effect e scipy signal has many window functions scipy signal hamming scipy signal bartlettQ scipy signal blackman e scipy signal has filtering median filter scipy signal medfilt Wiener scipy signal wiener but we will discuss this in the image section 5 9 Signal processing scipy signal 119 Scipy lecture notes Edition 2015 2 5 10 Image processing scipy ndimage The submodule dedicated to image processing in scipy is scipy ndimage
96. or a different method to inject an exception into the generator The first of the new methods is send value which is similar to next but passes value into the generator to be used for the value of the yield expression In fact g next and g send None are equivalent The second of the new methods is throw type value None traceback None which is equivalent to 7 1 Iterators generator expressions and generators 145 Scipy lecture notes Edition 2015 2 raise type value traceback at the point of the yield statement Unlike raise which immediately raises an exception from the current execution point throw first resumes the generator and only then raises the exception The word throw was picked because it is suggestive of putting the exception in another location and is associated with exceptions in other languages What happens when an exception is raised inside the generator It can be either raised explicitly or when executing some statements or it can be injected at the point of a yield statement by means of the throw method In either case such an exception propagates in the standard manner it can be intercepted by an except or finally clause or otherwise it causes the execution of the generator function to be aborted and propagates in the caller For completeness sake it s worth mentioning that generator iterators also have aclose method which can be used to force a generator that would otherwise be ab
97. permit to compute the position and some characteristics of the hit target When the footprint of the laser beam is around 1m on the Earth surface the beam can hit multiple targets during the two way propagation for example the ground and the top of a tree or building The sum of the contributions of each target hit by the laser beam then produces a complex signal with multiple peaks each one containing information about one target One state of the art method to extract information from these data is to decompose them in a sum of Gaussian functions where each function represents the contribution of a target hit by the laser beam Therefore we use the scipy optimize module to fit a waveform to one or a sum of Gaussian functions Loading and visualization Load the first waveform using gt gt gt import numpy as np gt gt gt waveform_1 np load data waveform_1 npy and visualize it gt gt gt import matplotlib pyplot as plt gt gt gt t np arange len waveform_1 gt gt gt plt plot t waveform_1 lt matplotlib lines Line2D object at gt gt gt gt plt show Intensity bins 0 10 20 30 AQ 50 60 70 80 Time ns As you can notice this waveform is a 80 bin length signal with a single peak The data used for this tutorial are part of the demonstration data available for the FullAnalyze software and were kindly provided by the GIS DRAIX 5 11 Summary exercises on scientific computing 129 Sci
98. pylab pcolor VL i reshape N N pylab axis equal pylab axis off pylab show examples pyamg_with_lobpcg py example by Nils Wagner examples lobpcg_sakurai py e output python examples lobpcg_sakurai py Results by LOBPCG for n 2500 0 06250083 0 06250028 0 06250007 Exact eigenvalues 0 06250005 0 0625002 0 06250044 Elapsed time 7 01 Eigenvalue distribution 1012 1912 102 10 10 10 10 lt 10 10 10 10 10 10 10 6 66666 107 10 103 10 107 10 Number z 11 3 Linear System Solvers 228 Scipy lecture notes Edition 2015 2 11 4 Other Interesting Packages e PyAMG algebraic multigrid solvers https github com pyamg pyamg e Pysparse own sparse matrix classes matrix and eigenvalue problem solvers http pysparse sourceforge net 11 4 Other Interesting Packages 229 CHAPTER 12 Image manipulation and processing using Numpy and Scipy Authors Emmanuelle Gouillart Ga l Varoquaux This section addresses basic image manipulation and processing using the core scientific modules NumPy and SciPy Some of the operations covered by this tutorial may be useful for other kinds of multidimensional array processing than image processing In particular the submodule scipy ndimage provides functions operating on n dimensional NumPy arrays See also For more advanced image processing and image specific routines see the tutori
99. releases_ts np array 4 5 3 5 3 5 5 3 2 1 3 3 view ReservoirEvolution reservoir reservoir inflows inflows_ts releases releases_ts view configure_traits Edit properties Plot Editor 45 27 97 27 Fi 4 10 4 D 2 4 B E 10 month A See also References e ETS repositories e Traits manual e Traits UI manual e Mailing list enthought dev enthought com 18 3 What are Traits 339 CHAPTER 19 3D plotting with Mayavi Author Ga l Varoquaux Mayavi is an interactive 3D plotting package matplotlib can also do simple 3D plotting but Mayavi relies on a more powerful engine VTK and is more suited to displaying large or complex data Chapters contents e Mlab the scripting interface 3D plotting functions Points Lines Elevation surface Arbitrary regular mesh Volumetric data Figures and decorations x Figure management Changing plot properties Decorations e Interactive work The pipeline dialog The script recording button e Slicing and dicing data sources modules and filters An example inspecting magnetic fields Different views on data sources and modules Different sources scatters and fields Transforming data filters x mlab pipeline the scripting layer e Animating the data e Making interactive dialogs A simple dialog Making it interactive e Putting it together 19 1 Mlab the script
100. samples position value array Or West TEOT 02d Gl On 51S Glo O37 4 Woe 0 eo Oele dtype position lt f8 value lt f8 Fancy indexing works as usual gt gt gt samples samples sensor_code ALFA array CLC ALFA 1 5 0 37 ALFA 3 0 0 11 dtype sensor_code S4 position lt f8 value lt f8 There are a bunch of other syntaxes for constructing structured arrays see here and here 3 3 3 maskedarray dealing with propagation of missing data e For floats one could use NaN s but masks work for all types gt gt gt x np ma array 1 2 3 4 mask 0 1 O 1 gt gt gt xX masked_array data 1 3 mask False True False True fill _value 999999 gt gt gt y np ma array 1 2 3 4 mask 0 1 1 1 gt gt gt x y masked_array data 2 mask False True True True fill_value 999999 e Masking versions of common functions gt gt Np ma sqrt it 1 2 21 masked_array data 1 0 1 41421356237 mask False True False True fill value 1e 20 There are other useful array siblings While it is off topic in a chapter on numpy let s take a moment to recall good coding practice which really do pay off in the long run 3 3 More elaborate arrays 71 Scipy lecture notes Edition 2015 2 Good practices e Explicit variable names no need of a comment to explain what is in the variable
101. scientific computing In Python x y Ipython x y executes the following imports at startup gt gt gt import numpy gt gt gt import numpy as np gt gt gt from pylab import gt gt gt import scipy and it is not necessary to re import these modules 2 5 3 Creating modules If we want to write larger and better organized programs compared to simple scripts where some objects are defined variables functions classes and that we want to reuse several times we have to create our own modules Let us create a module demo contained in the file demo py A demo module def print_b VPranGs Dii print b def print_a Q Prints a print a c 2 d 2 In this file we defined two functions print_a and print_b Suppose we want to call the print_a function from the interpreter We could execute the file as a script but since we just want to have access to the function print_a we are rather going to import it as a module The syntax is as follows In 1 import demo In 2 demo print_a a In 3 demo print_b b 2 5 Reusing code scripts and modules 29 Scipy lecture notes Edition 2015 2 Importing the module gives access to its objects using the module object syntax Don t forget to put the module s name before the object s name otherwise Python won t recognize the instruction Introspection In 4 demo Type module Base Class lt type module gt o
102. shared W1l soname libcos_doubles so o libcos_doubles so cos_doubles o ls cos_doubles c cos_doubles o libcos_doubles so test_cos_doubles py cos_doubles h cos_doubles py makefile Now we can proceed to wrap this library via ctypes with direct support for certain kinds of Numpy arrays nn Egample of wrapping a C Library function that accepts a C double array as nnnm input using the numpy ctypeslib import numpy as np import numpy ctypeslib as npct from ctypes import c_int input type for the cos_doubles function must be a double array with single dimension that is contiguous array_1d_double npct ndpointer dtype np double ndim 1 flags CONTIGUOUS load the library using numpy mechanisms libcd npct load_library libcos_doubles setup the return types and argument types libcd cos_doubles restype None libcd cos_doubles argtypes Larray_1d_double array_1d_double c_int def cos_doubles_func in_array out_array return libcd cos_doubles in_array out_array len in_array e Note the inherent limitation of contiguous single dimensional Numpy arrays since the C functions re quires this kind of buffer e Also note that the output array must be preallocated for example with numpy zeros and the function will write into it s buffer e Although the original signature of the cos_doubles function is ARRAY ARRAY int the final cos_doubles_func takes only two Numpy arrays as argum
103. site packages grin 1 1 py2 5 egg Users cburns local lib python2 5 site packages argparse 0 8 0 py2 5 egg Users cburns local lib python2 5 site packages urwid 0 9 7 1 py2 5 egg Users cburns local lib python2 5 site packages yolk 0 4 1 py2 5 egg Users cburns local lib python2 5 site packages virtualenv 1 2 py2 5 egg 2 5 pickle easy persistence Useful to store arbitrary objects to a file Not safe or fast In 1 import pickle In 2 1 1 None Stan In 3 pickle dump 1 file test pkl w In 4 pickle load file test pkl1 Gut 4 1 None Stan Exercise Write a program to search your PYTHONPATH for the module site py path_site 2 8 Exception handling in Python It is likely that you have raised Exceptions if you have typed all the previous commands of the tutorial For example you may have raised an exception if you entered a command with a typo Exceptions are raised by different kinds of errors arising when executing Python code In your own code you may also catch errors or define custom error types You may want to look at the descriptions of the the built in Exceptions when looking for the right exception type 2 8 Exception handling in Python 39 Scipy lecture notes Edition 2015 2 2 8 1 Exceptions Exceptions are raised by errors in Python TypeError unsupported operand type s for int To 3 d 1 1 2 2 In 4 d 3 KeyError 3 Ta 5 ILS 22 3
104. slices but also with boolean or integer arrays masks This method is called fancy indexing It creates copies not views Using boolean masks gt gt gt np random seed 3 gt gt gt a np random random_integers 0 20 15 gt gt gt a array tO Os S Ole Onn Onecd disses O scar O gn 20nu 1 7 nan gt gt gt a 4 3 0 array False True False True False False False True False True True False True False False dtype bool 3 1 The Numpy array object 53 Scipy lecture notes Edition 2015 2 gt gt gt mask a 3 0 gt gt gt extract_from_a al mask or a aZ3 0 gt gt gt extract_from_a extract a sub array with the mask array 3 0 9 6 20 121 Indexing with a mask can be very useful to assign a new value to a sub array gt gt gt ala 3 0 1 gt gt gt a array O i 2 ly 19ed Tone A 205 l Fy a Indexing with an array of integers gt gt gt a np arange 0 100 10 gt gt gt a array CLO 10 20 30 40 50 60 70 80 901 Indexing can be done with an array of integers where the same index is repeated several time eo all2 3s 2 Ar Qi Znote Wey te se ts a Python list array 20 30 20 40 20 New values can be assigned with this kind of indexing gt gt gt a 9 7 100 gt gt gt a array 0 10 20 305 40 50 60 100 80 100 When a new array is created by indexing with an array of integers the new ar
105. something gt gt gt np lookfor create array Search results for create array numpy array Create an array numpy memmap Create a memory map to an array stored in a binary file on disk 6 np con concatenate conj conjugate convolve Import conventions The recommended convention to import numpy is gt gt gt import numpy as np 3 1 The Numpy array object 44 Scipy lecture notes Edition 2015 2 3 1 2 Creating arrays Manual construction of arrays e 1 D gt gt gt a np array 0 1 2 3 gt gt gt a array Oss 2 oll gt gt gt a ndim 1 gt gt gt a shape 4 gt gt gt len a 4 e 2 D 3 D gt gt gt b np array LLO 1 2 3 4 5 2 x 3 array gt gt gt b array CLO 1 21 3 4 511 gt gt gt b ndim 2 gt gt gt b shape Qi 3 gt gt gt len b returns the size of the first dimension 2 gt gt gt c np array I t 2 3 MII gt gt gt c array 1 Pape ESI 4 gt gt gt c shape 2 591 Exercise Simple arrays e Create a simple two dimensional array First redo the examples from above And then create your own how about odd numbers counting backwards on the first row and even numbers on the second e Use the functions len numpy shape on these arrays How do they relate to each other And to the ndin attribute of the arrays Functions for creating arrays In practice we rarely en
106. sparse csc_matrix 3 4 dtype np int8 gt gt gt mtx todense Matrix LO 0 0 0 LOR O07 KO 0 0 0 OJ dtype ints e create using data ij tuple gt gt gt row np array 0 O 1 2 2 2 gt gt gt col np array 0 2 2 0 1 2 gt gt gt data np array 1 2 3 4 5 6 gt gt gt mtx sparse csc_matrix data row col shape 3 3 gt gt gt mtx lt 3x3 sparse matrix of type lt numpy int64 gt with 6 stored elements in Compressed Sparse Column format gt gt gt gt mtx todense matrix 1 0 21 O Or 45s Gaa gt gt gt mtx data array i4 552 Se Clic gt gt gt mtx indices array 0 2 2 0 1 2 dtype int32 gt gt gt mtx indptr array 0 2 3 6 dtype int32 e create using data indices indptr tuple gt gt gt data np array 1 4 5 2 3 6 gt gt gt indices np array 0 2 2 0 1 2 gt gt gt indptr np array 0 2 3 6 gt gt gt mtx sparse csc_matrix data indices indptr shape 3 3 gt gt gt mtx todense matrix 1 0 2 O Oa 45 75 6 Block Compressed Row Format BSR e basically a CSR with dense sub matrices of fixed shape instead of scalar items block size R C must evenly divide the shape of the matrix M N three NumPy arrays indices indptr data indices is array of column indices for each block x data is array of corresponding nonzero values of shape nnz R C subclass of _cs_m
107. subplot 1 1 1 X np linspace np pi np pi 256 endpoint True C S np cos X np sin X Plot cosine with a blue continuous line of width 1 pizels plt o plot X 3 C color blue 3 linewidth 1 0 linestyle Plot sine with a green continuous line of width 1 pizels plt plot X S color green linewidth 1 0 linestyle Set T limits plt xlim 4 0 4 0 Set x ticks plt xticks np linspace 4 4 9 endpoint True Set y limits plt ylim 1 0 1 0 4 2 Simple plot 84 Set y ticks plt yticks np linspace 1 1 5 endpoint True Save figure using 72 dots per inch plt savefig exercice_2 png dpi 72 Show result on screen plt show 4 2 3 Changing colors and line widths Documentation 1 0 Scipy lecture notes Edition 2015 2 e Controlling line properties e Line API First step we want to have the cosine in blue and the sine in red and a slighty thicker line for both of them We ll also slightly alter the figure size to make it more horizontal plt figure figsize 10 6 dpi 80 plt plot X C i color blue F linewidth 2 5 linestyle plt plot X S color red linewidth 2 5 linestyle 4 2 4 Setting limits ma al oo Documentation e xlim command e ylim command Current limits of the figure are a bit too tight and we want to make some space in order to clearly see all data points plt xlim X min 1 1
108. term This is almost always a good default choice however it is possible to specify different encodings for categorical variables http statsmodels sourceforge net devel contrasts html 15 3 Linear models multiple factors and analysis of variance 294 Scipy lecture notes Edition 2015 2 Link to t tests between different FSIQ and PIQ To compare different type of IQ we need to create a long form table listing IQs where the type of IQ is indicated by a categorical variable gt gt gt data_fisq pandas DataFrame iq data FSIQ type fsiq gt gt gt data_piq pandas DataFrame iq data PIQ type piq gt gt gt data_long pandas concat data_fisg data_pigq gt gt gt print data_long iq type O 133 fsig 1 140 fsiq 2 19mg ol 137 pig 32 110 pig 33 86 piq gt gt gt model ols iq type data_long fit gt gt gt print model summary OLS Regression Results coef std err t P gt t 95 0 Conf Int Intercept 113 4500 3 683 30 807 0 000 106 119 120 731 type T piq 2 4250 5 208 0 466 0 643 12 793 7 943 We can see that we retrieve the same values for t test and corresponding p values for the effect of the type ofiq than the previous t test gt gt gt stats ttest_ind data FSIQ data PIQ 0 46563759638 0 64277 250 15 3 2 Multiple Regression including multiple factors Consider a linear model explaining a variable z the depend
109. the Scipy lectures to learn the ecosystem you can directly skip to the next chapter NumPy creating and manipulating numerical data The remainder of this chapter is not necessary to follow the rest of the intro part But be sure to come back and finish this chapter later 2 6 Input and Output To be exhaustive here are some information about input and output in Python Since we will use the Numpy methods to read and write files you may skip this chapter at first reading We write or read strings to from files other types must be converted to strings To write in a file gt gt gt f open workfile w opens the workfile file gt gt gt type f lt type file gt gt gt gt f write This is a test nand another test gt gt gt f close To read from a file In 1 f open workfile r In 2 s f read In 3 print s This is a test and another test In 4 f close See also For more details https docs python org tutorial inputoutput html 2 6 Input and Output 34 Scipy lecture notes Edition 2015 2 2 6 1 Iterating over a file In 6 f open workfile r In 7 for line in f print line This is a test and another test In 8 f close File modes e Read only r e Write only w Note Create a new file or overwrite existing file e Append a file a e Read and Write r e Binary mode b Note Use for binary files especially on Windows
110. the mathematical or scientific understanding of the code Efficient code that executes quickly but needless to say that a very fast code becomes useless if we spend too much time writing it So we need both a quick development time and a quick execution time A single environment language for everything if possible to avoid learning a new software for each new problem 1 1 3 Existing solutions Which solutions do scientists use to work Compiled languages C C Fortran etc Advantages Very fast Very optimized compilers For heavy computations it s difficult to outperform these languages Scipy lecture notes Edition 2015 2 Some very optimized scientific libraries have been written for these languages Example BLAS vector matrix operations e Drawbacks Painful usage no interactivity during development mandatory compilation steps verbose syntax amp etc manual memory management tricky in C These are difficult languages for non computer scientists Scripting languages Matlab e Advantages Very rich collection of libraries with numerous algorithms for many different domains Fast exe cution because these libraries are often written in a compiled language Pleasant development environment comprehensive and well organized help integrated editor etc Commercial support is available e Drawbacks Base language is quite poor and can become restrictive for
111. the script is called Instructions may be e g copied and pasted from the interpreter but take care to respect indentation rules The extension for Python files is py Write or copy and paste the following lines in a file called test py message Hello how are you for word in message split print word Let us now execute the script interactively that is inside the Ipython interpreter This is maybe the most com mon use of scripts in scientific computing in Ipython the syntax to execute a script is run script py For example in 1 run test py Hello how are 2 5 Reusing code scripts and modules 27 Scipy lecture notes Edition 2015 2 you In 2 message Out 2 Hello how are you The script has been executed Moreover the variables defined in the script such as message are now available inside the interpreter s namespace Other interpreters also offer the possibility to execute scripts e g execfile in the plain Python interpreter etc It is also possible In order to execute this script as a standalone program by executing the script inside a shell terminal Linux Mac console or cmd Windows console For example if we are in the same directory as the test py file we can execute this in a console python test py Hello how are you Standalone scripts may also take command line arguments In file py import sys print sys argv python file py test arguments file py t
112. typing magic Furthermore Python ships with various aliases which emulate common UNIX command line tools such as 1s to list files cp to copy files and rm to remove files A list of aliases is shown when typing alias In 1 alias Total number of aliases 16 Gite En Creat eat clear clear mep sn CDi Es Idir ls F o color 41l l grep CVecev Miess i Cir le F o color lil erep Iik hs F o color l grep 1 11 ls F o color C ls ls F color Clg do Eo color 1il rep nx man man mkdir mkdir more more mv my i rm rm i rmdir rmdir Lastly we would like to mention the tab completion feature whose description we cite directly from the Python manual Tab completion especially for attributes is a convenient way to explore the structure of any object youre dealing with Simply type object_name lt TAB gt to view the object s attributes Besides Python objects and keywords tab completion also works on file and directory names In 1 x 10 In 2 x lt TAB gt x bit_length x conjugate x denominator x imag x numerator X real in 3 x real x real bit_length x real denominator x real numerator x real conjugate x real imag x real real In 4 x real 1 3 The interactive workflow IPython and a text editor 9 CHAPTER 2 The Python language Authors Chris Burns Christophe Combelles Emmanuel
113. use the following expression to create the array gt gt gt np arange 6 np arange 0 51 10 np newaxis array ILO js 2 aa A 5 10 e 14 15l 20 s 23 24 2514 130 cy oe Sell 40 43 44 45 50 oo B4 ool 3 1 The Numpy array object 51 Exercise Array creation Create the following arrays with correct data types lB Ste Sted lies legs gs oes 1 ey Bil Ste Dili i 6 1 AJ MO O Ora O PDA HO Oa Ooa 0T OSs Ol O sO se Oe Os 4 One Wali Oe Oe SOS a Onl Or tO ne Ones One 26 sil Par on course 3 statements for each Scipy lecture notes Edition 2015 2 Hint Individual array elements can be accessed similarly to a list e g aL 1 ora 1 2 Hint Examine the docstring for diag Exercise Tiling for array creation Skim through the documentation for np tile and use this function to construct the array i434 3 4 0 31 De Oe esos alle 4 73 e E 25 1 a 3 1 6 Copies and views A slicing operation creates a view on the original array which is just a way of accessing array data Thus the original array is not copied in memory You can use np may_share_memory to check if two arrays share the same memory block Note however that this uses heuristics and may give you false positives When modifying the view the original array is modified as well gt gt gt a np arange 10 gt gt gt a array O dhs 22 25 di 55 Ou gt a g gt gt gt b al 2 gt gt gt
114. which doesn t require this e property is the pythonic answer to the problem of getters and setters A method decorated with property becomes a getter which is automatically called on attribute access gt gt gt class A object property def a self an important attribute return a value gt gt gt A a lt property object at 0x gt gt gt gt AQ a a value In this example A a is an read only attribute It is also documented help A includes the docstring for attribute a taken from the getter method Defining a as a property allows it to be a calculated on the fly and has the side effect of making it read only because no setter is defined To have a setter and a getter two methods are required obviously Since Python 2 6 the following syntax is preferred class Rectangle object def _init__ self edge self edge edge property def area self Computed area setting this updates the edge length to the proper value nnnm return self edge 2 area setter def area self area self edge area 0 5 The way that this works is that the property decorator replaces the getter method with a property object This object in turn has three methods getter setter and deleter which can be used as decorators Their job is to set the getter setter and deleter of the property object stored as attributes fget fset and fdel The getter can be set like in the example above when creating th
115. 0 efficiency 0 8 state ReservoirState reservoir projectA storage 25 state release 4 state inflows state print_state state configure_traits F a am gm Edit properties Name Project A Storage 21 0 Spillage 0 Inflows 0 0 Release 0 0 r 5 0 4 0 Some use cases need the delegation mechanism to be broken by the user when setting the value of the trait The PrototypeFrom trait implements this behaviour from traits api import HasTraits Str Float Range PrototypedFrom Instance class Turbine HasTraits turbine_type Str power Float 1 0 desc Maximal power delivered by the turbine Mw class Reservoir HasTraits name Str max_storage Float 1e6 desc Maximal storage hm3 18 3 What are Traits 332 Scipy lecture notes Edition 2015 2 max_release Float 10 desc Maximal release m3 s head Float 10 desc Hydraulic head m efficiency Range 0O 1 turbine Instance Turbine installed_capacity PrototypedFrom turbine power if name _ main _ turbine Turbine turbine_type typel power 5 0 reservoir Reservoir name Project A max_storage 30 max release 100 0 head 60 efficiency 0 8 turbine turbine print installed capacity is initialised with turbine power print reservoir installed_capacity print 2 15 print updating the turbine power updates the installed capacity turbine power 10 print reservoir installed_
116. 02 x00 x00 x00 x03 x00 x00 x00 Memory address of the data gt gt gt x __array_interface__ data 0 64803824 The whole __array_interface__ gt gt gt X __array_interface__ data 35828928 False descr es LCG lt in shape 4 strides None typestr lt i4 yersion 3 Reminder two ndarrays may share the same memory gt gt gt x np array 1 2 3 4 gt gt gt y x1 8 1 Life of ndarray 161 Scipy lecture notes Edition 2015 2 gt gt gt x 0 9 gt gt gt y array 9 2 3l Memory does not need to be owned by an ndarray gt gt gt x b 1234 The b ts for bytes necessary in Python 3 x is a string in Python 3 a bytes we can represent its data as an array of ints gt gt gt y np frombuffer x dtype np int8 gt gt gt y data Se Ole wae gt gt gt y base is x True gt gt gt y flags C_CONTIGUOUS True F CONTIGUOUS True OWNDATA False WRITEABLE False ALIGNED True UPDATEIFCOPY False The owndata and writeable flags indicate status of the memory block 8 1 3 Data types The descriptor dtype describes a single item in the array type scalar type of the data one of int8 int16 float64 et al fixed size str unicode void flexible size itemsize size of the data block byte byte order big endian gt little endian lt not applicable order sub dtypes if it s a structured data type shape
117. 14 7 Further Reading and References i44 lt 4186e4i i6 5 64 6444 Gee eX SEHR Se SEH OC 280 14 8 EROIC S 6 ca Su eee RAE OEE EERE EERO eee ee 280 IHI Packages and applications 282 15 Statistics in Python 284 15 1 Data representation and interaction 2202265644445 Re eee RE ee ERE REE 285 15 2 Hypothesis testing comparing two groups 1 ee es 289 15 3 Linear models multiple factors and analysis of variance 2 000 eee eeee 292 15 4 More visualization seaborn for statistical exploration 0 eee eee eens 297 oo Tes np for Merac HONS e gt oo doy 2 6 eG is Hh GS Hw ee e ee 299 16 Sympy Symbolic Mathematics in Python 301 1G Pirst Steps wiih Sy yas ace Gee Kee eee eH eke He Ete ee ES eee we 302 16 2 Algebraic mamipulationSs iss s aretace a Sw SS eo ae EEE E 303 Do CU eener rrea eee ereere neen EEE 304 16 4 Equation solving aoaaa aaa e a 305 Foo Liner Apea 6 bs bee Bo ee ee AAE 306 17 Scikit image image processing 308 17 1 Introduction and Concepts cco ceeedeaede td adeeed wee dsRNA wend ead deeas 308 17 2 Input output datatypes and colorspaces 0 ee es 310 17 3 Image preprocessing enhancement 24 6 ecew ddada eka te aabhundddewegune ds 312 bye Mma e secmentaniOl ce awe eee ee hee eae a ed ee RARE SECEDE Eee eee 315 17 5 Measuring regions properties 2 4 4e4 082240445 ae Sandee bees ea td ewe seen 318 17 6 Data visualization and interaction 2 864622 464084 0 e4 Ee dA Dee OE
118. 16 4 1 Exercises 1 Solve the system of equations x y 2 2 x y 0 2 Are there boolean values x y that make x y amp y x true 16 5 Linear Algebra 16 5 1 Matrices Matrices are created as instances from the Matrix class gt gt gt from sympy import Matrix gt gt gt Matrix 1 0 0 1 i 0 L 0 1 unlike a NumPy array you can also put Symbols in it gt gt gt x Symbol x gt gt gt y Symbol y gt gt gt A Matrix 1 x y 1 16 5 Linear Algebra 306 Scipy lecture notes Edition 2015 2 gt gt gt A COR L y 1 gt gt gt Ax 2 x y 1 2 x L 2 y x y 1 16 5 2 Differential Equations SymPy is capable of solving some Ordinary Differential To solve differential equations use dsolve First create an undefined function by passing cls Function to the symbols function gt gt gt f g symbols f g cls Function f and g are now undefined functions We can call f x and it will represent an unknown function gt gt gt f x f x gt gt gt x diff x x f x 2 d Oe ae CeCe 2 dx gt gt gt dsolve f x diff x x f x f x f x Ci sin x C2 cos x Keyword arguments can be given to this function in order to help if find the best possible resolution system For example if you know that it is a separable equations you can use keyword hint separable to force dsolve to resolve it as a separable e
119. 1e6 dtype np float64 aor OI b np zeros 1e6 dtype np float32 In 3 timeit a a 1000 loops best of 3 1 78 ms per loop In 4 timeit b b 1000 loops best of 3 1 07 ms per loop e But bigger rounding errors sometimes in surprising places i e don t use them unless you really need them 3 3 2 Structured data types 4 character string gt gt gt samples np zeros 6 dtype sensor_code S4 en position float value float gt gt gt samples ndim 1 gt gt gt samples shape 65 gt gt gt samples dtype names sensor_code position value gt gt gt samples ALFA LO no CO BETA Aly Ori COTA i 0 13 Tr CUALP AY ico O 3r AEPA 3 0211 ATAU 2 0 13 gt gt gt samples 3 3 More elaborate arrays 70 Scipy lecture notes Edition 2015 2 array C ALFA 1 0 0 37 BETA 10 0 11 Ct AU 1 ALBA 255 037 ALBA 3 0 OT COTA 41 0 0 13 2 0 13 1 value lt f8 dtype sensor_code S4 position lt f8 Field access works by indexing with field names gt gt gt samples sensor_code array L ALFA BETA TAU ALFA ALFA TAU dtype S4 gt gt gt samples value array POS Otte Ontgo OnGr 0 10i1 07131 gt gt gt samples 0 CUA A 1 0 0 37 gt gt gt samples 0 sensor_code TAU gt gt gt samples 0 GTA 10 Oar Multiple fields at once gt gt gt
120. 2 Standard Library Reference document for this section e The Python Standard Library documentation https docs python org library index html e Python Essential Reference David Beazley Addison Wesley Professional 2 1 os module operating system functionality A portable way of using operating system dependent functionality Directory and file manipulation Current directory List a directory In 31 os listdir os curdir Ghote eel e andek rst SVO python_language rst swp view_array py swp static _templates basic_types rst conf py control_flow rst 2 7 Standard Library 35 Scipy lecture notes Edition 2015 2 debugging rst Make a directory In 32 os mkdir junkdir In 33 junkdir in os listdir os curdir Out 33 True Rename the directory In 36 os rename junkdir foodir In 37 junkdir in os listdir os curdir Out 37 False In 38 foodir in os listdir os curdir Out 38 True In 41 os rmdir foodir In 42 foodir in os listdir os curdir Out 42 False Delete a file In 44 fp open junk txt w In 45 fp close In 46 junk txt in os listdir os curdir Out 46 True In 47 os remove junk txt In 48 junk txt in os listdir os curdir Out 48 False os path path manipulations os path provides common operations on pathnames In 70 fp open junk txt
121. 2 x centerx 2 gt 230 2 circle then we assign the value 0 to the pixels of the image corresponding to the mask The syntax is extremely simple and intuitive gt gt gt face mask 0 gt gt gt plt imshow face lt matplotlib image AxesImage object at Ox gt Follow up copy all instructions of this exercise in a script called face_locket py then execute this script in Python with 4run face_locket py Change the circle to an ellipsoid 3 5 3 Data statistics The data in populations txt describes the populations of hares and lynxes and carrots in northern Canada during 20 years gt gt gt data np loadtxt data populations txt gt gt gt year hares lynxes carrots data T trick columns to variables gt gt gt import matplotlib pyplot as plt Pe plt axes 10 2701 10 55 075 lt matplotlib axes Axes object at gt gt gt gt plt plot year hares year lynxes year carrots lt matplotlib lines Line2D object at gt gt gt gt plt legend Hare Lynx Carrot loc 1 05 0 5 lt matplotlib legend Legend object at gt 3 5 Some exercises 78 Scipy lecture notes Edition 2015 2 80000 70000 60000 Hare Lynx 50000 l Carrot 40000 30000 20000 10000 O 1900 1905 1910 1915 1920 Computes and print based on the data in populations txt 1 The mean and std of the populations of each species for the years in the period 2 Which year ea
122. 2 4 173 0 000 0 159 0 441 gender T male 0 2750 0 093 2 972 0 003 0 093 0 457 education 0 0415 0 005 7 647 0 000 0 031 0 052 education gender T male 0 0134 0 007 1 919 0 056 70 027 702000 Can we conclude that education benefits males more than females Take home messages e Hypothesis testing and p value give you the significance of an effect difference e Formulas with categorical variables enable you to express rich links in your data e Visualizing your data and simple model fits matters e Conditionning adding factors that can explain all or part of the variation is important modeling aspect that changes the interpretation 15 5 Testing for interactions 300 CHAPTER 16 Sympy Symbolic Mathematics in Python Author Fabian Pedregosa Objectives 1 Evaluate expressions with arbitrary precision Perform algebraic manipulations on symbolic expressions Perform basic calculus tasks limits differentiation and integration with symbolic expressions Solve polynomial and transcendental equations Solve some differential equations om A W N What is SymPy SymPy is a Python library for symbolic mathematics It aims to be an alternative to systems such as Mathematica or Maple while keeping the code as simple as possible and easily extensible SymPy is written entirely in Python and does not require any external libraries Sympy documentation and packages for installation can be found on http www sympy org C
123. 20 01 5 10 4 Measurements on images Let us first generate a nice synthetic binary image gt gt gt x y np indices 100 100 gt gt gt sig np sin 2 np pi x 50 np sin 2 np pi y 50 1 x y 50 2 2 gt gt gt mask sig gt 1 Now we look for various information about the objects in the image gt gt gt labels nb ndimage label mask gt gt gt nb 8 gt gt gt areas ndimage sum mask labels range 1i labels max 1 5 10 Image processing scipy ndimage 123 Scipy lecture notes Edition 2015 2 gt gt gt areas array 190 45 424 278 459 190 549 424 gt gt gt maxima ndimage maximum sig labels range i labels max 1 gt gt gt maxima array 1 80238238 113527605 5 51954079 2 49611818 G7 1673619 iy 60236236 16 76547 217 5 51954079 gt gt gt ndimage find_objects labels 4 Cslice 30L 48L None slice 30L 48L None gt gt gt sl ndimage find_objects labels 4 gt gt gt import pylab as pl gt gt gt pl imshow sig s1 0 lt matplotlib image AxesImage object at gt sig mask labels See the summary exercise on Image processing application counting bubbles and unmolten grains for a more advanced example 5 11 Summary exercises on scientific computing The summary exercises use mainly Numpy Scipy and Matplotlib They provide some real life examples of sci entific computing with Python Now that the basics of wor
124. 6 double _it 3 Out 86 6 Keyword arguments allow you to specify default values Default values are evaluated when the function is defined not when it is called This can be problematic when using mutable types e g dictionary or list and modifying them in the function body since the modifications will be persistent across invocations of the function Using an immutable type in a keyword argument In 124 bigx 10 In 125 def double_it x bigx TETT return x 2 In 126 bigx 1e9 Now really big In 128 double_itQ Out 128 20 Using an mutable type in a keyword argument and modifying it inside the function body In 2 def add_to dict args a 1 b 2 for i in args keys args i 1 print args In 3 add to dict Out 3 lt function __main__ add_to_dict gt In 4 add to dict a 2 p i 3 In 5 add_to_dict SOW e ye idee a In 6 add_to_dict Cal 4A bp 5 2 4 Defining functions 23 Scipy lecture notes Edition 2015 2 More involved example implementing python s slicing In 98 def slicer seq start None stop None step None wes TImptlement basic python slicing TE return seq start stop step In 101 rhyme one fish two fish red fish blue fish split In 102 rhyme Out 102 lene fish twoly fish red fish bite fash In 103 slicer rhyme Cue ltos 2 lone fish two s fishy
125. 6 13 2 These operations are of course much faster than if you did them in pure python gt gt gt a np arange 10000 gt gt gt timeit a 1 10000 loops best of 3 24 3 us per loop gt gt gt 1 range 10000 gt gt gt timeit itl for i in 1 1000 loops best of 3 861 us per loop 3 2 Numerical operations on arrays 55 Scipy lecture notes Edition 2015 2 A Array multiplication is not matrix multiplication gt gt gt c np ones 3 3 gt gt gt c eG NOT matrix multiplication array l des dle AN Pela e als ave Aas Sly Matrix multiplication gt gt gt c dot c array Laa Ba Foil eS eee eee Desa Ses 8 Exercise Elementwise operations e Try simple arithmetic elementwise operations add even elements with odd elements e Time them against their pure python counterparts using timeit e Generate 2 0 2 1 2 x 2 2 3 2 4 a_j 2 3 j j Other operations Comparisons gt gt gt a np array 1 2 3 4 gt gt gt b np array 4 2 2 4 gt gt gt a array False True False True dtype bool1 gt gt gt a gt b array False False True False dtype bool1 Array wise comparisons gt gt gt a np array 1 2 3 4 gt gt gt b np array 4 2 2 4 gt gt gt c np array 1 2 3 4 gt gt gt np array_equal a b False gt gt gt np array_equal a c True Logical operations gt gt gt a
126. 7 lt mayavi modules surface Surface object at Oxde6f08c gt In 8 mlab mesh x y z extent 0 1 O 1 O 1 representation wireframe line_width 1 color 0 5 0 5 0 5 Out 8 lt mayavi modules surface Surface object at Oxdd6a71c gt Decorations Different items can be added to the figure to carry extra information such as a colorbar or a title In 9 mlab colorbar Out 7 orientation vertical Outl9 lt tvtk_classes scalar_bar_actor ScalarBarActor object at Oxd897f8c gt In 10 mlab title polar mesh Outl10 lt enthought mayavi modules text Text object at Oxd8ed38c gt In 11 mlab outline Out 7 Outlii lt enthought mayavi modules outline Outline object at Oxdd21b6c gt In 12 mlab axes Out 7 Out 12 lt enthought mayavi modules axes Axes object at Oxd2e4bcc gt 19 1 Mlab the scripting interface 345 Scipy lecture notes Edition 2015 2 1 60e 16 0 546 O 379 0 251 a 703 0 0442 0 192 A extent If we specified extents for a plotting object mlab outline and mlab axes don t get them by de fault 19 2 Interactive work The quickest way to create beautiful visualization with Mayavi is probably to interactively tweak the various settings 19 2 1 The pipeline dialog Click on the Mayavi button in the scene and you can control properties of objects with dialogs WEE JSG 284 G amp G OD Pipeline Scalar
127. 73 us per loop In 4 timeit a 2 1 1000 loops best of 3 154 us per loop In 5 timeit a a 100000 loops best of 3 5 56 us per loop Use this to guide your choice between strategies For long running calls using time instead of timeit it is less precise but faster 10 2 2 Profiler Useful when you have a large program to profile for example the following file For this example to run you also need the tca py file import numpy as np from scipy import linalg from ica import fastica def test data np random random 5000 100 u s v linalg svd data pea np dot ul 10 T data results fastica pca T whiten False if __name main __ test This is a combination of two unsupervised learning techniques principal component analysis PCA and in dependent component analysis ICA PCA is a technique for dimensionality reduction i e an algorithm to explain the observed variance in your data using less dimensions ICA is a source seperation technique for ex ample to unmix multiple signals that have been recorded through multiple sensors Doing a PCA first and then an ICA can be useful if you have more sensors than signals For more information see the FastlCA example from scikits learn To run it you also need to download the ica module In Python we can time the script In 1 run t demo py IPython CPU timings estimated 10 2 Profiling Python code 204 Scipy lectu
128. Date February 11 2010 SciPy pronounced Sigh Pie is open source software for mathematics science and engineering e SciPy Tutorial o Introduction f o Basic functions in Numpy and top level scipy Table Of Contents o Special functions Cay aeia j SciPy o Integration scipy integrate Reference o Optimization optimize Next topic Interpolation scipy interpolate o Signal Processing signal SciPy Tutorial o Linear Algebra This Page o Statistics o Multi dimensional image processing ndimage Suede o File IO scipy io Resources Scipy org website o Weave e Release Notes Edt rade Reference Quick search e Clustering package scipy cluster Go ate Enter s terms or a modul if fi ion name scipy opy e Miscellaneous routines scipy misc Find help Previous amp Next Highlight all Match case t Reached end of page continued from top e Numpy s and Scipy s documentations can be browsed online on_http docs scipy org doc The search button is quite useful inside the reference documentation of the two packages http docs scipy org doc numpy reference and http docs scipy org doc scipy reference Tutorials on various topics as well as the complete API with all docstrings are found on this website Sdpyidiacunentstionteditan Back to Numpy Scipy documentation Numpy documentation editor scipy org editor Wiki Docstrings Changes Milestones Search Stats Patch
129. Deprecation of functions Awhile loop removing decorator A plugin registration system e Context managers Catching exceptions Using generators to define context managers 7 1 iterators generator expressions and generators 7 1 1 Iterators 142 Scipy lecture notes Edition 2015 2 Simplicity Duplication of effort is wasteful and replacing the various home grown approaches with a standard fea ture usually ends up making things more readable and interoperable as well Guido van Rossum Adding Optional Static Typing to Python An iterator is an object adhering to the iterator protocol basically this means that it has a next method which when called returns the next item in the sequence and when there s nothing to return raises the StopIteration exception An iterator object allows to loop just once It holds the state position of a single iteration or from the other side each loop over a sequence requires a single iterator object This means that we can iterate over the same sequence more than once concurrently Separating the iteration logic from the sequence allows us to have more than one way of iteration Calling the __iter__ method on a container to create an iterator object is the most straightforward way to get hold of an iterator The iter function does that for us saving a few keystrokes gt gt gt nums 1 2 3 note that varies these are different objects gt gt gt iter nums
130. File home esc git working scipy lecture notes advanced interfacing with_c python_c_api cos_module so Deestring lt no docstring gt In 3 dir cos_module Out 3 __doc__ __file__ __name__ __package__ cos_func In 4 cos_module cos_func 1 0 Out 4 0 5403023058681398 In 5 cos_module cos_func 0 0 Ove ts 1 6 In 6 cos_module cos_func 3 14159265359 Out 7 1 0 Now let s see how robust this is In 10 cos_module cos_func foo TypeError Traceback most recent call last lt ipython input 10 11bee483665d gt in lt module gt gt 1 cos_module cos_func foo TypeError a float is required 14 2 2 Numpy Support Analog to the Python C API Numpy which is itself implemented as a C extension comes with the Numpy C API This API can be used to create and manipulate Numpy arrays from C when writing a custom C extension See also ref advanced_numpy _ If you do ever need to use the Numpy C API refer to the documentation about Arrays and Iterators The following example shows how to pass Numpy arrays as arguments to functions and how to iterate over Numpy arrays using the old Numpy C API It simply takes an array as argument applies the cosine function from the math h and returns a resulting new array Example of wrapping the cos function from math h using the Numpy C API 14 2 Python C Api 266 Scipy lecture notes Edition 2015 2 include lt Python h
131. H_INIT Includes the header in the wrapper code include cos_doubles h ht include the numpy typemaps include numpy i need this for correct module initialization Ainit import_array ht typemaps for the two arrays the second will be modified in place apply double IN_ARRAY1 int DIM1 double in_array int size_in apply double INPLACE_ARRAY1 int DIM1 double out_array int size_out Wrapper for cos_doubles that massages the types inline takes as input two numpy arrays void cos_doubles_func double in_array int size_in double out_array int size_out calls the original funcion providing only the size of the first cos_doubles in_array out_array size_in 14 4 SWIG 274 Scipy lecture notes Edition 2015 2 a To use the Numpy typemaps we need include the numpy i file Observe the call to import_array which we encountered already in the Numpy C API example Since the type maps only support the signature ARRAY SIZE we need to wrap the cos_doubles as cos_doubles_func which takes two arrays including sizes as input e As opposed to the simple SWIG example we don t include the cos_doubles h header There is nothing there that we wish to expose to Python since we expose the functionality through cos_doubles_func And as before we can use distutils to wrap this from distutils core import setup Extension import numpy setup ext_m
132. R RA ee eS 318 17 7 Feature extraction for computer vision 6 2 444 sien dueea cbdedeskwdwheeend dane 320 18 Traits building interactive dialogs 322 TOE TW OGQCGOW se gea bb eee eee ee Reha Pee REE Bee 323 fo E gic eee eee ee ee eee ae BA eee ee eee Behe eek ee Ee SESS 323 189 Whatare TRIIS se yop eaea hey ee chen ae ee eee need eee ee Cee eee 324 19 3D plotting with Mayavi 340 FO Mlab Wie scripting interlace s lt se becws so beh oda neve dee bee kaw ede ws a so 340 I2 CIAO sasssa w Aa ee Shea ee Gd eee eee eee See ee ee eee 346 19 3 Slicing and dicing data sources modules and filters 2 2 ee ee en 347 POs Sie Me a sg kk ee ee eee a ee eee eRe ee 6 ee 349 19 5 Making interactive dialogs 6464 5 866 h4562465 644 66 08s e deta SGHe Ge OS de 350 196 Une Ie OCCIIE 5 54 8 8 a Boo oe ee oe ee Bw eA EER ee OR SRS 351 20 scikit learn machine learning in Python 353 20 1 Loading an example dataset ooo 4 oe 4 HG eS SS os bo ee EEE amp A HE Bok Goes 354 202 MVS NCA lt 6 446 See ee ee eee eh et Be See ee aE eee SORES eae ee ee 355 20 3 Clustering grouping observations together 1 645 sia4s eaee aoe dew doe es maces 357 20 4 Dimension Reduction with Principal Component Analysis 00 0000s 359 20 5 Putting it all together face recognition 5 4 402s a eek Ca D ae ERP EEE ROYER REE ER RS 360 20 6 Linear model from regression to sparsity 1 0 ee ee ee 361 20 7 Model selection choosing estimators and their parameters
133. RE Gwe oe ee ee ae 155 8 Advanced Numpy 159 GI DICOlNdaTay serrera u ea Eee eee eee a Lee E e E e 160 Oe RVers cee ee ea Hee eRe RA e e a e aa a eee ee eee 173 8 3 Inter operapilty catures 2446 bays deed be P hee woe edad es ekeed detegetun 182 8 4 Array siblings chararray maskedarray matrix 4 2664 44 e UPA ekee eee Ewes 185 Oo DUMMY 2a oteteuee ee tbe eae eee oe ae vee eee eee se aoe oe ee 188 86 Contributing 16 Numpy Scipy 2 4 544 444 605668446 6 Se REa HR EH EE RER MEE GRO 188 9 Debugging code 192 Oak AVON US 2225486668 eRe oe eee eA eee Geew Chee eehee eR a sae ee 192 92 Debugging WOrKHOW gt 645 6 Geechee 4 ad SORGHE EE EER EERE MAES ESE REESE ES EDS 195 99 Using the Pyihon deDuggEr ros 445 4 ough ewe 4 eG ee oe ee eee ee ee eee 195 9 4 Debugging segmentation faults using gdb 1 1 o 200 10 Optimizing code 203 10 1 Ge OI ANGERTIOVE ice GA op aren oe oe ee oo SB Gs oe ee a 203 102 Pronling Py ihOMnCede rsss dane Bee ke Bee we Bees ee Oe eR eae ee eee ee 203 10 3 Making code goastet lt 4 o 4 44 006 OO HE OED SO Se HS OOS RES ERE 206 10 4 Writing taster numerical code 2466 64 lt 4 heo Fee htae debe eed ee decaddwseac 207 11 Sparse Matrices in SciPy 210 Pi TWOGiCnG oso ae we eee eee oe eee eee eee eee SSS 210 Ll2 SOE CEMS noc oo 44 eo oO SOG a ee oe eee eRe OK eee Se SS 212 11 3 Linear System Solvers ou 6844 54 844 oa whee ew ESAS EES ES EHE SE CSRS GASH 224 11 4 Other Interesting Packages 0 nae nga ek iae
134. S append plugin WordProcessor plugin class CleanMdashesExtension object def cleanup self text return text replace amp mdash u N em dash Here we use a decorator to decentralise the registration of plugins We call our decorator with a noun instead of a verb because we use it to declare that our class is a plugin for WordProcessor Method plugin simply appends the class to the list of plugins A word about the plugin itself it replaces HTML entity for em dash with a real Unicode em dash character It exploits the unicode literal notation to insert a character by using its name in the unicode database EM DASH If the Unicode character was inserted directly it would be impossible to distinguish it from an en dash in the source of a program See also More examples and reading e PEP 318 function and method decorator syntax PEP 3129 class decorator syntax http wiki python org moin PythonDecoratorLibrary https docs python org dev library functools html http pypi python org pypi decorator Bruce Eckel Decorators I Introduction to Python Decorators Python Decorators II Decorator Arguments Python Decorators III A Decorator Based Build System 7 3 Context managers A context manager is an object with __enter__ and __exit__ methods which can be used in the with state ment with manager as var do_something var is in the simplest case equivalent to var manager __enter__
135. SNIE IPython SCI Edited by DY Gael Varoquaux Emmanuelle Gouillart _e ctu e N otes Olea elalarcrs www Scipy lectures org oO Ga l Varoquaux Emmanuelle Gouillart e Olav Vahtras Valentin Haenel Nicolas P Rougier Ralf Gommers Fabian Pedregosa Zbigniew Jedrzejewski Szmek Pauli Virtanen Christophe Combelles Didrik Pinte Robert Cimrman Andr Espaze Adrian Chauve Christopher Burns creative commons Getting started with Python for science Scientific computing with tools and workflow LL Wy POM sirere 64ne h0 es oboe ee heme wee ce amp os 1 2 Scientific Python building blocks 2 4 24 64 88e e064a408 1 3 The interactive workflow Python and a text editor The Python language Polk PNGESIOOS 64 oe ee eee eee eke ee eo ee ee 22 DC PES to 6 kos eee ie o oe oe E ee Oo ee Eaa 2 COWUORTIOW mesie rpe 6 OS eee ROR E Eh He 24 Denning MIMCuOns 24 404 42 45 48 B64 460465 amaii 2 5 Reusing code scripts and modules 00005 260 Inputand OUI ssia 6s 4 SHOR EOE SHEA EOS ERO GEES 2 7 Standard Lary gt ee a RR RS ee ee we ES ee a G 2 8 Exception handling in Python 2244466244 2842408 lt e aes 2 9 Object oriented programming OOP 0 NumPy creating and manipulating numerical data 3 1 The Numpy array object 2c eta caee shaded d ch baw dx 3 2 Numerical operations on arrays 3 4 44 lt 6 64h440 80dce bbe 3 3 More elaborate arrays ee oe e4ne oe bed
136. Sennen eheearaee ete ewae ade e dean 229 12 Image manipulation and processing using Numpy and Scipy 230 12 1 Opening and writing to image files asuaan e 231 122 Displaying images a a ee arra eek Coe aieeaa HEE ESSE BEER 292 129 Basic manipa ODS s 2s aan oe eee Oe Oe a ee a ee e a a 233 LA mMare INE seppe ee ew doe ed oe eww 6 S a ee He eee we EE EY S A 235 120 Feamre estacion esaerea ee Ges oa aeea e E 240 12 6 Measuring objects properties ndimage measurements assas 2 0 e eee ee enee 243 13 Mathematical optimization finding minima of functions 248 13 1 Knowing your proDlenn s e s a s recrea do e o a eva O E ae a E a e R a a aaa a a 249 13 2 A review of the different optimizers 2 64 oS Ge Se Sw ee Ew ow a ed a 251 13 3 Practical guide to optimization with scipy lt 444 5 46 5446446 5 646 ooo ow OS ew ls 258 13 4 Special case non linear least squares 254262 5Aa 08 Kh 6s a bb hee dere we anges 260 13 5 Optimization with constraints 4 choo ee eee oe So Se oa SESS Oe RHE Ses SSS 261 14 Interfacing with C 263 HAT TO AUCMOW see eee he ae bos SORE Ae eee eee PSs BEES ate ae tess 263 I2 PDOn CAP so 6s pos oo eee Ee Re ROE eee eA EES 264 Po et aa ce ee he AS OLE eee eee SORE a eee eee eS ee Cee ee ee Oe 268 Id SG eee Ee G kee ee EE Re eek ee ee ee ea REAPS eRe eee 6 OES 212 iS Caen eee chee es eee eee eee eee eae eed neta seeadaahe ge E st eae 276 120 OOM 2a ee cede eee neeee de eens thee de bee SAS eeteee ee eek eee ee uH 279
137. Sle cli 1254 img save test2 png 8 3 Interoperability features 183 Scipy lecture notes Edition 2015 2 8 3 4 Array interface protocol e Multidimensional buffers e Data type information present e Numpy specific approach slowly deprecated but not going away e Not integrated in Python otherwise See also Documentation http docs scipy org doc numpy reference arrays interface html gt gt gt x np array 1 2 3 4 gt gt gt x __array_interface__ data 171694552 False memory address of data is readonly descr C s eid data type descriptor typestr lt i4 same in another form strides None strides or None if in C order shape 2 2 version 3 i 8 3 Interoperability features 184 Scipy lecture notes Edition 2015 2 gt gt gt from PIL import Image gt gt gt img Image open data test png gt gt gt img __array_interface__ Heder 2 asus shape 200 200 4 typestr u1 gt gt gt x np asarray img gt gt gt x shape 200 200 4 A more C friendly variant of the array interface is also defined 8 4 Array siblings chararray maskedarray matrix 8 4 1 chararray vectorized string operations gt gt gt x np array a bbb ccc view np chararray gt gt gt x lstrip 7 chararray a bbb Tece dtype gt gt gt x upper chararray A C BBB CCC dtype view has a seco
138. What they are Exercise building an ufunc from scratch Solution building an ufunc from scratch Generalized ufuncs e Interoperability features Sharing multidimensional typed data The old buffer protocol The old buffer protocol Array interface protocol e Array siblings chararray maskedarray matrix chararray vectorized string operations masked_array missing data recarray purely convenience matrix convenience e Summary e Contributing to Numpy Scipy Why Reporting bugs Contributing to documentation Contributing features How to help in general In this section numpy will be imported as follows gt gt gt import numpy as np 8 1 Life of ndarray 8 1 1 It s ndarray block of memory indexing scheme data type descriptor e raw data e how to locate an element e how to interpret an element 8 1 Life of ndarray 160 Scipy lecture notes Edition 2015 2 array scalar iie U ndarray typedef struct PyArrayObject PyObject _HEAD Block of memory char data Data type descriptor PyArray_Descr descr Indexing scheme int nd npy_intp dimensions npy_intp strides Other stuff PyObject base int flags PyObject weakreflist PyArrayObject 8 1 2 Block of memory gt gt gt x np array 1 2 3 dtype np int32 gt gt gt x data eee ata gt gt gt gt str x data x01 x00 x00 x00 x
139. Z0 for k in range iterations Zz Zz Zz c say 100 iterations or until z real 2 z imag 2 gt 1000 Use it to determine which c are in the Mandelbrot set e Our function is a simple one so make use of the PyUFunc_ helpers e Write it in Cython See also mandel pyx mandelplot py Fix the parts marked by TODO Compile this file by Cython gt 0 12 required because of the complex vars cython mandel pyx python setup py build_ext i and try it out with in this directory gt gt gt import mandel H gt gt gt mandel mandel 0 1 2j The elementwise function cdef void mandel_single_point double complex z_in double complex c_in double complex z_out nogil The Mandelbrot iteration Some points of note It s NOT allowed to call any Python functions here The Ufunc loop runs with the Python Global Interpreter Lock released Hence the nogil And so all local variables must be declared with cdef Note also that this function receives pointers to the data 8 2 Universal functions 175 Scipy lecture notes Edition 2015 2 cdef double complex z z_in 0 cdef double complex c c_in 0 cdef int k the integer we use in the for loop TODO write the Mandelbrot iteration for one point here as you would write it in Python say use 100 as the maximum number of iterations and 1000 as
140. a 3 gt gt gt A np ones 2 array l2 ae A Few Notes on Preconditioning e problem specific e often hard to develop e if not sure try ILU available in dsolve as spilu 11 3 3 Eigenvalue Problem Solvers The eigen module e arpack a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems e lobpcg Locally Optimal Block Preconditioned Conjugate Gradient Method works very well in combi nation with PyAMG example by Nathan Bell nnnm Compute eigenvectors and eigenvalues using a preconditioned eigensolver In this example Smoothed Aggregation SA is used to precondition the LOBPCG eztgensolver on a two dimensional Poisson problem with Dirichlet boundary conditions nnm import scipy from scipy sparse linalg import lobpcg from pyamg import smoothed_aggregation_solver from pyamg gallery import poisson N 100 K 9 A poisson N N format csr create the AMG hierarchy ml smoothed_aggregation_solver A initial approximation to the K etgenvectors X scipy rand A shape l0 K preconditioner based on ml M ml aspreconditioner compute eigenvalues and eigenvectors with LOBPCG W V lobpcg A X M M tol 1e 8 largest False plot the eigenvectors import pylab 11 3 Linear System Solvers 227 Scipy lecture notes Edition 2015 2 pylab figure figsize 9 9 for i in range K pylab subplot 3 3 i 1 pylab title Eigenvector 4d i
141. a continuous function Optimizing smooth functions is easier true in the context of black box optimization otherwise Linear Pro gramming is an example of methods which deal very efficiently with piece wise linear functions 13 1 Knowing your problem 250 Scipy lecture notes Edition 2015 2 13 1 3 Noisy versus exact cost functions Noisy blue and non noisy green functions Noisy gradients Many optimization methods rely on gradients of the objective function If the gradient function is not given they are computed numerically which induces errors In such situation even if the objective function is not noisy a gradient based optimization may be a noisy optimization 13 1 4 Constraints Optimizations under constraints Here l lt x lt 1 l lt x lt l 13 2 A review of the different optimizers 13 2 1 Getting started 1D optimization Use scipy optimize brent to minimize 1D functions It combines a bracketing strategy with a parabolic approximation 13 2 A review of the different optimizers 251 Scipy lecture notes Edition 2015 2 Brent s method on a quadratic function it converges in 3 iterations as the quadratic approximation is then exact 10 15 20 25 Iterati Brent s method on a non convex function note that the fact that the optimizer avoided the local minimum is a matter of luck gt gt gt from scipy import optimize gt gt gt def f x return np exp x
142. akes init letx temp file flymake init create temp buffer copy flymake create temp inplace local file file relative name temp file 9 1 Avoiding bugs 194 Scipy lecture notes Edition 2015 2 file name directory buffer file name list pyflakes list local file add to list flymake allowed file name masks C py flymake pyflakes init add hook find file hook flymake find file hook 9 2 Debugging workflow If you do have a non trivial bug this is when debugging strategies kickin There is no silver bullet Yet strategies help For debugging a given problem the favorable situation is when the problem is isolated in a small number of lines of code outside framework or application code with short modify run fail cycles Make it fail reliably Find a test case that makes the code fail every time Divide and Conquer Once you have a failing test case isolate the failing code 3 4 5 e Which module e Which function e Which line of code gt isolate a small reproducible failure a test case Change one thing at a time and re run the failing test case Use the debugger to understand what is going wrong Take notes and be patient It may take a while Once you have gone through this process isolated a tight piece of code reproducing the bug and fix the bug using this piece of code add the corresponding code to your test suite 9 3 Using the Python debugger The pyth
143. al Scikit image image pro cessing dedicated to the skimage module Image 2 D numerical array or 3 D CT MRI 2D time 4 D Here image Numpy array np array Tools used in this tutorial e numpy basic array manipulation e scipy scipy ndimage submodule dedicated to image processing n dimensional images See the documentation gt gt gt from scipy import ndimage Common tasks in image processing e Input Output displaying images e Basic manipulations cropping flipping rotating e Image filtering denoising sharpening e Image segmentation labeling pixels corresponding to different objects Classification e Feature extraction Registration 230 Scipy lecture notes Edition 2015 2 Chapters contents e Opening and writing to image files Displaying images e Basic manipulations Statistical information Geometrical transformations Image filtering Blurring smoothing Sharpening Denoising Mathematical morphology e Feature extraction Edge detection Segmentation e Measuring objects properties ndimage measurements 12 1 Opening and writing to image files Writing an array to a file from scipy import misc f misc face misc imsave face png f uses the Image module PIL import matplotlib pyplot as plt plt imshow f plt show Creating a numpy array from an image file gt gt gt from scipy import misc gt gt gt face misc face
144. am filtdat bins np arange 256 5 11 Summary exercises on scientific computing 133 Scipy lecture notes Edition 2015 2 140000 120000 100000 80000 60000 40000 20000 0 50 100 150 200 250 4 Using the histogram of the filtered image determine thresholds that allow to define masks for sand pix els glass pixels and bubble pixels Other option homework write a function that determines automat ically the thresholds from the minima of the histogram gt gt gt void filtdat lt 50 gt gt gt sand np logical_and filtdat gt 50 filtdat lt 114 gt gt gt glass filtdat gt 114 5 Display an image in which the three phases are colored with three different colors gt gt gt phases void astype np int 2 glass astype np int 3 sand astype np int 5 11 Summary exercises on scientific computing 134 Scipy lecture notes Edition 2015 2 6 Use mathematical morphology to clean the different phases gt gt gt sand_op ndimage binary_opening sand iterations 2 7 Attribute labels to all bubbles and sand grains and remove from the sand mask grains that are smaller than 10 pixels To do so use ndimage sum or np bincount to compute the grain sizes gt gt gt sand_labels sand_nb ndimage label sand_op gt gt gt sand_areas np array ndimage sum sand_op sand_labels np arange sand_labels max 1 gt gt gt mask sand_areas gt 100 gt gt gt remove_small_sand
145. an argument to another function In 38 va variable_args In 39 va three x 1 y 2 args is three kyargs is y 2 r 2 4 Defining functions 26 Scipy lecture notes Edition 2015 2 2 4 9 Methods Methods are functions attached to objects You ve seen these in our examples on lists dictionaries strings etc 2 4 10 Exercises Exercise Fibonacci sequence Write a function that displays the n first terms of the Fibonacci sequence defined by euO 1 uie tl e u_ n 2 u_ n 1 u_n Exercise Quicksort Implement the quicksort algorithm as defined by wikipedia function quicksort array var list less greater if length array lt 2 return array select and remove a pivot value pivot from array for each x in array if x lt pivot 1 then append x to less else append x to greater return concatenate quicksort less pivot quicksort greater 2 5 Reusing code scripts and modules For now we have typed all instructions in the interpreter For longer sets of instructions we need to change track and write the code in text files using a text editor that we will call either scripts or modules Use your fa vorite text editor provided it offers syntax highlighting for Python or the editor that comes with the Scientific Python Suite you may be using e g Scite with Python x y 2 5 1 Scripts Let us first write a script that is a file with a sequence of instructions that are executed each time
146. and modify this image according to how the shape locally fits or misses the image Structuring element 12 4 Image filtering 237 Scipy lecture notes Edition 2015 2 gt gt gt el ndimage generate_binary_structure 2 1 gt gt gt el array False True False True True Truel False True False dtype bo01 gt gt gt el astype np int array 0 1 O EE a OR Erosion minimum filter Replace the value of a pixel by the minimal value covered by the structuring ele ment gt gt gt a np zeros 7 7 dtype np int gt gt gt a 1 6 2 5 1 gt gt gt a array 0 Kew LO LO LO Lo 0 ol iL 0l the Ol ie 0l i Ol il 0l 0 0 gt gt gt ndimage binary_erosion a astype a dtype array 0 005205 01 0 Ol i 0l ih 0l LO i OF LO 0 0 oF Onis 0 4 Os 0 gt gt gt Erosion removes objects smaller than the structure gt gt gt ndimage binary_erosion a structure np ones 5 5 astype a dtype array I0 0 Os 0 0 0 0l LO 0 0 LO or O ol iO Ol or or LO 0 3 O LO 0 Tey 0 or 0 O 3 2 12 4 Image filtering 238 Scipy lecture notes Edition 2015 2 Erosion Dilation Opening Closing aL a Dilation maximum filter gt gt gt a np zeros 5 5 gt gt gt ald 2N 1 gt gt gt a array GUL One Oly On 10 5 0l 084 WORs Og Os O Oise Ora lag Oa Oal LOr Ole Oak Ong Oss L
147. arnings e All of these techniques may crash segmentation fault the Python interpreter which is usually due to bugs in the C code e All the examples have been done on Linux they should be possible on other operating systems e You will need a C compiler for most of the examples 14 2 Python C Api The Python C API is the backbone of the standard Python interpreter a k a CPython Using this API it is possible to write Python extension module in C and C Obviously these extension modules can by virtue of language compatibility call any function written in C or C When using the Python C API one usually writes much boilerplate code first to parse the arguments that were given to a function and later to construct the return type Advantages e Requires no additional libraries e Lots of low level control e Entirely usable from C Disadvantages e May require a substantial amount of effort e Much overhead in the code e Must be compiled e High maintenance cost e No forward compatibility across Python versions as C API changes e Reference count bugs are easy to create and very hard to track down The Python C Api example here serves mainly for didactic reasons Many of the other techniques actually depend on this so it is good to have a high level understanding of how it works In 99 of the use cases you will be better off using an alternative technique Since refernce counting bugs are easy to create and hard to track do
148. aroquau dev scipy lecture notes advanced optimizing index_error py 5 index_error gt print lst len 1st Pdb Step by step execution Situation You believe a bug exists in a module but are not sure where For instance we are trying to debug wiener_filtering py Indeed the code runs but the filtering does not work well 9 3 e Run the script in Python with the debugger using run d wiener_filtering p In i run d wiener_filtering py xxx Blank or comment xxx Blank or comment xxx Blank or comment Breakpoint 1 at home varoquau dev scipy lecture notes advanced optimizing wiener_filtering py 4 NOTE Enter c at the apdb gt prompt to start your script gt lt string gt 1 lt module gt Set a break point at line 34 using b 34 ipdb gt n gt home varoquau dev scipy lecture notes advanced optimizing wiener_filtering py 4 lt module gt 3 1 gt 4 import numpy as np 5 import Scipy as sp ipdb gt b 34 Breakpoint 2 at home varoquau dev scipy lecture notes advanced optimizing wiener_filtering py 34 Continue execution to next breakpoint with c ont inue ipdb gt c gt home varoquau dev scipy lecture notes advanced optimizing wiener_filtering py 34 iterated_wiener 33 unn Using the Python debugger 197 Scipy lecture notes Edition 2015 2 2 gt 34 noisy_img noisy_img 35 denoised_img local_mean noisy_img size size e Step into code with n ext and s tep next jumps to
149. arse matrix classes with data attribute e fast matrix vector products and other arithmetics sparsetools e constructor accepts dense matrix array sparse matrix shape tuple create empty matrix data ij tuple data indices indptr tuple e efficient row slicing row oriented operations e slow column slicing expensive changes to the sparsity structure e use actual computations most linear solvers support this format Examples e create empty CSR matrix gt gt gt mtx sparse csr_matrix 3 4 dtype np int8 gt gt gt mtx todense matrix 0 0 0 0 O O O O 0 0 0 OJ dtype int8 e create using data ij tuple gt gt gt row np array 0 0 1 2 2 2 gt gt gt col np array 0 2 2 0 1 2 gt gt gt data np array 1 2 3 4 5 6 gt gt gt mtx sparse csr_matrix data row col shape 3 3 gt gt gt mtx 11 2 Storage Schemes 219 Scipy lecture notes Edition 2015 2 lt 3x3 sparse matrix of type lt numpy int64 gt with 6 stored elements in Compressed Sparse Row format gt gt gt gt mtx todense matrix CUI 0 2215 KOR 0 cull C Oldies gt gt gt mtx data array L 2 Sy lt 4 55 Gla gt gt gt mtx indices array 0 2 2 0 1 2 dtype int32 gt gt gt mtx indptr array 0 2 3 6 dtype int32 e create using data indices indptr tuple gt gt gt data np array 1 2 3 4 5 6 gt gt gt indic
150. ary 279 Scipy lecture notes Edition 2015 2 Of all three presented techniques Cython is the most modern and advanced In particular the ability to opti mize code incrementally by adding types to your Python code is unique 14 7 Further Reading and References e Ga l Varoquaux s blog post about avoiding data copies provides some insight on how to handle memory management cleverly If you ever run into issues with large datasets this is a reference to come back to for some inspiration 14 8 Exercises Since this is a brand new section the exercises are considered more as pointers as to what to look at next so pick the ones that you find more interesting If you have good ideas for exercises please let us know l 2 3 Download the source code for each example and compile and run them on your machine Make trivial changes to each example and convince yourself that this works E g change cos for sin Most of the examples especially the ones involving Numpy may still be fragile and respond badly to input errors Look for ways to crash the examples figure what the problem is and devise a potential solution Here are some ideas a Numerical overflow b Input and output arrays that have different lengths c Multidimensional array d Empty array e Arrays with non double types 4 Use the timeit Python magic to measure the execution time of the various solutions 14 8 1 Python C API l Modify the Nump
151. ary rst control_flow rst first_steps rst python_language rst demo2 py functions rst python logo png demo py 10 TSt reusing code rst In 22 print com exit_code 0 In 23 type com Out 23 sh RunningCommand Walking a directory os path walk generates a list of filenames in a directory tree In 10 for dirpath dirnames filenames in os walk os curdir for fp in filenames print os path abspath fp Users cburns src scipy2009 scipy_2009_tutorial source index rst swo Users cburns src scipy2009 scipy_2009_tutorial source view_array py swp Users cburns src scipy2009 scipy_2009_tutorial source basic_types rst Users cburns src scipy2009 scipy_2009_tutorial source conf py Users cburns src scipy2009 scipy_2009_tutorial source control_flow rst 2 7 Standard Library 37 Scipy lecture notes Edition 2015 2 Environment variables In 9 import os In 11 os environ keys Out tii ees FSLDIR TERM_PROGRAM_VERSION FSLREMOTECALL USER HOMET PATH PSdar SHELL EDETORS WORKON_HOME PYTHONPATH In 12 os environ PYTHONPATH Out 12 Users cburns src utils Users cburns src nitools Users cburns local lib python2 5 site packages usr local lib python2 5 site packages Library Frameworks Python framework Versions 2 5 lib python2 5 In 16 os getenv PYTHONPATH Outl16 Users cburns src utils Users cburns src nitools Users cburns local lib python2 5 site pa
152. as well gt gt gt gauss_denoised ndimage gaussian_filter noisy 2 Most local linear isotropic filters blur the image ndimage uniform_filter A median filter preserves better the edges gt gt gt med_denoised ndimage median_filter noisy 3 Gaussian filter Median filter Median filter better result for straight boundaries low curvature im np zeros 20 20 im 5 5 5 5 1 im ndimage distance_transform_bf im im_noise im 0 2 np random randn im shape im_med ndimage median_filter im_noise 3 Original image Noisy image Median filter Error Other rank filter ndimage maximum_filter ndimage percentile_filter Other local non linear filters Wiener scipy signal wiener etc Non local filters Exercise denoising e Create a binary image of Os and 1s with several objects circles ellipses squares or random shapes e Add some noise e g 20 of noise e Try two different denoising methods for denoising the image gaussian filtering and median filter ing e Compare the histograms of the two different denoised images Which one is the closest to the histogram of the original noise free image See also More denoising filters are available in skimage denoising see the Scikit image image processing tutorial 12 4 4 Mathematical morphology See wikipedia for a definition of mathematical morphology Probe an image with a simple shape a structuring element
153. at the beginning of the notebook the fol lowing magic matplotlib inline 82 Scipy lecture notes Edition 2015 2 4 1 2 pyplot pyplot provides a procedural interface to the matplotlib object oriented plotting library It is modeled closely after Matlab Therefore the majority of plotting commands in pyplot have Matlab analogs with similar arguments Important commands are explained with interactive examples from matplotlitb import pyplot as plt 4 2 Simple plot In this section we want to draw the cosine and sine functions on the same plot Starting from the default settings we ll enrich the figure step by step to make it nicer First step is to get the data for the sine and cosine functions import numpy as np X np linspace np pi np pi 256 endpoint True C S np cos X np sin X X is now a numpy array with 256 values ranging from x to z included C is the cosine 256 values and S is the sine 256 values To run the example you can type them in an Python interactive session ipython pylab This brings us to the Python prompt IPython 0 13 An enhanced Interactive Python gt Introduction to Python s features magic gt Information about IPython s magic functions help gt Python s own help system object gt Details about object object also works prints more Welcome to pylab a matplotlib based Python environment For more information type help pylab
154. at they can be called with a single argument This means that deco rators can be implemented as normal functions or as classes with a__call__ method or in theory even as lambda functions Let s compare the function and class approaches The decorator expression the part after can be either just a name or a call The bare name approach is nice less to type looks cleaner etc but is only possible when no arguments are needed to customise the decorator Decorators written as functions can be used in those two cases gt gt gt def simple_decorator function print doing decoration return function gt gt gt simple_decorator def function print inside function doing decoration gt gt gt function inside function gt gt gt def decorator_with_arguments arg print defining the decorator def _decorator function in this inner function arg ts available too print doing decoration r arg return function 7 2 Decorators 148 Scipy lecture notes Edition 2015 2 return _decorator gt gt gt decorator_with_arguments abc def function print inside function defining the decorator doing decoration abc gt gt gt function inside function The two trivial decorators above fall into the category of decorators which return the original function If they were to return a new function an extra level of nestedness would be required In the worst case three levels of
155. ate line in an easy to read way 7 2 1 Replacing or tweaking the original object Decorators can either return the same function or class object or they can return a completely different ob ject In the first case the decorator can exploit the fact that function and class objects are mutable and add attributes e g add a docstring to a class A decorator might do something useful even without modifying the object for example register the decorated class in a global registry In the second case virtually anything is possible when something different is substituted for the original function or class the new object can be com pletely different Nevertheless such behaviour is not the purpose of decorators they are intended to tweak the decorated object not do something unpredictable Therefore when a function is decorated by replacing it with a different function the new function usually calls the original function after doing some preparatory work Likewise when a class is decorated by replacing if with a new class the new class is usually derived from the original class When the purpose of the decorator is to do something every time like to log every call to a decorated function only the second type of decorators can be used On the other hand if the first type is sufficient it is better to use it because it is simpler 7 2 2 Decorators implemented as classes and as functions The only requirement on decorators is th
156. atrix common CSR CSC functionality subclass of _data_matrix sparse matrix classes with data attribute e fast matrix vector products and other arithmetics sparsetools e constructor accepts dense matrix array sparse matrix 11 2 Storage Schemes 221 Scipy lecture notes Edition 2015 2 shape tuple create empty matrix data ij tuple data indices indptr tuple e many arithmetic operations considerably more efficient than CSR for sparse matrices with dense sub matrices e use like CSR vector valued finite element discretizations Examples e create empty BSR matrix with 1 1 block size like CSR gt gt gt mtx sparse bsr_matrix 3 4 dtype np int8 gt gt gt mtx lt 3x4 sparse matrix of type lt numpy ints gt with 0 stored elements blocksize 1x1 in Block Sparse Row format gt gt gt gt mtx todense Matrix 0 0 0 0 1072070 Oil 0 0 0 OJ dtype ints e create empty BSR matrix with 3 2 block size gt gt gt mtx sparse bsr_matrix 3 4 blocksize 3 2 dtype np int8 gt gt gt mtx lt 3x4 sparse matrix of type lt numpy ints gt with 0 stored elements blocksize 3x2 in Block Sparse Row format gt gt gt gt mtx todense matrix 0 0 0 0 105205 0O 0 0 0 OJ dtype int8 abug e create using data ij tuple with 1 1 block size like CSR gt gt gt row np array 0 O 1 2 2 21 gt
157. b array lO 2 4 6 31 gt gt gt np may_share_memory a b True gt gt gt b 0 12 gt gt gt b array Gl 2 4 6 2 gt gt gt a C1 array 2 jt 24 35 A Bhs Os y con J gt gt gt a np arange 10 gt gt gt c al 2 copyQ force a copy gt gt gt c 0 12 gt gt gt a array Os 1 2 3 4 5 6 7 3 9J gt gt gt np may_share_memory a c False 3 1 The Numpy array object 52 Scipy lecture notes Edition 2015 2 This behavior can be surprising at first sight but it allows to save both memory and time Worked example Prime number sieve 0123456 78 910111213 VN Compute prime numbers in 0 99 with a sieve e Construct a shape 100 boolean array is_prime filled with True in the beginning gt gt gt is_prime np ones 100 dtype bool e Cross out 0 and 1 which are not primes gt gt gt is_prime 2 0 e For each integer j starting from 2 cross out its higher multiples gt gt gt N_max int np sqrt len is_prime gt gt gt for j in range 2 N_max is_prime 2 j j False e Skim through help np nonzero and print the prime numbers e Follow up Move the above code into a script file named prime_sieve py Run it to check it works Use the optimization suggested in the sieve of Eratosthenes 1 Skip j which are already known to not be primes 2 The first number to cross out is j 3 1 7 Fancy indexing Numpy arrays can be indexed with
158. been defined above Note that one does not declare the type of an variable before assigning its value In C conversely one should write int a 3 In addition the type of a variable may change in the sense that at one point in time it can be equal to a value of a certain type and a second point in time it can be equal to a value of a different type b was first equal to an integer but it became equal to a string when it was assigned the value hello Operations on integers b 2 a are coded natively in Python and so are some operations on strings such as additions and multiplications which amount respectively to concatenation and repetition 2 2 Basic types 2 2 1 Numerical types Python supports the following numerical scalar types Integer gt gt gt F i 2 gt gt gt a 4 gt gt gt type a lt type int gt Floats gt gt gt c 2 1 gt gt gt type c lt type float gt 2 2 Basic types 11 Scipy lecture notes Edition 2015 2 1 5 020 gt gt gt a real 1 5 gt gt gt a imag Os gt gt gt type 1 0j lt type complex gt Booleans gt gt gt 3 gt 4 False gt gt gt test 3 gt 4 gt gt gt test False gt gt gt type test lt type bool gt A Python shell can therefore replace your pocket calculator with the basic arithmetic operations 7 modulo natively implemented gt gt gt T x 3 21 0 gt gt gt 2 10 1024
159. ble True Fire up the dialog Visualization configure_traits Let us read a bit the code above examples mlab_dialog py First the curve function is used to compute the coordinate of the curve we want to plot Second the dialog is defined by an object inheriting from HasTraits as it is done with Traits The important point here is that a Mayavi scene is added as a specific Traits attribute Instance This is important for embedding it in the dialog The view of this dialog is defined by the view attribute of the object In the init of this object we populate the 3D scene with a curve Finally the configure_traits method creates the dialog and starts the event loop See also There are a few things to be aware of when doing dialogs with Mayavi Please read the Mayavi documentation 19 5 Making interactive dialogs 350 Scipy lecture notes Edition 2015 2 19 5 2 Making it interactive We can combine the Traits events handler with the mlab_source to modify the visualization with the dialog We will enable the user to vary the n_turns parameter in the definition of the curve For this we need e to define an n_turns attribute on our visualization object so that it can appear in the dialog We use a Range type e to wire modification of this attribute to a recomputation of the curve For this we use the on_traits_change decorator o Edit properties BRR MMe 46u HG N turns 0 gt 30 11 from traits api import
160. by the structuring ele ment gt gt gt a np zeros 7 7 dtype np int gt gt gt a 1 6 2 5 1 gt gt gt a array 0 Oo Lor oi O Cle O Ole O OE O Cle O 0119 17 3 Image preprocessing enhancement 313 Scipy lecture notes Edition 2015 2 gt gt gt morphology binary_erosion a morphology diamond 1 astype np uint8 array CUO 0 On 0 0 0 0 Oe 0 0 0 0O 0 0 O 0s Ol d Oat 0 OT Or On Onn ee 0 0 Or LO Oe Om L 0 00r O0 0 0 0 O0 Olle 0 0 0 0 0 0 Oll dtype nines gt gt gt Erosion removes objects smaller than the structure gt gt gt morphology binary_erosion a morphology diamond 2 astype np uint8 array LIO 0 0 0 0 04 0 lOr 0 O0 O O O lll LO Ore 0 1002 0 Ol LOR Ones OO 0 Ou Olin 0 OO 0 0 0 0 0 00 0s 0 0p 0 0 0 0 0 0 0 OJ dtype uints Dilation maximum filter gt gt gt a np zeros 5 5 oo gt a2 2 r 4 gt gt gt a array IL 07 20h jg Onn Ora O Or Oraa Oa 0ra 0ds E Ors Oa i Oa Oaa BO Oa Ora Oan Ols Oe Oe O O OD gt gt gt morphology binary_dilation a morphology diamond 1 astype np uint8 array LO Oy 0 0 0l LOR 0 1 0 0l One E a ON 0O 0 l0 Oly 0 0 07 0 0O dtype uint8g Opening erosion dilation gt gt gt a np zeros 5 5 dtype np int gt gt gt alil 4 1 4 1 al4 4 1 gt gt gt a array cl lO 0 0 0 0l Oe Wes a IEA Gast aie dy Ol Ce ted a o LOC O
161. bytes to find the next column e Similarly to higher dimensions C last dimensions vary fastest smaller strides F first dimensions vary fastest shape dj d2 dn strides S1 S2 Sp s dj 1dj 2 dn x itemsize s d d2 dj x itemsize Now we can understand the behavior of view gt gt gt y np array 1 3 2 4 dtype np uint8 transpose gt gt gt x y copy Transposition does not affect the memory layout of the data only strides gt gt gt x strides 2 1 gt gt gt y strides Ci 2 gt gt gt str x data x01 x02 x03 x04 gt gt gt str y data x01 x03 x02 x04 e the results are different when interpreted as 2 of int16 e copy creates new arrays in the C order by default Slicing with integers e Everything can be represented by changing only shape strides and possibly adjusting the data pointer e Never makes copies of the data gt gt gt x np array 1 2 3 4 5 6 dtype np int32 gt gt gt y xi 1 gt gt gt y array l6 6 4 3 2 I dtype int32 gt gt gt y strides 4 8 1 Life of ndarray 168 Scipy lecture notes Edition 2015 2 gt gt gt y x 2 gt gt gt y __array_interface__ data 0 x __array_interface__ data 0 8 gt gt gt x np zeros 10 10 10 dtype np float gt gt gt x strides 800 80 8 Bo gt xl 2 s 3 A strides 1600 240 32 e Similarly tran
162. capacity print gt 15 print setting the installed capacity breaks the link between turbine power print and the installed_capacity trait reservoir installed_capacity 8 print turbine power reservoir installed_capacity 18 3 6 Notification Traits implements a Listener pattern For each trait a list of static and dynamic listeners can be fed with call backs When the trait does change all the listeners are called Static listeners are defined using the _XXX_changed magic methods from traits api import HasTraits Instance DelegatesTo Float Range from reservoir import Reservoir class ReservoirState HasTraits Keeps track of the reservoir state given the initial storage nnm reservoir Instance Reservoir min_storage Float max_storage DelegatesTo reservoir min_release Float max_release DelegatesTo reservoir state attributes storage Range low min_storage high max_storage control attributes inflows Float desc Inflows hm3 release Range low min_release high max_release spillage Float desc Spillage hm3 18 3 What are Traits 333 def print_state self print Storage tRelease tInflows tSpillage stroformat t join 7 2f for i in range 4 Scipy lecture notes Edition 2015 2 print str_format format self storage self release self inflows self spillage print 79 Traits listeners H HHHHHHHH def _release_changed self
163. ce Generator e Cython These four techniques are perhaps the most well known ones of which Cython is probably the most advanced one and the one you should consider using first The others are also important if you want to understand the wrapping problem from different angles Having said that there are other alternatives out there but having understood the basics of the ones above you will be in a position to evaluate the technique of your choice to see if it fits your needs The following criteria may be useful when evaluating a technology e Are additional libraries required e Is the code autogenerated e Does it need to be compiled e Is there good support for interacting with Numpy arrays 263 Scipy lecture notes Edition 2015 2 e Does it support C Before you set out you should consider your use case When interfacing with native code there are usually two use cases that come up e Existing code in C C that needs to be leveraged either because it already exists or because it is faster e Python code too slow push inner loops to native code Each technology is demonstrated by wrapping the cos function from math h While this is a mostly a trivial example it should serve us well to demonstrate the basics of the wrapping solution Since each technique also includes some form of Numpy support this is also demonstrated using an example where the cosine is computed on some kind of array Last but not least two small w
164. ch more difficult and ill posed interpolation problems the data structure itself does not tell us what are the neighbors of a data point We call such data a scatter Unstructured and unconnected data a scatter Structured and connected data a field mlab points3d mlab quiver3d mlab contour3d Data sources corresponding to scatters can be created with mayavi mlab pipeline scalar_scatter or mayavi mlab pipeline vector_scatter field data sources can be created with mlab pipeline scalar_field ormlab pipeline vector_field Exercice 1 Create a contour for instance of the magnetic field norm by using one of those functions and adding the right module by clicking on the GUI dialog 2 Create the right source to apply a vector_cut_plane and reproduce the picture of the magnetic field shown previously Note that one of the difficulties is providing the data in the right form number of arrays shape to the functions This is often the case with real life data See also Sources are described in details in the Mayavi manual Transforming data filters If you create a vector field you may want to visualize the iso contours of its magnitude But the isosurface module can only be applied to scalar data and not vector data We can use a filter ExtractVectorNorm to add this scalar value to the vector field Filters apply a transformation to data and can be added between sources and modules 19 3 Slicing and dicing data so
165. ch species had the largest population 3 Which species has the largest population for each year Hint argsort amp fancy indexing of np array H L C 4 Which years any of the populations is above 50000 Hint comparisons and np any 5 The top 2 years for each species when they had the lowest populations Hint argsort fancy indexing 6 Compare plot the change in hare population see help np gradient and the number of lynxes Check correlation see help np corrcoef all without for loops Solution Python source file 3 5 4 Crude integral approximations Write a function f a b c that returns a ranges 0 1 x 0 1 x 0 1 c Form a 24x12x6 array containing its values in parameter Approximate the 3 d integral l pl pl a c dadbdc 0 Jo Jo over this volume with the mean The exact result is In2 5 z 0 1931 what is your relative error Hints use elementwise operations and broadcasting You can make np ogrid give a number of points in given range with np ogridl0 1 20j 3 5 Some exercises 79 Scipy lecture notes Edition 2015 2 Reminder Python functions def f a b c return some_result Solution Python source file 3 5 5 Mandelbrot set 1 5 1 0 0 5 0 0 0 5 1 0 1 5 2 0 1 5 1 0 0 5 0 0 0 5 1 0 Write a script that computes the Mandelbrot fractal The Mandelbrot iteration N_max 50 some_threshold 50
166. checking is going to change In case a 2D spline is wanted the BivariateSpline class family is provided All those classes for 1D and 2D splines use the FITPACK For tran subroutines that s why a lower library access is available through the splrep and splev functions for respectively representing and evaluating a spline Moreover interpolation functions without the use of FIT PACK parameters are also provided for simpler use see interpid interp2d barycentric_interpolate and so on For the Sprog maxima wind speeds the UnivariateSpline will be used because a spline of degree 3 seems to correctly fit the data gt gt gt from scipy interpolate import UnivariatesSpline gt gt gt quantile_func UnivariateSpline cprob sorted_max_speeds The quantile function is now going to be evaluated from the full range of probabilities gt gt gt nprob np linspace 0O 1 1e2 gt gt gt fitted_max_speeds quantile_func nprob In the current model the maximum wind speed occurring every 50 years is defined as the upper 2 quantile As a result the cumulative probability value will be gt gt gt Litty prob 1 0 02 So the storm wind speed occurring every 50 years can be guessed by gt gt gt fifty_wind quantile_func fifty_prob gt gt gt fifty_wind array 32 97989825 The results are now gathered on a Matplotlib figure Exercise with the Gumbell distribution The interested readers are now invited to make an
167. city of the example the release is considered in hm3 timestep and not in m3 s reservoir Instance Reservoir min_storage Float max_storage DelegatesTo reservoir min release Float max_release DelegatesTo reservoir state attributes storage Range low min_storage high max_storage control attributes inflows Float desc Inflows hm3 release Range low min_release high max_release spillage Float desc Spillage hm3 update_storage Event desc Updates the storage to the next time step def _update_storage_fired self update storage state new_storage self storage self release self inflows self storage min new_storage self max_storage overflow new_storage self max_storage self spillage max overflow 0 def print_state self print Storage tRelease tInflows tSpillage str_format t join 7 2f for i in range 4 print str_format format self storage self release self inflows self spillage print gt 79 if oname _main __ projectA Reservoir name Project A max_storage 30 max_release 5 0 hydraulic_head 60 efficiency 0 3 state ReservoirState reservoir projectA storage 15 state release 5 state inflows 0 release the maximum amount of water during 3 time steps state update_storage True state print_state state update_storage True state print_state 18 3 What are Traits 329 Scipy lecture not
168. ckages usr local lib python2 5 site packages Library Frameworks Python framework Versions 2 5 lib python2 5 2 2 shutil high level file operations The shutil provides useful file operations e shutil rmtree Recursively delete a directory tree e shutil move Recursively move a file or directory to another location e shutil copy Copy files or directories 2 3 glob Pattern matching on files The glob module provides convenient file pattern matching Find all files ending in txt In 18 import glob In 19 glob glob txt Qut 19 holy grail txt junk txt newfile txt 2 1 4 sys module system specific information System specific information related to the Python interpreter e Which version of python are you running and where is it installed 2 7 Standard Library 38 Scipy lecture notes Edition 2015 2 In 117 sys platform Out 117 darwin In 118 sys version Hut 118 2 5 2 252 6091 Feb 22 2008 07 57 53 An GCC 4 0 1 Apple Computer Inc build 5363 In 119 sys prefix Out 119 Library Frameworks Python framework Versions 2 5 e List of command line arguments passed to a Python script In 100 sys argv Out 100 Users cburns local bin ipython sys path is a list of strings that specifies the search path for modules Initialized from PYTHONPATH In 121 sys path Out 21 p Users cburns local bin Users cburns local lib python2 5
169. color blue alpha 1 00 Click on the figure for solution 4 4 2 Scatter Plots o 0 8 e gt Pg 8 0 J vo ee ge oa 009 o ODO og ef ee pid 008 o 000 8 de othe MF ede og of Color is given by angle of X Y Starting from the code below try to reproduce the graphic on the right taking care of marker size color and transparency 1024 np random normal 0 1 n np random normal 0 1 n plt scatter X Y Click on figure for solution 4 4 Other Types of Plots examples and exercises 91 Scipy lecture notes Edition 2015 2 4 4 3 Bar Plots You need to take care of text alignment Starting from the code below try to reproduce the graphic on the right by adding labels for red bars n 12 X np arange n 1 X float n np random uniform 0 5 1 0 n Y2 1 X float m np random uniform 0 5 1 0 n plt bar X Y1 facecolor 9999ff edgecolor white plt bar X Y2 facecolor ff 9999 edgecolor white for x yim zip X Yi plt text x 0 4 y 0 05 2f h y ha center va bottom plt ylim 1 25 1 25 Click on figure for solution 4 4 4 Contour Plots You need to use the clabel command Starting from the code below try to reproduce the graphic on the right taking care of the colormap see Col ormaps below ities y return 1 x 2 x 5 y 3 np exp x 2 y 2 256 np linspace 3 3 n np lins
170. comes with its source 4 5 5 Mailing lists Finally there is a user mailing list where you can ask for help and a developers mailing list that is more techni cal 4 6 Quick references Here is a set of tables that show main properties and styles 4 6 1 Line properties alpha or a alpha transparency on 0 1 scale antialiased True or False use antialised Aliased rendering Anti alilased matplotlib color arg linesiyle orl RiAiSAAALLS ee eee essere osmana TEV T ETS SS a p salami a a ee o e eee 7 z Oo oO oO Oo Oo Oo Oo Oo oO marker see Markers markeredgewidth line width around the marker mew symbol markeredgecolor edge color if a marker is used mec markerfacecolor face color if a marker is used mfc markersize ms size of the marker in points 4 6 Quick references 98 Scipy lecture notes Edition 2015 2 4 6 2 Line styles 14 O S F gt FVAWNUTFTKTDTANMTSTOONM IAL Ioco Yva Ttar EeETAYyYyIi Yw jj wy TER AVIPH XOyY Ax 00 90 SESH AVdP H xX yr4Ax gt G6 00e I le OAVIAPHH HXOYA 001 HO Efe avVrqarPHc x r rA lt k gt ee ite AVIPH X y4A x gt 88 e ibe eave mo xoyrixreei eo ilo AVIPB XOy4A4x gt 001 0 TIE eavVrvarprH x yrAaxk gt e e EEF eavrvarPrHiit x yr4A lt x gt oe itn mwAwWRe wd eA ie Oe 4 6 3 Markers CHNMNTtTNOROSIIANMTbO OC SVAD D HE LE KOAT gt gt lt lt lt lt lt lt 4 6 4 Co
171. containers e Tick containers e Path tutorial e Introduction e B zier example e Compound paths e Transforms tutorial e Introduction e Data coordinates e Axes coordinates e Blended transformations e Using offset transforms to create a shadow effect e The transformation pipeline 4 5 2 Matplotlib documentation e User guide e FAQ Installation Usage How To Troubleshooting Environment Variables e Screenshots 4 5 3 Code documentation The code is well documented and you can quickly access a specific command from within a python session gt gt gt import matplotlib pyplot as plt gt gt gt help plt plot Help on function plot in module matplotiib pyplot plot xargs kwargs Plot lines and or markers to the class matplotlib axes Axes args is a variable length argument allowing for multiple x y pairs with an optional format string For example each of the following is legal plot x y plot x and y using default line style and color plot x y bo plot x and y using blue circle markers plot y plot y using x as index array 0 N 1 plot y TF ditto but with red plusses If x and or y is 2 dimensional then the corresponding columns will be plotted 4 5 Beyond this tutorial 97 Scipy lecture notes Edition 2015 2 4 5 4 Galleries The matplotlib gallery is also incredibly useful when you search how to render a given graphic Each example
172. cts 1 0 000 0 000 0 008 0 008 ica py 97 fastica Clearly the svd in decomp py is what takes most of our time a k a the bottleneck We have to find a way to make this step go faster or to avoid this step algorithmic optimization Spending time on the rest of the code is useless Profiling outside of IPython running cProfile Similar profiling can be done outside of IPython simply calling the built in Python profilers cProfile and profile python m cProfile o demo prof demo py Using the o switch will output the profiler results to the file demo prof to view with an external tool This can be useful if you wish to process the profiler output with a visualization tool 10 2 3 Line profiler The profiler tells us which function takes most of the time but not where it is called For this we use the line_profiler in the source file we decorate a few functions that we want to inspect with profile no need to import it profile def test data np random random 5000 100 10 2 Profiling Python code 205 Scipy lecture notes Edition 2015 2 u s v linalg svd data pea np dot uL 10 data results fastica pca T whiten False Then we run the script using the kernprof py program with switches 1 line by line and v view to use the line by line profiler and view the results in addition to saving them kernprof py l v demo py Wrote profile results to demo py lprof Timer unit le 06
173. cubic_interp interpid measured_time measures kind cubic gt gt gt cubic_results cubic_interp computed_time The results are now gathered on the following Matplotlib figure 1 0 measures linear interp cubic interp 0 5 0 0 0 5 1 0 0 0 0 2 0 4 0 6 0 8 1 0 scipy interpolate interp2d is similar to scipy interpolate interpid but for 2 D arrays Note that for the interp family the computed time must stay within the measured time range See the summary exercise on Maximum wind speed prediction at the Sprog station for a more advance spline interpolation example 5 8 Numerical integration scipy integrate The most generic integration routine is scipy integrate quad gt gt gt from scipy integrate import quad gt gt gt res err quad np sin 0 np pi 2 gt gt gt np allclose res 1 True gt gt gt np allclose err 1 res True Others integration schemes are available with fixed_quad quadrature romberg scipy integrate also features routines for integrating Ordinary Differential Equations ODE In particular scipy integrate odeint is a general purpose integrator using LSODA Livermore Solver for Ordinary Differential equations with Automatic method switching for stiff and non stiff problems see the ODEPACK Fortran library for more details odeint solves first order ODE systems of the form dy dt rhs y1 y2 t0 5 8 Numerical integration scipy integ
174. d PyUFunc_ff_f_As_dd_d char args npy_intp dimensions npy_intp steps void func void PyUFunc_ff_f char args npy_intp dimensions npy_intp steps void func void PyUFunc_dd_d char x args npy_intp dimensions npy_intp steps void func void PyUFunc_gg g char x args npy_intp dimensions npy_intp steps void func void PyUFunc_FF_F_As_DD_D char args npy_intp dimensions npy_intp steps void func void PyUFunc_DD_D char x args npy_intp dimensions npy_intp steps void func void PyUFunc_FF_F char args npy_intp dimensions npy_intp steps void func void PyUFunc_GG_G char x args npy_intp dimensions npy_intp steps void func Required module initialization import_array 8 2 Universal functions 176 Scipy lecture notes Edition 2015 2 import_ufunc The actual ufunc declaration cdef PyUFuncGenericFunction loop_func 1 cdef char input_output_types 3 cdef void elementwise_funcs 1 Reminder some pre made Ufunc loops 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 55 5 5 55 5 5 5 5 5 PyUfunc_f_f gt float elementwise_func float input_1 PyUfunc_ff_f float elementwise_func float input_1 float input_2 PyUfunc_d_d double elementwise_func double input_1 PyUfunc_dd_d double elementwise_func double input_1 double input_2 PyUfunc_D_D elementwise_func complex_double input complex_double complex_
175. d to statistics Python is a general purpose language with statistics modules R has more statistical analysis features than Python and specialized syntaxes However when it comes to building complex analysis pipelines that mix statistics with e g image analysis text mining or control of a physical experiment the richness of Python is an invaluable asset 284 Scipy lecture notes Edition 2015 2 Contents e Data representation and interaction Data asa table The panda data frame e Hypothesis testing comparing two groups Student s t test the simplest statistical test Paired tests repeated measurements on the same indivuals Linear models multiple factors and analysis of variance formulas to specify statistical models in Python Multiple Regression including multiple factors Post hoc hypothesis testing analysis of variance ANOVA More visualization seaborn for statistical exploration Pairplot scatter matrices Implot plotting a univariate regression Testing for interactions In this document the Python inputs are represented with the sign gt gt gt Disclaimer Gender questions Some of the examples of this tutorial are chosen around gender questions The reason is that on such questions controlling the truth of a claim actually matters to many people 15 1 Data representation and interaction 15 1 1 Data as a table The setting that we consider for statistical analysis
176. dcasting e Basic operations on numpy arrays addition etc are elementwise e This works on arrays of the same size Nevertheless It s also possible to do operations on arrays of different sizes if Numpy can transform these arrays so that they all have the same size this conversion is called broadcasting The image below gives an example of broadcasting BEEBE NK Let s verify gt gt gt a np tile np arange 0 40 10 3 1 T gt gt gt a array 0 0 Ol 10 10 10 20 20 201 30 30 3011 gt gt gt b np array 0 1 2 gt gt gt a b array IL 0 ie 2 ELR iio 2O 2r 221 EOR CEED We have already used broadcasting without knowing it gt gt gt a np ones 4 5 gt gt gt a 0 2 we assign an array of dimension O to an array of dimension 1 gt gt gt a array l 2a 2 E l 3 2 Numerical operations on arrays 62 Scipy lecture notes Edition 2015 2 al il e v v v v v p v i v p v An useful trick gt gt gt a np arange 0 40 10 gt gt gt a shape 4 gt gt gt a al np newaxis adds a new axis gt 2D array gt gt gt a shape 4 1 gt gt gt a array 0 Flex 20 230 J gt gt gt a b array Cit 0 1 2 O a2 20 21 22 rome PTA Broadcasting seems a bit magical but it is actually quite natural to use it when we want to solve a problem whose output data is a
177. de ea bx 3 4 Advanced operations 2 66442 2e82s 088 eee heb bbs Re ed Oo SOMECXEIC SE o e eaaa pe eee ee ee Eee eee OS Matplotlib plotting 4 1 4 2 4 3 4 4 4 5 4 6 TT OGUCHON 2240s eueke dc doe ee oe ey bw Ode oe Se SPEI 25 445 24065 Bow he oe eo oe Oe ee bee Figures Subplots Axes and Ticks i52260442 46600084044 Other Types of Plots examples and exercises Beyond this tutorial 2266644465 685 264528 aS oo S eee A Quick references 4 444444 245 456 6 dS Soh KS REESE ew Scipy high level scientific computing 5 1 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 File input output Scipy i siai sessi a oe oe Oe dasri Special functions scipy special saas aeea Linear algebra operations scipy linalg Fast Fourier transforms scipy fftpack Optimization and fit scipy optimize Statistics and random numbers scipy stats Interpolation scipy interpolate Numerical integration scipy integrate Signal processing scipy signal 2 00000 5 10 Image processing scipy ndimage Contents 5 11 Summary exercises on scientific computing 46 Kad dee eo RK OE RM OE EEE 124 6 Getting help and finding documentation 137 II Advanced topics 140 7 Advanced Python Constructs 142 7 1 Iterators generator expressions and generators 1 ee e 142 Te MICO GES x hk he Se SR a A eee a 147 To Come 2 peek ei RR AERA OO ee BEE EE
178. do so using array computing tricks we are going to create a 2D array with the stories each walker has a story in one direction and the time in the other shuffled jumps Position cumulated jumps sum time time gt gt gt n_stories 1000 number of walkers gt gt gt t_max 200 time during which we follow the walker We randomly choose all the steps 1 or 1 of the walk gt gt gt t np arange t_max gt gt gt steps 2 np random random_integers 0 1 n_stories t_max 1 gt gt gt np unique steps Verification all steps are 1 or 1 array TLD We build the walks by summing steps along the time gt gt gt positions np cumsum steps axis 1 aris 1 dimension of time gt gt gt sq_distance positions 2 We get the mean in the axis of the stories gt gt gt mean_sq_distance np mean sq_distance axis 0 Plot the results gt gt gt plt figure figsize 4 3 lt matplotlib figure Figure object at gt gt gt gt plt plot t np sqrt mean_sq_distance g t np sqrt t y lt matplotlib lines Line2D object at gt lt matplotlib lines Line2D object at gt gt gt gt plt xlabel r t lt matplotlib text Text object at gt gt gt gt plt ylabel r sqrt langle delta x 2 rangle lt matplotlib text Text opject at gt 16 14 12 3 2 gt N gferations on arrays 61 Scipy lecture notes Edition 2015 2 3 2 3 Broa
179. double PyUfunc_DD_D elementwise_func complex_double in1 complex_double in2 complex_double out gt 5 5 5 5 5 5 5 5 5 5 5 5 5 5 S 5 55 5 S5 5 55 5 5 5 The full list is above Type codes NPY_BOOL NPY_BYTE NPY_UBYTE NPY_SHORT NPY_USHORT NPY_INT NPY_UINT NPY_LONG NPY_ULONG NPY_LONGLONG NPY_ULONGLONG NPY_FLOAT NPY_DOUBLE NPY_LONGDOUBLE NPY_CFLOAT NPY_CDOUBLE NPY_CLONGDOUBLE NPY_DATETIME NPY_TIMEDELTA NPY_OBJECT NPY_STRING NPY_UNICODE NPY_VOID loop_funcl0O TODO suitable PyUFunc_ input_output_types 0O TODO TODO fill in rest of input_output_types This thing is passed as the data parameter for the generic PyUFunc_ loop to let it know which function it should call elementwise_funcs 0 lt void gt mandel_single_point Construct the ufunc mandel PyUFunc_FromFuncAndData loop_func elementwise_funcs input_output_types 1 number of supported input types TODO number of input args TODO number of output args O identity element never mind this mandel function name mandel z c gt computes z z c docstring O unused Reminder some pre made Ufunc loops 8 2 Universal functions 177 Scipy lecture notes Edition 2015 2 PyUfunc_f_f float elementwise_func float input_1 PyUfunc_ff_f float elementwise_func float input_1 float input_2 P
180. ds and https docs python org library string html new string formatting 2 2 Basic types 16 Scipy lecture notes Edition 2015 2 String formatting gt gt gt An integer 42 a float Zf another string 7s 4 1 0 1 string An integer 1 a float 0 100000 another string string gt gt gt i 102 gt gt gt filename processing of_dataset_ d txt i gt gt gt filename processing of _dataset_102 txt Dictionaries A dictionary is basically an efficient table that maps keys to values It is an unordered container gt gt gt tel emmanuelle 5752 sebastian 5578 gt gt gt tel francis 5915 gt gt gt tel sebastian 5578 francis 5915 emmanuelle 5752 gt gt gt tel sebastian 5578 gt gt gt tel keys Sebastian francis emmanuelle gt gt gt tel values 5578 5915 5752 gt gt gt francis in tel True It can be used to conveniently store and retrieve values associated with a name a string for a date a name etc See https docs python org tutorial datastructures html dictionaries for more information A dictionary can have keys resp values with different types gt gt gt d a 1 b 2 3 hello gt gt gt d ta l 3 Phello TD 2 More container types Tuples Tuples are basically immutable lists The elements of a tuple are written between parentheses or just separated by commas gt
181. e the arguments and return types into place and for the module initialisation Although some of this is amortised as the extension grows the boilerplate required for each function s remains The standard python build system distutils supports compiling C extensions from a setup py which is rather convenient from distutils core import setup Extension define the extension module cos_module Extension cos_module sources cos_module c run the setup setup ext_modules cos_module This can be compiled cd advanced interfacing with_c python_c_api 1s 14 2 Python C Api 265 Scipy lecture notes Edition 2015 2 cos_module c setup py python setup py build_ext inplace running build_ext building cos_module extension creating build creating build temp linux x86_64 2 7 gcc pthread fno strict aliasing g 02 DNDEBUG g fwrapv 03 Wall Wstrict prototypes fPIC I home esc an gcc pthread shared build temp linux x86_64 2 7 cos_module o L home esc anaconda lib lpython2 7 o home esc 1s build cos_module c cos_module so setup py e build_ext is to build extension modules e inplace will output the compiled extension module into the current directory The file cos_module so contains the compiled extension which we can now load in the Python interpreter In 1 import cos_module Ine 21 cos_module Type module string Form lt module cos_module from cos_module so gt
182. e from reservoir import Reservoir class ReservoirState HasTraits Keeps track of the reservoir state given the initial storage nnn reservoir Instance Reservoir min_storage Float max_storage DelegatesTo reservoir min release Float max_release DelegatesTo reservoir state attributes storage Range low min_storage high max_storage control attributes inflows Float desc Inflows hm3 release Range low min_release high max_release spillage Float desc Spillage hm3 def print_state self print Storage tRelease tInflows tSpillage str_format t join 7 2f for i in range 4 print str_format format self storage self release self inflows self spillage print 79 if name _ main _ projectA Reservoir name Project A max_storage 30 max release 100 0 hydraulic_head 60 efficiency 0 8 state ReservoirState reservoir projectA storage 10 state release 90 state inflows state print_state 18 3 What are Traits 328 Scipy lecture notes Edition 2015 2 print How do we update the current storage A special trait allows to manage events and trigger function calls using the magic _xxxx_fired method from traits api import HasTraits Instance DelegatesTo Float Range Event from reservoir import Reservoir class ReservoirState HasTraits Keeps track of the reservoir state given the initial storage For the simpli
183. e Numpy array object 48 Scipy lecture notes Edition 2015 2 gt gt gt image np random rand 30 30 gt gt gt plt imshow image cmap plt cm hot gt gt gt plt colorbar lt matplotlib colorbar Colorbar instance at See also More in the matplotlib chapter Exercise Simple visualizations e Plot some simple arrays a cosine as a function of time and a 2D matrix e Try using the gray colormap on the 2D matrix 3 1 5 Indexing and slicing The items of an array can be accessed and assigned to the same way as other Python sequences e g lists gt gt gt a np arange 10 gt gt gt a array CLO 1 2 3 4 5 6 75 S91 gt gt gt afO al2 al 0 2 9 A Indices begin at 0 like other Python sequences and C C In contrast in Fortran or Matlab indices begin at 1 The usual python idiom for reversing a sequence is supported 3 1 The Numpy array object 49 Scipy lecture notes Edition 2015 2 gt gt gt a 1 array l9 Sr Tr Oy 5473 2571 0 For multidimensional arrays indexes are tuples of integers gt gt gt a np diag np arange 3 gt gt gt a array 0 0 0l lO Ole O0 2b gt gt gt ali 1 1 gt gt gt al2 1 10 third line second column gt gt gt a array 0 0 0 EO as Ol 20 210 2109 gt gt gt ali array 0 1 0 e In 2D the first dimension corresponds to rows the second to columns e for multidimensional a a L0 i
184. e Pick diagonal entries of the matrix assume C memory order gt gt gt x np array 1 2 3 8 Sle 7 8 9 dtype np int32 gt gt gt x_diag as_strided x shape 3 strides 77 e Pick the first super diagonal entries 2 6 e And the sub diagonals Hint to the last two slicing first moves the point where striding starts from Solution Pick diagonals gt gt gt x_diag as_strided x shape 3 strides 3 1 x itemsize gt gt gt x_diag array 1 5 9 dtype int32 Slice first to adjust the data pointer gt gt gt as_strided xL0 1 shape 2 strides 3 1 x itemsize array 2 6 dtype int32 gt gt gt as_strided x 1 0 shape 2 strides 3 1 x itemsize array 4 8 dtype int32 Using np diag gt gt gt y np diag x k 1 gt gt gt y array 2 6 dtype int32 However gt gt gt y flags owndata False Note This behavior has changed before numpy 1 9 np diag would make a copy See also stride diagonals py Challenge Compute the tensor trace gt gt gt x np arange 5 5 5 5 reshape 5 5 5 5 gt gt gt s 0 gt gt gt for i in range 5 for j in range 5 s x j i j il by striding and using sum on the result gt gt gt y as_strided x shape 5 5 strides TODO TODO gt gt gt s2 gt gt gt assert s s2 Solution 8 1 Life of ndarray 171 Scipy lecture notes Edition 2015 2 gt
185. e can find the corresponding Python code gdb pyframe home varoquau usr lib python2 6 site packages numpy core arrayprint py 158 _leading_trailing gdb This is numpy code we need to go up until we find code that we have written gdb up 34 0x080dc97a in PyEval_EvalFrameEx f Frame 0x82f064c for file segfault py line 11 in print_big_array small_array lt numpy ndarray at remote Ox 1630 Python ceval c No such file or directory in Python ceval c gdb pyframe segfault py 12 print_big_array The corresponding code is 9 4 Debugging segmentation faults using gdb 201 Scipy lecture notes Edition 2015 2 def make_big_array small_array big_array stride_tricks as_strided small_array shape 2e6 2e6 strides 32 32 return big_array def print_big_array small_array big_array make_big array small_array Thus the segfault happens when printing big_array 10 The reason is simply that big_array has been allocated with its end outside the program memory For a list of Python specific commands defined in the gdbinit read the source of this file Wrap up exercise The following script is well documented and hopefully legible It seeks to answer a problem of actual interest for numerical computing but it does not work Can you debug it Python source code to_debug py 9 4 Debugging segmentation faults using gdb 202 CHAPTER 10 Optimizing code Donald Knuth Premature optimization is t
186. e each suitable for some tasks e many employ sparsetools C module by Nathan Bell e assume the following is imported gt gt gt import numpy as np gt gt gt import scipy sparse as sps gt gt gt import matplotlib pyplot as plt e warning for NumPy users the multiplication with is the matrix multiplication dot product not part of NumPy 11 2 Storage Schemes 212 Scipy lecture notes Edition 2015 2 passing a sparse matrix object to NumPy functions expecting ndarray matrix does not work 11 2 1 Common Methods e all scipy sparse classes are subclasses of spmatrix default implementation of arithmetic operations always converts to CSR subclasses override for efficiency shape data type set get nonzero indices format conversion interaction with NumPy toarray todense e attributes mtx A same as mtx toarray mtx T transpose same as mtx transpose mtx H Hermitian conjugate transpose mtx real real part of complex matrix mtx imag imaginary part of complex matrix mtx size the number of nonzeros same as self getnnz mtx shape the number of rows and columns tuple e data usually stored in NumPy arrays 11 2 2 Sparse Matrix Classes Diagonal Format DIA e very simple scheme e diagonals in dense NumPy array of shape n_diag length fixed length gt waste space a bit when far from main diagonal subclass
187. e following example demonstrate the usage of the Enum and List traits from traits api import HasTraits Str Float Range Enum List from traitsui api import View Item class IrrigationArea HasTraits name Str surface Float desc Surface ha crop Enum Alfalfa Wheat Cotton class Reservoir HasTraits name Str max_storage Float 1e6 desc Maximal storage hm3 max_release Float 10 desc Maximal release m3 s head Float 10 desc Hydraulic head m efficiency Range 0O 1 irrigated_areas List IrrigationArea def energy_production self release Returns the energy production Wh for the given release m3 s 18 3 What are Traits 336 Scipy lecture notes Edition 2015 2 power 1000 9 81 self head release self efficiency return power 3600 traits_view View Item name Item max_storage Ttem max release Item head Item efficiency Item irrigated_areas resizable True __name__ main __ upper_block IrrigationArea name Section C surface 2000 crop Wheat reservoir Reservoir name Project A max_storage 30 max _release 100 0 head 60 efficiency 0 8 irrigated_areas upper_block release 80 print Releasing m3 s produces kWh format release reservoir energy_production release Trait listeners can be used to listen to changes in the content of the list to e g keep track of the total crop sur
188. e loop Mandelbrot problem 72 gt Ae at A1 gt gt gt while abs z lt 100 Z Z 2 1 gt gt gt Z 134 352j More advanced features break out of enclosing for while loop gt gt gt z 1 1j gt gt gt while abs z lt 100 if z imag break z z 2 1 2 3 Control Flow 19 continue the next iteration of a loop SoS ar lO 2 4 gt gt gt for element in a if element continue print 1 element OROG N O O o1 2 3 4 Conditional Expressions if lt OBJECT gt Evaluates to False e any number equal to zero 0 0 0 0 0 e an empty container list tuple set dictionary e False None Evaluates to True e everything else a b Tests equality with logics gt gt gt True a is b Tests identity both sides are the same object gt gt gt i is 1 False gt gt gt a gt gt gt b gt gt gt a is b True e a in b For any collection b b contains a gt gt gt b 1 2 3 gt gt gt 2 in b True gt gt gt 5 in b False If b is a dictionary this tests that a is a key of b 2 3 5 Advanced iteration lterate over any sequence Scipy lecture notes Edition 2015 2 You can iterate over any sequence string list keys in a dictionary lines in a file gt gt gt vowels aeiouy gt gt gt for i in powerful if i in vowels print i 2 3 Control Flow 20 Scipy lecture notes Edition
189. e morphology In 6 from scipy ndimage import morphology in Lr morphology binary_dilation Type function Base Class lt type function gt String Form lt function binary_dilation at Ox9bedd84 gt Namespace Interactive File usr lib python2 6 dist packages scipy ndimage morphology py Definition morphology binary_dilation input structure None iterations 1 mask None output None border_value 0 origin 0 brute _force False Docstring Multi dimensional binary dilation with the given structure An output array can optionally be provided The origin parameter controls the placement of the filter If no structuring element is provided an element is generated with a squared connectivity equal to one The dilation operation is repeated iterations times If iterations is less than 1 the dilation is repeated until the result does not change anymore If a mask is given only those elements with a true value at the corresponding mask element are modified at each iteration 2 5 7 Good practices e Use meaningful object names e Indentation no choice 2 5 Indenting is compulsory in Python Every command block following a colon bears an additional indentation level with respect to the previous line with a colon One must therefore indent after def or while At the end of such logical blocks one decreases the indentation depth and re increases it if a new block is entered etc Strict respect of indentation
190. e object When defining the setter we already have the property object under area and we add the setter to it by using the setter method All this happens when we are creating the class Afterwards when an instance of the class has been created the property object is special When the interpreter executes attribute access assignment or deletion the job is delegated to the methods of the property object To make everything crystal clear let s define a debug example gt gt gt class D object property def a self print getting 1 return 1 a setter def a self value print setting r value a deleter def a self 7 2 Decorators 152 Scipy lecture notes Edition 2015 2 print deleting gt gt gt D a lt property object at Ox gt gt gt gt D a fget lt function gt gt gt gt D a fset lt function gt gt gt gt D a fdel lt function gt gt gt gt d D varies this is not the same a function gt gt gt d a getting 1 1 gt gt gt d a setting gt gt gt del deleting gt gt gt d a getting 1 1 Il N Q N w Properties are a bit of a stretch for the decorator syntax One of the premises of the decorator syntax that the name is not duplicated is violated but nothing better has been invented so far It is just good style to use the same name for the getter setter and deleter methods Some newer examples include e functools lr
191. ear approximation inequality con straints only gt gt gt optimize fmin_cobyla f np array 0 0 cons constraint array 1 25009622 0 24990378 A The above problem is known as the Lasso problem in statistics and there exists very efficient solvers for it for instance in scikit learn In general do not use generic solvers when specific ones exist Lagrange multipliers If you are ready to do a bit of math many constrained optimization problems can be converted to non constrained optimization problems using a mathematical trick known as Lagrange multipliers 13 5 Optimization with constraints 262 CHAPTER 14 Interfacing with C Author Valentin Haenel This chapter contains an introduction to the many different routes for making your native code primarily C C available from Python a process commonly referred to wrapping The goal of this chapter is to give you a flavour of what technologies exist and what their respective merits and shortcomings are so that you can select the appropriate one for your specific needs In any case once you do start wrapping you almost certainly will want to consult the respective documentation for your selected technique Chapters contents Introduction e Python C Api e Ctypes e SWIG e Cython e Summary e Further Reading and References e Exercises 14 1 Introduction This chapter covers the following techniques e Python C Api e Ctypes e SWIG Simplified Wrapper and Interfa
192. ed it s likely that gen send will become gen __send__ The last of generator meth ods close is pretty obviously named incorrectly because it is already invoked implicitly 7 1 Iterators generator expressions and generators 146 Scipy lecture notes Edition 2015 2 7 1 5 Chaining generators This is a preview of PEP 380 not yet implemented but accepted for Python 3 3 Let s say we are writing a generator and we want to yield a number of values generated by a second generator a subgenerator If yielding of values is the only concern this can be performed without much difficulty using a loop such as subgen some_other_generator for v in subgen yield v However if the subgenerator is to interact properly with the caller in the case of calls to send throw and close things become considerably more difficult The yield statement has to be guarded by a try except finally structure similar to the one defined in the previous section to debug the generator func tion Such code is provided in PEP 380 here it suffices to say that new syntax to properly yield from a subgen erator is being introduced in Python 3 3 yield from some_other_generator This behaves like the explicit loop above repeatedly yielding values from some_other_generator until it is exhausted but also forwards send throw and close to the subgenerator 1 2 Decorators Summary This amazing feature appeared in the language almost apolog
193. ed by any subclass derived from the class The common way of creating a traits class is by extending from the HasTraits base class and defining class traits from traits api import HasTraits Str Float class Reservoir HasTraits name Str max_storage Float 18 3 What are Traits 324 Scipy lecture notes Edition 2015 2 For Traits 3 x users If using Traits 3 x you need to adapt the namespace of the traits packages e traits api should be enthought traits api e traitsui api should be enthought traits ui api Using a traits class like that is as simple as any other Python class Note that the trait value are passed using keyword arguments reservoir Reservoir name Lac de Vouglans max_storage 605 18 3 1 Initialisation All the traits do have a default value that initialise the variables For example the basic python types do have the following trait equivalents Python Type Built in Default Value Boolean Complex number 0 0j Floating point number 0 0 Plain integer Long integer Unicode A number of other predefined trait type do exist Array Enum Range Event Dict List Color Set Expression Code Callable Type Tuple etc Custom default values can be defined in the code from traits api import HasTraits Str Float class Reservoir HasTraits name Str max_storage Float 100 reservoir Reservoir name Lac de Vouglans Complex initialisation When a complex initialisat
194. ed years Thus it will be possible to calculate the cumulative probability of every measured wind speed maxima From those experimental points the scipy interpolate module will be very useful for fitting the quantile function Finally the 50 years maxima is going to be evaluated from the cumulative probability of the 2 quantile 5 11 Summary exercises on scientific computing 124 Scipy lecture notes Edition 2015 2 Computing the cumulative probabilities The annual wind speeds maxima have already been computed and saved in the numpy format in the file examples max speeds npy thus they will be loaded by using numpy gt gt gt import numpy as np gt gt gt max_speeds np load intro summary exercises examples max speeds npy gt gt gt years_nb max_speeds shape 0 Following the cumulative probability definition p_i from the previous section the corresponding values will be gt gt gt cprob np arange years_nb dtype np float32 1 years_nb 1 and they are assumed to fit the given wind speeds gt gt gt sorted_max_speeds np sort max_speeds Prediction with UnivariateSpline In this section the quantile function will be estimated by using the UnivariateSpline class which can represent a spline from points The default behavior is to build a spline of degree 3 and points can have different weights according to their reliability Variants are InterpolatedUnivariateSpline and LSQUnivariateSpline on which errors
195. een black A Note that 1 start stop contains the elements with indices i such as start lt i lt stop i ranging from start to stop 1 Therefore 1 start stop has stop start elements Slicing syntax 1 start stop stride All slicing parameters are optional gt gt gt 1 red blue green black white gt HS black white gt gt gt 1173 red blue green ooo 2 red green white Lists are mutable objects and can be modified gt gt gt 1 0 yellow gt gt gt 1 yellow blue green black white gt gt gt 1 2 4 gray purple gt gt gt 1 yellow blue gray purple white The elements of a list may have different types gt gt gt 1 3 200 hello gt gt gt 1 3 200 hello gt gt gt Lit E12 200 hello For collections of numerical data that all have the same type it is often more efficient to use the array type provided by the numpy module A NumPy array is a chunk of memory containing fixed sized items With NumPy arrays operations on elements can be faster because elements are regularly spaced in memory and more operations are performed through specialized C functions instead of Python loops Python offers a large panel of functions to modify lists or query them Here are a few examples for more details see https docs python org tutorial datastructures html more on lists Add and re
196. eful to return slices on object in an image 17 4 2 Marker based methods If you have markers inside a set of regions you can use these to segment the regions 17 4 Image segmentation 316 Scipy lecture notes Edition 2015 2 Watershed segmentation The Watershed skimage morphology watershed is a region growing approach that fills basins in the image gt gt gt from skimage morphology import watershed gt gt gt from skimage feature import peak_local_max gt gt gt gt gt gt Generate an initial image with two overlapping circles gt gt gt x y np indices 80 80 gt gt gt x1 yl x2 y2 28 28 44 52 gt gt gt ri ro 20 gt gt gt mask_circle1 x x1 2 y yl 2 lt r1 2 gt gt gt mask_circle2 x x2 2 y y2 2 lt r2 2 gt gt gt image np logical_or mask_circle1 mask_circle2 gt gt gt Now we want to separate the two objects in image gt gt gt Generate the markers as local maxima of the distance gt gt gt to the background gt gt gt from scipy import ndimage gt gt gt distance ndimage distance_transform_edt image gt gt gt local_maxi peak_local_max distance indices False footprint np ones 3 3 labels image gt gt gt markers morphology label local_maxi gt gt gt labels_ws watershed distance markers mask image Random walker segmentation The random walker algorithm skimage segmentation ra
197. eir histogram is an estimator of the random process s PDF proba bility density function 5 6 Statistics and random numbers scipy stats 113 gt gt gt a np random normal size 1000 gt gt gt bins np arange 4 5 gt gt gt bins array 4 Sei a lle o a 2 3 a gt gt gt histogram np histogram a bins bins normed True 0 gt gt gt bins 0 5 bins 1 bins 1 gt gt gt bins array 3 5 25 15 05 05 15 225 35 gt gt gt from scipy import stats gt gt gt b stats norm pdf bins norm is a distribution gt gt gt plt plot bins histogram lt matplotlib lines Line2D object at gt gt gt gt plt plot bins b lt matplotlib lines Line2D object at gt 0 40 0 35 0 30 0 25 0 20 0 15 0 10 6 2 Scipy lecture notes Edition 2015 2 If we know that the random process belongs to a given family of random processes such as normal processes we can do a maximum likelihood fit of the observations to estimate the parameters of the underlying distri bution Here we fit a normal process to the observed data gt gt gt loc std stats norm fit a gt gt gt loc 0 0314345570 gt gt gt std 0 9778613090 Exercise Probability distributions Generate 1000 random variates from a gamma distribution with a shape parameter of 1 then plot a histogram from those samples Can you plot the pdf on top it should match Extra the distrib
198. eneric optimizer BFGS gt gt gt def g x return np sum f x 2 gt gt gt optimize fmin_bfgs g x0 Optimization terminated successfully Current function value 0 000000 Iterations 11 Function evaluations 144 Gradient evaluations 12 array 7 4 09 1 1 6701 2 2 e 01 oro 26 01 4 4 e 01 5 5 56 04 6 6 4 6 01 Uae Oe Oso ue 0i 130 2 00 BFGS needs more function calls and gives a less precise result leastsq is interesting compared to BFGS only if the dimensionality of the output vector is large and larger than the number of parameters to optimize If the function is linear this is a linear algebra problem and should be solved with scipy linalg 1stsq 13 4 Special case non linear least squares 260 Scipy lecture notes Edition 2015 2 13 4 2 Curve fitting 1 0 0 5 0 0 0 0 0 5 1 0 1 5 2 0 2 5 3 0 Least square problems occur often when fitting a non linear to data While it is possible to construct our optimization problem ourselves scipy provides a helper function for this purpose scipy optimize curve_fit gt gt gt def f t omega phi return np cos omega t phi gt gt gt x np linspace 0 3 50 gt gt gt y f x 1 5 1 1 np random normal size 50 gt gt gt optimize curve_fit f x y array 1 51854577 0 92665541 array L 0 00037994 0 00056796 0 00056796 0 00123972e Exercise Do the same with omega
199. ent frame with the same name use different names than your local variable when typing code in the debugger Getting help when in the debugger Type h or help to access the interactive help ipdb gt help Documented commands type help lt topic gt EOF bt cont enable jump pdef r tbreak w a C continue exit 1 pdoc restart u whatis alias cl d h list pinfo return unalias where args clear debug help n pp run unt b commands disable ignore next q S until break condition down j p quit step up Miscellaneous help topics retval rv 9 4 Debugging segmentation faults using gdb If you have a segmentation fault you cannot debug it with pdb as it crashes the Python interpreter before it can drop in the debugger Similarly if you have a bug in C code embedded in Python pdb is useless For this we turn to the gnu debugger gdb available on Linux Before we start with gdb let us add a few Python specific tools to it For this we add a few macros to our gbdinit The optimal choice of macro depends on your Python version and your gdb version I have added a simplified version in gdbinit but feel free to read DebuggingWithGdb 9 4 Debugging segmentation faults using gdb 200 Scipy lecture notes Edition 2015 2 To debug with gdb the Python script segfault py we can run the script in gdb as follows gdb python gdb run segfault py Starting program usr bin python segfault py Thread debugging using libthread_db enabled
200. ent variable with 2 variables x and y Z XC ycot it e Such a model can be seen in 3D as fitting a plane to a cloud of x y z points 15 3 Linear models multiple factors and analysis of variance 295 Scipy lecture notes Edition 2015 2 Example the iris data examples iris csv Sepal and petal size tend to be related bigger flowers are bigger But is there in addition a systematic effect of species blue setosa green versicolor red virginica o sepal_width sepal_length p5 2 4 6 5 2a 7 3 Q 2 24 2 0 pe S15 10 Ea V os nononono A O NA O A N M F N O Fe A O A Q n n N g O KNK KM w N mi m N N tal length sepal_length sepal_width perareng petal width gt gt gt data pandas read_csv examples iris csv gt gt gt model ols sepal_width name petal_length data fit gt gt gt print model summary OLS Regression Results Dep Variable sepal_width R squared 0 478 Model OLS Adj R squared 0 468 Method Least Squares F statistic 44 63 Date Sec Prob F statistic 1 58e 20 Time TE Log Likelihood 38 185 No Observations AIC 84 37 Df Residuals 96 41 Df Model 95 0 Conf Int Intercept 29 989 2 185 1 8 name T versicolor 1 4821 8 190 1 840 124 name T virginica 1 6635 6 502 2 169 ES petal_length 4 920 0 178 418 Omnibus Durbin Watson Prob Omnibus Jarque Bera JB Skew Prob JB Kurtosis Cond No
201. entation and interaction 287 Scipy lecture notes Edition 2015 2 Exercise e What is the mean value for VIQ for the full population e How many males females were included in this study Hint use tab completion to find out the methods that can be called instead of mean in the above example e What is the average value of MRI counts expressed in log units for males and females groupby_gender boxplot is used for the plots above see this example Plotting data Pandas comes with some plotting tools pandas tools plotting using matplotlib behind the scene to dis play statistics of the data in dataframes Scatter matrices gt gt gt from pandas tools import plotting gt gt gt plotting scatter_matrix datal Weight Height MRI_Count 180 160 140 Weight Ao 62 1050000 1000000 950000 900000 MRI Count 850000 800000 120 140 160 180 6 800000 850000 900000 950000 1000000 1050000 gt gt gt plotting scatter_matrix dataL PIQ VIQ FSIQ Two populations The IQ metrics are bimodal as if there are 2 sub populations 15 1 Data representation and interaction 288 Scipy lecture notes Edition 2015 2 150 140 130 120 oO om 110 aA 100 90 80 150 140 130 120 g 110 100 90 80 70 140 130 120 o 110 Ww 100 90 80 Reseegesesessgeses 8B B amp B a 4 A A S amp F e amp PIQ e Exercise
202. ents And as before we convince ourselves that it worked import numpy as np import pylab import cos_doubles x np arange O 2 np pi 0 1 y np empty_like x cos_doubles cos_doubles_func x y pylab plot x y pylab show 14 3 Ctypes 271 Scipy lecture notes Edition 2015 2 14 4 SWIG SWIG the Simplified Wrapper Interface Generator is a software development tool that connects programs written in C and C with a variety of high level programming languages including Python The important thing with SWIG is that it can autogenerate the wrapper code for you While this is an advantage in terms of development time it can also be a burden The generated file tend to be quite large and may not be too human readable and the multiple levels of indirection which are a result of the wrapping process may be a bit tricky to understand The autogenerated C code uses the Python C Api Advantages e Can automatically wrap entire libraries given the headers e Works nicely with C Disadvantages e Autogenerates enormous files e Hard to debug if something goes wrong e Steep learning curve 14 4 1 Example Let s imagine that our cos function lives in a cos_module which has been written in c and consists of the source file cos_module c Hinclude lt math h gt double cos_func double arg return cos arg and the header file cos_module h double cos_func double arg And our goal is to expose
203. er right corner But you can close a figure programmatically by calling close Depending on the argument it closes 1 the current figure no argument 2 a specific figure figure number or figure instance as argument or 3 all figures al1 as argument plt close 1 Closes figure 1 4 3 2 Subplots With subplot you can arrange plots in a regular grid You need to specify the number of rows and columns and the number of the plot Note that the gridspec command is a more powerful alternative Axes 1 subplot 2 1 1 subplot 2 2 1 subplot 2 2 2 subplot 1 2 1 subplot 1 2 2 Axes 2 Axes 3 subplot 2 1 2 subplot 2 2 3 subplot 2 2 4 Axes 4 Axes 5 4 3 3 Axes Axes are very similar to subplots but allow placement of plots at any location in the fig ure So if we want to put a smaller plot inside a bigger one we do so with axes 4 3 Figures Subplots Axes and Ticks 89 Scipy lecture notes Edition 2015 2 axes 0 1 0 1 8 8 axes 0 4 0 4 5 5 xes 0 2 0 2 3 3 axes 0 3 0 3 5 5 axes 0 2 0 2 5 5 axes 0 1 0 1 8 8 4 3 4 Ticks Well formatted ticks are an important part of publishing ready figures Matplotlib provides a totally config urable system for ticks There are tick locators to specify where ticks should appear and tick formatters to give ticks the appearance you want Major and mi
204. ervals and triangles to high dimensional spaces to bracket the minimum Strong points it is robust to noise as it does not rely on computing gradients Thus it can work on functions that are not locally smooth such as experimental data points as long as they display a large scale bell shape behavior However it is slower than gradient based methods on smooth non noisy functions iterations function calls Error on f x e e o 50 100 150 200 250 An ill conditionned non quadratic function iterations function calls Error on f x 150 200 250 An ill conditionned very non quadratic function In scipy scipy optimize fmin implements the Nelder Mead approach gt gt gt def f x The rosenbrock function return 5 1 x 0 2 x 1 x 0 2 2 gt gt gt optimize fmin f 2 2 Optimization terminated successfully Current function value 0 000000 Iterations 46 Function evaluations 91 array 0 99998568 0 99996682 13 2 A review of the different optimizers 257 Scipy lecture notes Edition 2015 2 13 2 5 Global optimizers If your problem does not admit a unique local minimum which can be hard to test unless the function is convex and you do not have prior information to initialize the optimization close to the solution you may need a global optimizer Brute force a grid search scipy optimize brute evaluates the function on a given grid of parameters a
205. es measure regionprops labels_rw gt gt gt prop area for prop in properties 770 0 1168 0 gt gt gt prop perimeter for prop in properties MOOr Ole eset 26261 See also for some properties functions are available as well in scipy ndimage measurements with a different API a list is returned Exercise continued e Use the binary image of the coins and background from the previous exercise e Compute an image of labels for the different coins e Compute the size and eccentricity of all coins 17 6 Data visualization and interaction Meaningful visualizations are useful when testing a given processing pipeline Some image processing operations gt gt gt coins data coins gt gt gt mask coins gt filters threshold_otsu coins gt gt gt clean_border segmentation clear_border mask Visualize binary result gt gt gt plt figure lt matplotiib tipume Figure object at 0x gt gt gt gt plt imshow clean_border cmap gray lt matplotlib image AxesImage object at Ox gt Visualize contour gt gt gt plt figure lt matplotiib figure Figure object at 0x gt gt gt gt plt imshow coins cmap gray lt matplotlib image AxesImage object at Ox gt gt gt gt plt contour clean_border 0 5 lt matplotiib contour QuadContourset gt 17 5 Measuring regions properties 318 Use skimage dedicated utility function Scipy lecture notes Edition 2015 2 gt
206. es np array 0 2 2 0 1 2 gt gt gt indptr np array LO 2 3 6 gt gt gt mtx sparse csr_matrix data indices indptr shape 3 3 gt gt gt mtx todense MatraxChilt 0 2 O On a3 4 5 ei Compressed Sparse Column Format CSC e column oriented three NumPy arrays indices indptr data indices is array of row indices data is array of corresponding nonzero values indptr points to column starts in indices and data lengthisn_col 1 last item number of values length of both indices and data nonzero values of the i th column are data indptr i indptr lit 1 with row in dices indices Lindptr Li indptr i 1 item i j can be accessed as dataLlindptr j k where k is position of i in indicesLindptr j indptr jt 1 subclass of _cs_matrix common CSR CSC functionality subclass of _data_matrix sparse matrix classes with data attribute e fast matrix vector products and other arithmetics sparsetools e constructor accepts dense matrix array sparse matrix shape tuple create empty matrix data ij tuple data indices indptr tuple e efficient column slicing column oriented operations e slow row slicing expensive changes to the sparsity structure e use actual computations most linear solvers support this format 11 2 Storage Schemes 220 Scipy lecture notes Edition 2015 2 Examples e create empty CSC matrix gt gt gt mtx
207. es Edition 2015 2 state update_storage True state print_state Dependency between objects can be made automatic using the trait Property The depends_on attribute expresses the dependency between the property and other traits When the other traits gets changed the property is invalidated Again Traits uses magic method names for the property e _get_XXX for the getter of the XXX Property trait e _set_XXX for the setter of the XXX Property trait from traits api import HasTraits Instance DelegatesTo Float Range from traits api import Property from reservoir import Reservoir class ReservoirState HasTraits Keeps track of the reservoir state given the initial storage For the simplicity of the example the release ts considered in hm3 timestep and not in m3 s reservoir Instance Reservoir max_storage DelegatesTo reservoir min release Float max_release DelegatesTo reservoir state attributes storage Property depends_on inflows release control attributes inflows Float desc Inflows hm3 release Range low min_release high max_release spillage Property desc Spillage hm3 depends_on storage inflows release Private traits _storage Float Traits property implementation def _get_storage self new_storage self _storage self release self inflows return min new_storage self max_storage def _set_storage self storage_value
208. es tools to understand better your code base debugging to find and fix bugs It is not specific to the scientific Python community but the strategies that we will employ are tailored to its needs Prerequisites e Numpy e Python e nosetests http readthedocs org docs nose en latest e pyflakes http pypi python org pypi pyflakes e gdb for the C debugging part Chapter contents e Avoiding bugs Coding best practices to avoid getting in trouble pyflakes fast static analysis e Debugging workflow e Using the Python debugger Invoking the debugger Debugger commands and interaction e Debugging segmentation faults using gdb 9 1 Avoiding bugs 9 1 1 Coding best practices to avoid getting in trouble Brian Kernighan Everyone knows that debugging is twice as hard as writing a program in the first place So if you re as clever as you can be when you write it how will you ever debug it e We all write buggy code Accept it Deal with it e Write your code with testing and debugging in mind e Keep It Simple Stupid KISS 192 Scipy lecture notes Edition 2015 2 What is the simplest thing that could possibly work e Don t Repeat Yourself DRY Every piece of knowledge must have a single unambiguous authoritative representation within a system Constants algorithms etc e Try to limit interdependencies of your code Loose Coupling e Give your variables functions and m
209. est arguments A Don t implement option parsing yourself Use modules such as optparse argparse or docopt 2 5 2 Importing objects from modules In 1 import os In 2 os Out 2 lt module os from usr lib python2 6 os pyc gt In 3 oc listdir Mat Su conf py os basic_types rst control_flow rst functions rst python_language rst reusing rst file_io rst exceptions rst workflow rst index rst And also In 4 from os import listdir Importing shorthands In 5 import numpy as np 2 5 Reusing code scripts and modules 28 Scipy lecture notes Edition 2015 2 A from os import This is called the star import and please Use it with caution e Makes the code harder to read and understand where do symbols come from e Makes it impossible to guess the functionality by the context and the name hint os name is the name of the OS and to profit usefully from tab completion e Restricts the variable names you can use os name might override name or vise versa e Creates possible name clashes between modules e Makes the code impossible to statically check for undefined symbols Modules are thus a good way to organize code in a hierarchical way Actually all the scientific computing tools we are going to use are modules gt gt gt import numpy as np data arrays gt gt gt np linspace 0O 10 6 array CL 0 Zex an 6 Se TO D gt gt gt import scipy
210. est_ind female_viq male_viq vee ON 72611617232 244 0 4445287677858 15 2 Hypothesis testing comparing two groups 290 Scipy lecture notes Edition 2015 2 15 2 2 Paired tests repeated measurements on the same indivuals 150 140 130 120 110 100 90 80 70 FSIQ PIQ PIQ VIQ and FSIQ give 3 measures of IQ Let us test if FISQ and PIQ are significantly different We can use a 2 sample test gt gt gt stats ttest_ind data FSIQ data PIQ 0 46563759638 0 64277250 The problem with this approach is that it forgets that there are links between observations FSIQ and PIQ are measured on the same individuals Thus the variance due to inter subject variability is confounding and can be removed using a paired test or repeated measures test gt gt gt stats ttest_rel data FSIQ data PIQ 1 784201940 0 082172638183 20 15 10 ie Relies This is equivalent to a 1 sample test on the difference gt gt gt stats ttest_1samp data FSIQ data PIQ 0 1 784201940 0 082172638 T tests assume Gaussian errors We can use a Wilcoxon signed rank test that relaxes this assumption gt gt gt stats wilcoxon data FSIQ data PIQ 274 5 0 106594927 The corresponding test in the non paired case is the Mann Whitney U test scipy stats mannwhitneyu Exercice e Test the difference between weights in males and females e Use non para
211. estimator chose automatically its lambda gt gt gt lasso alpha_ OnO12 These estimators are called similarly to their counterparts with CV appended to their name Exercise On the diabetes dataset find the optimal regularization parameter alpha 20 7 Model selection choosing estimators and their parameters 362 D diff 304 307 differentiation 304 dsolve 307 E equations algebraic 305 differential 307 integration 305 M Matrix 306 P Python Enhancement Proposals PEP 255 145 PEP 3118 182 PEP 3129 155 PEP 318 147 155 PEP 342 145 PEP 343 155 PEP 380 147 PEP 380 id13 147 PFP 8 150 S solve 305 Index 363
212. etically and with concern that it might not be that useful Bruce Eckel An Introduction to Python Decorators Since a function or a class are objects they can be passed around Since they are mutable objects they can be modified The act of altering a function or class object after it has been constructed but before is is bound to its name is called decorating There are two things hiding behind the name decorator one is the function which does the work of deco rating i e performs the real work and the other one is the expression adhering to the decorator syntax i e an at symbol and the name of the decorating function Function can be decorated by using the decorator syntax for functions decorator def function 0 pass e A function is defined in the standard way e An expression starting with placed before the function definition is the decorator The part after must be a simple expression usually this is just the name of a function or class This part is evaluated first and after the function defined below is ready the decorator is called with the newly defined function object as the single argument The value returned by the decorator is attached to the original name of the function Decorators can be applied to functions and to classes For classes the semantics are identical the original class definition is used as an argument to call the decorator and whatever is returned is assigned under t
213. ey For instance an eigenvalue problem with a symmetric matrix is easier to solve than with a general matrix Also most often you can avoid inverting a matrix and use a less costly and more numerically stable operation Know your computational linear algebra When in doubt explore scipy linalg and use timeit to try out different alternatives on your data 10 4 Writing faster numerical code A complete discussion on advanced use of numpy is found in chapter Advanced Numpy or in the article The NumPy array a structure for efficient numerical computation by van der Walt et al Here we discuss only some commonly encountered tricks to make code faster e Vectorizing for loops Find tricks to avoid for loops using numpy arrays For this masks and indices arrays can be useful e Broadcasting Use broadcasting to do operations on arrays as small as possible before combining them e In place operations In 1 a np zeros 1e7 In 2 timeit global a a Oxa 10 loops best of 3 111 ms per Loop 10 4 Writing faster numerical code 207 Scipy lecture notes Edition 2015 2 In 3 timeit global a a 0 10 loops best of 3 48 4 ms per loop note we need global a in the timeit so that it work as it is assigning to a and thus considers it as a local variable e Be easy on the memory use views and not copies Copying big arrays is as costly as making simple numerical operations on them In 1 a np zeros 1e7 In 2
214. face on linked to a given reservoir from traits api import HasTraits Str Float Range Enum List Property from traitsui api import View Item class IrrigationArea HasTraits name Str surface Float desc Surface ha crop Enum Alfalfa Wheat Cotton class Reservoir HasTraits name Str max_storage Float 1e6 desc Maximal storage hm3 max_release Float 10 desc Maximal release m3 s head Float 10 desc Hydraulic head m efficiency Range 0O 1 irrigated_areas List IrrigationArea total_crop_surface Property depends_on irrigated_areas surface def _get_total_crop_surface self return sum iarea surface for iarea in self irrigated_areas def energy_production self release Returns the energy production Wh for the given release m3 s or power 1000 9 81 self head release self efficiency return power 3600 18 3 What are Traits 337 Scipy lecture notes Edition 2015 2 traits view View Item name Item max_storage Item max_release Item head Item efficiency Item irrigated_areas Item total_crop_surface resizable True __name main _ upper_block IrrigationArea name Section C surface 2000 crop Wheat reservoir Reservoir name Project A max_storage 30 max_release 100 0 head 60 efficiency 0 8 irrigated_areas upper_block release 80 print Releasing m3 s produces kWh
215. filled with a given value to get an usual array back gt gt gt x2 mx filled 1 gt gt gt x2 array de 95 gt 8 1 5l The mask can also be cleared gt gt gt mx mask np ma nomask gt gt gt mx masked_array data i 9 3 99 5 mask False False False False False fill_value 999999 Domain aware functions The masked array package also contains domain aware functions gt gt gt np ma log np array 1 2 1 2 3 5 masked_array data 0 0 0 6931471805599453 1 0986122886681098 mask False False True True False Truel fill_value 1e 20 Streamlined and more seamless support for dealing with missing data in arrays is making its way into Numpy 1 7 Stay tuned 8 4 Array siblings chararray maskedarray matrix 186 Scipy lecture notes Edition 2015 2 Example Masked statistics Canadian rangers were distracted when counting hares and lynxes in 1903 1910 and 1917 1918 and got the numbers are wrong Carrot farmers stayed alert though Compute the mean populations over time ignoring the invalid numbers gt gt gt data np loadtxt data populations txt gt gt gt populations np ma masked_array datalL 1 gt gt gt year datal 0 gt gt gt bad_years year gt 1903 amp year lt 1910 Cyear gt 1917 amp year lt 1918 gt gt gt amp means and and means or gt gt gt populations bad_years 0 np ma masked gt
216. filtering Local filters replace the value of pixels by a function of the values of neighboring pixels Neighbourhood square choose size disk or more complicated structuring element Ed 9 15 is 12 4 Image filtering 235 Scipy lecture notes Edition 2015 2 12 4 1 Blurring smoothing Gaussian filter from scipy ndimage gt gt gt from scipy import misc gt gt gt face misc face gray True gt gt gt blurred_face ndimage gaussian_filter face sigma 3 gt gt gt very_blurred ndimage gaussian_filter face sigma 5 Uniform filter gt gt gt local_mean ndimage uniform_filter face size 11 gt m f Sle 12 4 2 Sharpening Sharpen a blurred image gt gt gt from scipy import misc gt gt gt face misc face gray True astype float gt gt gt blurred_f ndimage gaussian_filter face 3 increase the weight of edges by adding an approximation of the Laplacian gt gt gt filter_blurred_f ndimage gaussian_filter blurred_f 1 gt gt gt alpha 30 gt gt gt sharpened blurred_f alpha blurred_f filter_blurred_f 12 4 3 Denoising Noisy face gt gt gt from scipy import misc gt gt gt f misc face gray True gt gt gt f f 230 290 220 320 gt gt gt noisy f 0 4 f std np random random f shape 12 4 Image filtering 230 Scipy lecture notes Edition 2015 2 A Gaussian filter smoothes the noise out and the edges
217. finally obj close Let s rewrite the assert_raises example as a generator contextlib contextmanager def assert_raises type try yield except type 7 3 Context managers 157 Scipy lecture notes Edition 2015 2 return except Exception as value raise AssertionError wrong exception type else raise AssertionError exception expected Here we use a decorator to turn generator functions into context managers 7 3 Context managers 158 CHAPTER 8 Advanced Numpy Author Pauli Virtanen Numpy is at the base of Python s scientific stack of tools Its purpose to implement efficient operations on many items in a block of memory Understanding how it works in detail helps in making efficient use of its flexibility taking useful shortcuts This section covers e Anatomy of Numpy arrays and its consequences Tips and tricks e Universal functions what why and what to do if you want a new one e Integration with other tools Numpy offers several ways to wrap any data in an ndarray without unnec essary copies e Recently added features and what s in them PEP 3118 buffers generalized ufuncs Prerequisites e Numpy e Cython e Pillow Python imaging library used in a couple of examples 159 Scipy lecture notes Edition 2015 2 Chapter contents e Life of ndarray It s Block of memory Data types Indexing scheme strides Findings in dissection e Universal functions
218. for historical reasons only there s usually no reason to use import scipy in your code 5 1 File input output scipy io e Loading and saving matlab files gt gt gt from scipy import io as spio gt gt gt a np ones 3 3 gt gt gt spio savemat file mat a a savemat expects a dictionary gt gt gt data spio loadmat file mat struct_as_record True gt gt gt data a array Clb die has de Pos Taa bale ha es to e Reading images gt gt gt from scipy import misc gt gt gt misc imread fname png array gt gt gt Matplotlib also has a similar function gt gt gt import matplotlib pyplot as plt gt gt gt plt imread fname png array See also e Load text files numpy loadtxt numpy savetxt e Clever loading of text csv files numpy genfromtxt numpy recfromcsv e Fast and efficient but numpy specific binary format numpy save numpy load 5 1 File input output scipy io 102 Scipy lecture notes Edition 2015 2 5 2 Special functions scipy special Special functions are transcendental functions The docstring of the scipy special module is well written so we wont list all functions here Frequently used ones are e Bessel function such as scipy special jn nth integer order Bessel function e Elliptic function scipy special ellipj for the Jacobian elliptic function e Gamma function scipy special gamma also no
219. for small arrays because of buffering but fail for large one in unpredictable ways Linear algebra The sub module numpy linalg implements basic linear algebra such as solving linear systems singular value decomposition etc However it is not guaranteed to be compiled using efficient routines and thus we recommend the use of scipy linalg as detailed in section Linear algebra operations scipy linalg Exercise other operations e Look at the help for np allclose When might this be useful e Look at the help fornp triuandnp tril 3 2 2 Basic reductions Computing sums gt gt gt x np array 1 2 3 4 gt gt gt np sum x 10 gt gt gt x sum 10 3 2 Numerical operations on arrays 57 Scipy lecture notes Edition 2015 2 axis 1 axis O Sum by rows and by columns gt gt gt x np array UPL P2 21 Por X array CULT 11 2 2D gt gt gt x sum axis 0 columns first dimension array 3 3 gt gt gt x O sum x 1 sum 3 3 gt gt gt x sum axis 1 rows second dimension array 2 4 gt gt gt x 0 sumQ x 1 sun0 ae Same idea in higher dimensions gt gt gt x np random rand 2 2 2 gt gt gt x sum axis 2 0 1 1 14764 gt gt gt x 0 1 sum 1 14764 Other reductions works the same way and take axis Extrema gt gt gt x np array 1 3 2 gt gt gt x min 1 gt gt gt x max 3 gt gt gt x
220. format release reservoir energy_production release The next example shows how the Array trait can be used to feed a specialised TraitsUI Item the ChacoPlotItem import numpy as np from traits api import HasTraits Array Instance Float Property from traits api import DelegatesTo from traitsui api import View Item Group from chaco chaco_plot_editor import ChacoPlotItem from reservoir import Reservoir class ReservoirEvolution HasTraits reservoir Instance Reservoir name DelegatesTo reservoir inflows Array dtype np float64 shape None releass Array dtype np float64 shape None initial_stock Float stock Property depends_on inflows releases initial_stock month Property depends_on stock HHH Traits view H FFAHFAFAA traits_view View Item name Group ChacoPlotiItem month stock show_label False J width 500 resizable True 18 3 What are Traits 338 Scipy lecture notes Edition 2015 2 Traits properties H def _get_stock self moe fixme should handle cases where we go over the max storage nnnm return self initial_stock self inflows self releases cumsum _get_month self return np arange self stock size if name _main_ reservoir Reservoir name Project A max_storage 30 max release 100 0 head 60 efficiency 0 8 initial_stock 10 inflows ts np array 6 6 4 4 1 2 0 0 3 1 5 3J
221. ge 121 Scipy lecture notes Edition 2015 2 0 0 0 0G 0 0 Ol On 005 0 0 00 gt gt gt Erosion removes objects smaller than the structure gt gt gt ndimage binary_erosion a structure np ones 5 5 astype a dtype array TOO OR CAs C0 52 Ol On 0 0540502 On a LOR 205 0 One On Os 0l Om Oe Oe 0 O LOns Ol 0 0 00 80 10 0 0 0 0 0 0 0 0 IO Os 00 00 OID e Dilation gt gt gt a np zeros 5 5 ooo all 2 1 gt gt gt a array l 0 0 5 0u Oe OF Ot Ole Okan Ora Od POs WO ea O O Olek JO 0e O Ome l O Ora Oke Oea T gt gt gt ndimage binary_dilation a astype a dtype array L 0 0a Ora Wage JO Ose Oea Tg Ora l E Ona ea dite lt i od EO Oa es COL o 20 5 Oen Oe Ox 0n 10 e Opening gt gt gt a np zeros 5 gt gt gt al1 4 1 4 1 gt gt gt al4 4 1 5 dtype np int gt gt gt a array CULO 0 10 0 0 O a aa o Oa he Tero Od de ol O oro O 14 gt gt gt Opening remove gt gt gt ndimage binary_opening a structure np ones 3 3 astype np int array L0 O O LO LO 0 OTi 1 1 s small objects Our 0l oli CHIE iy le 1E 1 Rae On 0 050 oD gt gt gt Opening can also smooth corners gt gt gt ndimage binary_opening a astype np int array LOO 0 0 0 EO 0 13 00 POr le dee ta Ole LO Oe Cie Oe Ol Oz 0 0 10 101 15 e Closing ndimage binary_closing Exercise Check
222. gt gt coins_edges segmentation mark_boundaries coins clean_border astype np int perimental scikit image viewer The ex skimage viewer matplotlib based canvas for displaying images experimental Qt based GUI toolkit gt gt gt from skimage import viewer gt gt gt new_viewer viewer ImageViewer coins gt gt gt new_viewer show Useful for displaying pixel values For more interaction plugins can be added to the viewer gt gt gt new_viewer viewer ImageViewer coins gt gt gt from skimage viewer plugins import lineprofile gt gt gt new_viewer lineprofile LineProfile gt gt gt new_viewer show 17 6 Data visualization and interaction 319 Scipy lecture notes Edition 2015 2 290 200 150 100 30 0 50 100 150 200 Image Viewer X File 28 160 231 17 7 Feature extraction for computer vision Geometric or textural descriptor can be extracted from images in order to e classify parts of the image e g sky vs buildings e match parts of different images e g for object detection e and many other applications of Computer Vision 17 7 Feature extraction for computer vision 320 Scipy lecture notes Edition 2015 2 gt gt gt from skimage import feature Example detecting corners using Harris detector from skimage feature import corner_harris corner_subpix corner_peaks from skimage transform import warp AffineTransform tform AffineTransform scale
223. gt gt col mp array 0 2 2 0 1 21 gt gt gt data np array 1 2 3 4 5 6 gt gt gt mtx sparse bsr_matrix data row col shape 3 3 gt gt gt mtx lt 3x3 sparse matrix of type lt numpy int64 gt with 6 stored elements blocksize 1x1 in Block Sparse Row format gt gt gt gt mtx todense matriz in eee Ihr OE E 4 Sy GI gt gt gt mtx data array 1 E LIne 4 LESIE 11 2 Storage Schemes 222 Scipy lecture notes Edition 2015 2 Heill gt gt gt mtx indices array 10 2 2 0 1 2 dtype int32 gt gt gt mtx indptr array 0 2 3 6 dtype int32 e create using data indices indptr tuple with 2 2 block size gt gt gt indptr np array 0 2 3 6 gt gt gt indices np array 0 2 2 O 1 2 gt gt gt data np array 1 2 3 4 5 6 repeat 4 reshape 6 2 2 gt gt gt mtx sparse bsr_matrix data indices indptr shape 6 6 gt gt gt mtx todense matrix 1l 1 0 PEN EO lOO 4 4 2 oe Doe 2l By Bilis Sro 6 6 v 2 2 v 2 2 4 4 gt gt gt data array ICL 11 EERIE 2 oll poll 2 2 v GOOO oao aooo O 2 2 v Ue Er 22 TIES ss ile Toe lle 4 4 4 4 She pile 1S Elle Hog Heep 11 2 3 Summary Table 11 1 Summary of storage schemes matrix sparse conversion has data array fast row wise ops has data array fast column wise ops
224. gt gt from scipy import misc gt gt gt face misc face gray True 2D grayscale image Here are a few images we will be able to obtain with our manipulations use different colormaps crop the image change some parts of the image e Lets use the imshow function of pylab to display the image gt gt gt import pylab as plt gt gt gt face misc face gray True gt gt gt plt imshow face lt matplotlib image AxesImage object at Ox gt 3 5 Some exercises 77 Scipy lecture notes Edition 2015 2 e The face is displayed in false colors Acolormap must be specified for it to be displayed in grey gt gt gt plt imshow face cmap plt cm gray lt matplotlib image AxesImage object at Ox gt e Create an array of the image with a narrower centering for example remove 100 pixels from all the borders of the image To check the result display this new array with imshow gt gt gt crop_face face 100 100 100 100 e We will now frame the face with a black locket For this we need to create a mask corresponding to the pixels we want to be black The center of the face is around 660 330 so we defined the mask by this condition y 300 2 x 660 2 gt gt gt sy sx face shape gt gt gt y x np ogrid 0O sy O sx x and y indices of pizels gt gt gt y shape x shape Cee 1 1022 gt gt gt centerx centery 660 300 center of the image gt gt gt mask y centery
225. gt gt square 10 10 10 10 1 gt gt gt np random seed 2 gt gt gt x y 32 np random random 2 20 astype np int gt gt gt square x y 1 gt gt gt open_square ndimage binary_opening square gt gt gt eroded_square ndimage binary_erosion square Scipy lecture notes Edition 2015 2 gt gt gt reconstruction ndimage binary_propagation eroded_square mask square Closing dilation erosion Many other mathematical morphology operations hit and miss transform tophat etc 12 5 Feature extraction 12 5 1 Edge detection Synthetic data gt gt gt im np zeros 256 256 gt gt gt im 64 64 64 64 1 gt gt gt gt gt gt im ndimage rotate im 15 mode constant gt gt gt im ndimage gaussian_filter im 8 Use a gradient operator Sobel to find high intensity variations gt gt gt sx ndimage sobel im axis 0 mode constant gt gt gt sy ndimage sobel im axis 1 mode constant gt gt gt sob np hypot sx sy square Sobel x direction Sobel filter 12 5 Feature extraction Sobel for noisy image 240 Scipy lecture notes Edition 2015 2 12 5 2 Segmentation e Histogram based segmentation no spatial information gt gt gt n 10 gt gt gt 1 256 gt gt gt im np zeros 1 1 gt gt gt np random seed 1 gt gt gt points 1 np random random 2 n 2 gt gt gt imL points 0 astype np int poi
226. guage with dynamic typing the same variable can contain objects of different types during the course of a program See https www python org about for more information about distinguishing features of Python 2 1 First steps Start the Ipython shell an enhanced interactive Python shell e by typing ipython from a Linux Mac terminal or from the Windows cmd shell 10 Scipy lecture notes Edition 2015 2 e or by starting the program from a menu e g in the Python x y or EPD menu if you have installed one of these scientific Python suites If you don t have Ipython installed on your computer other Python shells are available such as the plain Python shell started by typing python in a terminal or the Idle interpreter However we advise to use the Ipython shell because of its enhanced features especially for interactive scientific computing Once you have started the interpreter type gt gt gt print Hello world Hello world The message Hello world is then displayed You just executed your first Python instruction congratulations To get yourself started type the following stack of instructions gt gt gt a 3 gt gt gt b 2 a gt gt gt type b lt type int gt gt gt gt print b 6 gt gt gt ax b 18 gt gt gt b hello gt gt gt type b lt type str gt gt gt gt b b hellohello gt gt gt 2 b hellohello Two variables a and b have
227. hapters contents e First Steps with SymPy Using SymPy as a calculator Exercises Symbols e Algebraic manipulations Expand Simplify e Calculus Limits Differentiation Series expansion Integration Exercises e Equation solving Exercises e Linear Algebra Matrices Differential Equations 301 Scipy lecture notes Edition 2015 2 16 1 First Steps with SymPy 16 1 1 Using SymPy as a calculator SymPy defines three numerical types Real Rational and Integer The Rational class represents a rational number as a pair of two Integers the numerator and the denominator so Rational 1 2 represents 1 2 Rational 5 2 5 2 and so on gt gt gt from sympy import gt gt gt a Rational 1 2 gt gt gt a 1 2 gt gt gt ax2 1 SymPy uses mpmath in the background which makes it possible to perform computations using arbitrary precision arithmetic That way some special constants like e pi oo Infinity are treated as symbols and can be evaluated with arbitrary precision gt gt gt pi 2 2 pi gt gt gt pi evalf 3 14159265358979 gt gt gt pi exp 1 evalf 5 85987448204884 as you see evalf evaluates the expression to a floating point number There is also a class representing mathematical infinity called oo gt gt gt oo gt 99999 True gt gt gt oo 1 OO 10 1 2 Exercises 1 Calculate v2 with 100 decimals 2 Calculate 1 2 1 3 i
228. he original name Before the decorator syntax was implemented PEP 318 it was possible to achieve the same effect by assign ing the function or class object to a temporary variable and then invoking the decorator explicitly and then assigning the return value to the name of the function This sounds like more typing and it is and also the 7 2 Decorators 147 Scipy lecture notes Edition 2015 2 name of the decorated function doubling as a temporary variable must be used at least three times which is prone to errors Nevertheless the example above is equivalent to def function 0 pass function decorator function Decorators can be stacked the order of application is bottom to top or inside out The semantics are such that the originally defined function is used as an argument for the first decorator whatever is returned by the first decorator is used as an argument for the second decorator and whatever is returned by the last decorator is attached under the name of the original function The decorator syntax was chosen for its readability Since the decorator is specified before the header of the function it is obvious that its is not a part of the function body and its clear that it can only operate on the whole function Because the expression is prefixed with is stands out and is hard to miss in your face according to the PEP When more than one decorator is applied each one is placed on a separ
229. he filesystem with IPython In 1 pwd show current directory home user stuff 2011 numpy tutorial In 2 cd ex home user stuff 2011 numpy tutorial ex Ia aie Is populations txt species txt Images Using Matplotlib gt gt gt img plt imread data elephant png gt gt gt img shape img dtype 200 300 3 dtypeC float32 gt gt gt plt imshow img lt matplotlib image AxesImage object at gt gt gt gt plt savefig plot png gt gt gt plt imsave red_elephant imgL 0 cmap plt cm gray This saved only one channel of RGB gt gt gt plt imshow plt imread red_elephant png lt matplotlib image AxesImage object at gt Other libraries gt gt gt imsave tiny_elephant png imgL 6 6 Scipy lecture notes Edition 2015 2 gt gt gt plt imshow plt imread tiny_elephant png interpolation nearest lt matplotlib image AxesImage object at gt 3 4 Advanced operations 15 Scipy lecture notes Edition 2015 2 es mare aii s 0 10 20 30 40 Numpy s own format Numpy has its own binary format not portable but with efficient I O gt gt gt data np ones 3 3 gt gt gt np save pop npy data gt gt gt data3 np load pop npy Well known amp more obscure file formats e HDF5 h5py PyTables e NetCDF scipy io netcdf_file netcdf4 python e Matlab scipy io loadmat scipy io savemat e MatrixMarket scipy io mmread
230. he latter is installed as follows prepare a linear system gt gt gt import numpy as np gt gt gt from scipy import sparse gt gt gt mtx sparse spdiags 1 2 3 4 5 6 5 8 9 10 10 1 5 5 gt gt gt mtx todense matriz ll ae 5 0 0 7 0l LOr 22 26 Oe Ols LO Ore a 9a O LO Ons 0 A o Os O O 0 E gt gt gt rhs np array 1 2 3 4 5 dtype np float32 11 3 Linear System Solvers 224 Scipy lecture notes Edition 2015 2 solve as single precision real gt gt gt mtx1 mtx astype np float32 gt gt gt x dsolve spsolve mtx1 rhs use_umfpack False gt gt gt print x 106 21 5 5 1 5 E gt gt gt print Error 7s 4 itxl x rhs Error 0 0 0 0 0 solve as double precision real gt gt gt mtx2 mtx astype np float64 gt gt gt x dsolve spsolve mtx2 rhs use_umfpack True gt gt gt print x P06 Daal ee gt 27 print Error 3 ntx2 x rhs Error WO On 705 sO g solve as single precision complex gt gt gt mtxi mtx astype np complex64 gt gt gt x dsolve spsolve mtx1 rhs use_umfpack False gt gt gt print x 106 00 J 24 0707 S SrO EO 1 0 0 j gt gt gt print Error Zs mtxi x rhs Error TO tO yO stOe 00 00 Oral solve as double precision complex gt gt gt mtx2 mtx astype np complexi28 gt gt gt x dsolve spsolve mtx2 rhs use_umfpack True
231. he root of all evil Author Ga l Varoquaux This chapter deals with strategies to make Python code go faster Prerequisites e line_profiler Chapters contents e Optimization workflow e Profiling Python code Timeit Profiler Line profiler e Making code go faster Algorithmic optimization x Example of the SVD e Writing faster numerical code Additional Links 10 1 Optimization workflow 1 Make it work write the code in a simple legible ways 2 Make it work reliably write automated test cases make really sure that your algorithm is right and that if you break it the tests will capture the breakage 3 Optimize the code by profiling simple use cases to find the bottlenecks and speeding up these bottle neck finding a better algorithm or implementation Keep in mind that a trade off should be found be tween profiling on a realistic example and the simplicity and speed of execution of the code For efficient work it is best to work with profiling runs lasting around 10s 10 2 Profiling Python code 203 Scipy lecture notes Edition 2015 2 No optimization without measuring e Measure profiling timing e You ll have surprises the fastest code is not always what you think 10 2 1 Timeit In IPython use timeit https docs python org library timeit html to time elementary operations In 1 import numpy as np In 2 a np arange 1000 In 3 timeit a 2 100000 loops best of 3 5
232. his can be used to posterize an image conversion of a continuous gradation of tone to several regions of fewer tones gt gt gt from scipy import misc gt gt gt face misc face gray True astype np float32 gt gt gt X face reshape 1 1 We need an n_sample n_feature array gt gt gt K k_means cluster KMeans n_clusters 5 5 clusters gt gt gt k_ means fit X KMeans gt gt gt values k_means cluster_centers_ squeeze gt gt gt labels k_means labels_ gt gt gt face_compressed np choose labels values gt gt gt face_compressed shape face shape TE T T NN r7 i Raw image K means quantization K 5 20 3 Clustering grouping observations together 358 Scipy lecture notes Edition 2015 2 20 4 Dimension Reduction with Principal Component Analysis The cloud of points spanned by the observations above is very flat in one direction so that one feature can almost be exactly computed using the 2 other PCA finds the directions in which the data is not flat and it can reduce the dimensionality of the data by projecting on a subspace A Depending on your version of scikit learn PCA will be in module decomposition or pca gt gt gt from sklearn import decomposition gt gt gt pca decomposition PCA n_components 2 gt gt gt pcea fit iris data PCA copy True n_components 2 whiten False gt gt gt X pca transform iris data
233. hon org tutorial modules html for more information about modules 2 5 6 Packages A directory that contains many modules is called a package A package is a module with submodules which can have submodules themselves etc A special file called __ init__ py which may be empty tells Python that the directory is a Python package from which modules can be imported ls cluster io README txt stsci __config _ py LATEST txt setup py __svn_version__ py Config _ pyc lib setup pyc __Svn_version__ pyc constants linalg setupscons py THANKS txt fftpack linsolve setupscons pyc TOCHANGE txt __init__ py maxentropy signal version py Beige pyc misc sparse version pyc INSTALL txt ndimage spatial weave integrate odr special interpolate optimize stats cd ndimage ls 2 5 Reusing code scripts and modules 32 Scipy lecture notes Edition 2015 2 doccer py fourier pyc interpolation py morphology pyc setup pyc doccer pyc info py interpolation pyc _nd_image so setupscons py filters py info pyc measurements py _ni_support py setupscons pyc filters pyc __init__ py measurements pyc _ni_support pyc tests fourier py __init__ pyc morphology py setup py From Ipython In 1 import scipy In 2 scipy __file__ Out 2 usr lib python2 6 dist packages scipy __init__ pyc In 3 import scipy version In 4 scipy version version Out l4 0 7 0 In 5 import scipy ndimag
234. i i i yt Tennene eee aS Starting from the code below try to reproduce the graphic on the right taking care of line styles axes plt gca axes set_xlim 0 4 axes set_ylim 0 3 axes set_xticklabels axes set_yticklabels Click on figure for solution 4 4 9 Multi Plots You can use several subplots with different partition Starting from the code below try to reproduce the graphic on the right 4 4 Other Types of Plots examples and exercises 94 Scipy lecture notes Edition 2015 2 plt subplot 2 2 1 plt subplot 2 2 3 plt subplot 2 2 4 Click on figure for solution 4 4 10 Polar Axis You only need to modify the axes line Starting from the code below try to reproduce the graphic on the right plt axes 0 0 1 1 N 20 theta np arange 0O 2 np pi 2 np pi N radii 10 np random rand N width np pi 4 np random rand N bars plt bar theta radii width width bottom 0 0 for r bar in zip radii bars bar set_facecolor cm jet r 10 bar set_alpha 0 5 Click on figure for solution 4 4 11 3D Plots t 2 0 T 15 You need to use contourf Starting from the code below try to reproduce the graphic on the right from mpl_toolkits mplot3d import Axes3D fig plt figure ax Axes3D fig X np arange 4 4 0 25 Y np arange 4 4 0 25 X Y np meshgrid X Y R np sqrt X 2 Y 2 Z np sin R 4 4 Other
235. ia 10 20g Os 01 gt gt gt ndimage binary_dilation a astype a dtype array LI One TOn 20s On 2On 5 One Ome lee 230i Oe Oise siaa ie Gila Oe E Or Orr ar Our Oal Ors 0a O 2055 Old Also works for grey valued images gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt np random seed 2 im np zeros 64 64 x y 63 np random random 2 8 astype np int im x y np arange 8 bigger_points ndimage grey_dilation im size 5 5 structure np ones 5 5 Square np zeros 16 16 square 4 4 4 4 1 dist ndimage distance_transform_bf square dilate_dist ndimage grey_dilation dist size 3 3 structure np ones 3 3 Opening erosion dilation gt gt gt a np zeros 5 5 dtype np int gt gt gt a tes 1 4 a 4 4 i gt gt gt a array 0 0 07 0 Ol O a Ao Oa AE HO aoe os 05 0 0 LD gt gt gt Opening removes small objects gt gt gt ndimage binary_opening a structure np ones 3 3 astype np int Aiea Ll0 0 07 07 0l 12 4 Image filtering 239 LO Ee Cie is Ge ste ais ly Ol Or ta dy tsi O 0 0 0 0D gt gt gt Opening can also smooth corners gt gt gt ndimage binary_opening a astype np int array L0 0 05 0 0 0 0 se 0 0 Or es Ol O Oi 1 0 Ol On 0 Oe 0 oD Application remove noise gt gt gt square np zeros 32 32 gt
236. iced using the same syntax and rules Indexing gt gt gt a hello gt gt gt a 0 Hs 27 gt all ee gt gt gt a 1 lo Remember that negative indices correspond to counting from the right end Slicing gt gt gt a hello world gt gt gt al 3 6 3rd to 6th excluded elements elements 3 4 5 iow gt gt gt al2 10 2 Syntaz alstart stop step pile So gt gt gt al 3 every three characters from beginning to end Pla te Accents and special characters can also be handled in Unicode strings see https docs python org tutorial introduction html unicode strings A string is an immutable object and it is not possible to modify its contents One may however create new strings from the original one In 53 a hello world In 54 al2 z Traceback most recent call last File lt stdin gt line 1 in lt module gt TypeError str object does not support item assignment In 55 a replace 1 2 1 Out 55 hezlo world In 56 a replace 1l z Out 56 hezzo worzd Strings have many useful methods such as a replace as seen above Remember the a object oriented notation and use tab completion or help str to search for new methods See also Python offers advanced possibilities for manipulating strings looking for patterns or formatting The interested reader is referred to https docs python org library stdtypes html string metho
237. ied by the environment variable PYTHONPATH The list of directories searched by Python is given by the sys path variable In 1 import sys In 2 sys path Oat l2 Des home varoquau local bin 1 7 ue lib pytouzs T home varoquau local lib python2 7 site packages usr 1lib python2 7 dist packages usr local lib python2 7 dist packages ei Modules must be located in the search path therefore you can e write your own modules within directories already defined in the search path e g HOME local lib python2 7 dist packages You may use symbolic links on Linux to keep the code somewhere else e modify the environment variable PYTHONPATH to include the directories containing the user defined modules On Linux Unix add the following line to a file read by the shell at startup e g etc profile profile export PYTHONPATH PYTHONPATH home emma user_defined_modules On Windows http support microsoft com kb 310519 explains how to handle environment variables e or modify the sys path variable itself within a Python script import sys new_path home emma user_defined_modules if new_path not in sys path sys path append new_path This method is not very robust however because it makes the code less portable user dependent path and because you have to add the directory to your sys path each time you want to import from a module in this directory See also See https docs pyt
238. ient way to find a minimum for this function is to conduct a gradient descent starting from a given initial point The BFGS algorithm is a good way of doing this gt gt gt optimize fmin_bfgs f 0 Optimization terminated successfully Current function value 7 945823 Iterations 5 Function evaluations 24 Gradient evaluations 8 array 1 30644003 A possible issue with this approach is that if the function has local minima the algorithm may find these local minima instead of the global minimum depending on the initial point gt gt gt optimize fmin_bfgs f 3 disp 0 array 3 83746663 If we don t know the neighborhood of the global minimum to choose the initial point we need to resort to costlier global optimization To find the global minimum we use scipy optimize basinhopping which combines a local optimizer with stochastic sampling of starting points for the local optimizer New in version 0 12 0 basinhopping was added in version 0 12 0 of Scipy gt gt gt optimize basinhopping f 0 nfev 1725 minimization_failures 0 fun 7 9458233756152845 x array 1 30644001 message requested number of basinhopping iterations completed successfully njev 575 nit 100 5 5 Optimization and fit scipy optimize 110 Scipy lecture notes Edition 2015 2 Another available but much less efficient global optimizer is scipy optimize brute brute force opti mization on a grid More efficient algorithms
239. iew of the different optimizers Getting started 1D optimization Gradient based methods Newton and quasi newton methods Gradient less methods Global optimizers e Practical guide to optimization with scipy Choosing a method Making your optimizer faster Computing gradients Synthetic exercices e Special case non linear least squares Minimizing the norm of a vector function Curve fitting e Optimization with constraints Box bounds General constraints 13 1 Knowing your problem Not all optimization problems are equal Knowing your problem enables you to choose the right tool Dimensionality of the problem The scale of an optimization problem is pretty much set by the dimensionality of the problem i e the number of scalar variables on which the search is performed 13 1 Knowing your problem 249 Scipy lecture notes Edition 2015 2 13 1 1 Convex versus non convex optimization A convex function A non convex function e f is above all its tangents e equivalently for two point A B f C lies below the segment f A f B ifA lt C lt B Optimizing convex functions is easy Optimizing non convex functions can be very hard It can be proven that for a convex function a local minimum is also a global minimum Then in some sense the minimum is unique 13 1 2 Smooth and non smooth problems A smooth function A non smooth function The gradient is defined everywhere and is
240. ild temp linux x86_64 2 7 cos_doubles o build temp linux x86_64 2 7 cos_doubles_wrap o i 1s build cos_doubles h cos_doubles py cos_doubles_wrap c setup py cos_doubles c cos_doubles i _cos_doubles so numpy i test_cos_doubles py And as before we convince ourselves that it worked import numpy as np import pylab import cos_doubles np arange 0 2 np pi 0 1 np empty_like x x y cos_doubles cos_doubles_func x y pylab plot x y pylab show 14 4 SWIG 275 Scipy lecture notes Edition 2015 2 14 5 Cython Cython is both a Python like language for writing C extensions and an advanced compiler for this language The Cython language is a superset of Python which comes with additional constructs that allow you call C functions and annotate variables and class attributes with c types In this sense one could also call it a Python with types In addition to the basic use case of wrapping native code Cython supports an additional use case namely interactive optimization Basically one starts out with a pure Python script and incrementally adds Cython types to the bottleneck code to optimize only those code paths that really matter In this sense it is quite similar to SWIG since the code can be autogenerated but in a sense it also quite similar to ctypes since the wrapping code can almost be written in Python While others solutions that autogenerate code can be quite difficult t
241. ing interface The mayavi mlab module provides simple plotting functions to apply to numpy arrays similar to matplotlib or matlab s plotting interface Try using them in IPython by starting Python with the switch gui wx 340 Scipy lecture notes Edition 2015 2 19 1 1 3D plotting functions Points points3 oe c E a e ee Points in 3D represented with markers or glyphs and optionaly different sizes X y Z value np random random 4 40 mlab points3d x y z value Lines plota A line connecting points in 3D with optional thickness and varying color mlab clf Clear the figure t np linspace 0 20 200 mlab plot3d np sin t np cos t 0 1 t t Elevation surface surf A surface given by its elevation coded as a 2D array 19 1 Mlab the scripting interface 341 Scipy lecture notes Edition 2015 2 mlab clf x y np mgrid 10 10 100j 10 10 100j r np sqrt x 2 y 2 z np sin r r mlab surf z warp_scale auto Arbitrary regular mesh A surface mesh given by x y z positions of its node points mlab clf phi theta np mgrid O np pi 11j 0 2 np pi 11j x np sin phi np cos theta y np sin phi np sin theta z np cos phi mlab mesh x y Z mlab mesh x y Zz representation wireframe color 0 0 0 A surface is defined by points connected to form triangles or polygones In mayavi mlab
242. interest 17 4 1 Binary segmentation foreground background Histogram based method Otsu thresholding The Otsu method is a simple heuristic to find a threshold to separate the foreground from the background Earlier scikit image versions skimage filters is called skimage filter in earlier versions of scikit image from skimage import data from skimage import filters camera data camera val filters threshold_otsu camera mask camera lt val 17 4 Image segmentation 315 Scipy lecture notes Edition 2015 2 7000 6000 5000 4000 3000 2000 1000 0 O 50 100 150 200 250 300 Labeling connected components of a discrete image Once you have separated foreground objects it is use to separate them from each other For this we can assign a different integer labels to each one Synthetic data gt gt gt n 20 gt gt gt 1 256 gt gt gt im np zeros 1 1 gt gt gt points 1 np random random 2 n 2 gt gt gt imL points 0 astype np int points 1 astype np int 1 gt gt gt im filters gaussian_filter im sigma l 4 n gt gt gt blobs im gt im mean Label all connected components gt gt gt from skimage import measure gt gt gt all_labels measure label blobs Label only foreground connected components gt gt gt blobs_labels measure label blobs background 0 See also scipy ndimage find_objects is us
243. ion is required for a trait a_XXX_default magic method can be implemented It will be lazily called when trying to access the XXX trait For example def _name default self u Complex initialisation of the reservoir name return Undefined 18 3 2 Validation Every trait does validation when the user tries to set its content reservoir Reservoir name Lac de Vouglans max_storage 605 reservoir max_storage 230 TraitError Traceback most recent call last scipy lecture notes advanced traits lt ipython input 7 979bdff 9974a gt in lt module gt gt 1 reservoir max_storage 230 18 3 What are Traits 325 Scipy lecture notes Edition 2015 2 traits trait_handlers pyc in error self object name value 166 ye 167 raise TraitError object name self full_info object name value gt 168 value 169 170 def arg_error self method arg_num object name value TraitError The max_storage trait of a Reservoir instance must be a float but a value of 23 lt type str gt w 18 3 3 Documentation By essence all the traits do provide documentation about the model itself The declarative approach to the creation of classes makes it self descriptive from traits api import HasTraits Str Float class Reservoir HasTraits name Str max_storage Float 100 The desc metadata of the traits can be used to provide a more descriptive information about the trait from traits
244. is on the left in memory 2 Create a new view gt gt gt y x view lt i4 gt gt gt y array 67305985 dtype int32 gt gt gt 0x04030201 67305985 8 1 Life of ndarray 165 Scipy lecture notes Edition 2015 2 e view makes views does not copy or alter the memory block e only changes the dtype and adjusts array shape gt gt gt x 1 5 gt gt gt y array 328193 dtype int32 gt gt gt y base is X True Mini exercise data re interpretation See also view colors py You have RGBA data in an array gt gt gt x np zeros 10 10 4 dtype np int8 gt gt gt x 0 1 So gt xy aT 2 gt gt gt xl 202 S383 eee lt 45 3 S24 where the last three dimensions are the R B and G and alpha channels How to make a 10 10 structured array with field names t g b a without copying data gt oy Ss gt gt gt assert y r 1 al10 gt gt gt assert y g 2 al10 gt gt gt assert y b 3 al1Q gt gt gt assert y a 4 al1Q Solution Poy xavier ClLGr sy ta Clg Maiden Chip Paley ta Pace Dil Ae te 8 1 Life of ndarray 166 Scipy lecture notes Edition 2015 2 A Another array taking exactly 4 bytes of memory gt gt gt y np array 1 3 2 4 dtype np uint8 transpose gt gt gt x y copyQ gt gt gt X array CL Eee 3 4 dtype uints8 gt gt gt y array 1 2 3 4
245. is that of multiple observations or samples described by a set of different attributes or features The data can than be seen as a 2D table or matrix with columns giving the different attributes of the data and rows the observations For instance the data contained in examples brain_size csv mm Gender FOIQ VIQ PIO Weight Height MRI Count vi Pema lie 1335 132 124 0118 64 57 66932 U2 Male 140 150 124 72 5 1001121 Wo 5 Malev 139 123 150 143 73 3 1038437 tat Male 133 129 125 172 69 97 965353 6 Female 137 132 134 147 65 0 951545 15 1 2 The panda data frame We will store and manipulate this data in a pandas DataFrame from the pandas module It is the Python equivalent of the spreadsheet table It is different from a 2D numpy array as it has named columns can contain a mixture of different data types by column and has elaborate selection and pivotal mechanisms Creating dataframes reading data files or converting arrays Separator It is a CSV file but the separator is 15 1 Data representation and interaction 285 Scipy lecture notes Edition 2015 2 Reading from a CSV file Using the above CSV file that gives observations of brain size and weight and IQ Willerman et al 1991 the data are a mixture of numerical and categorical values gt gt gt import pandas gt gt gt data
246. is the price to pay for getting rid of or characters that delineate logical blocks in other languages Improper indentation leads to errors such as IndentationError unexpected indent test py line 2 All this indentation business can be a bit confusing in the beginning However with the clear indentation and in the absence of extra characters the resulting code is very nice to read compared to other languages Reusing code scripts and modules 33 Scipy lecture notes Edition 2015 2 e Indentation depth Inside your text editor you may choose to indent with any positive number of spaces 1 2 3 4 However it is considered good practice to indent with 4 spaces You may configure your editor to map the Tab key to a 4 space indentation In Python x y the editor is already configured this way e Style guidelines Long lines you should not write very long lines that span over more than e g 80 characters Long lines can be broken with the character gt gt gt long_line Here is a very very long line that we break in two parts Spaces Write well spaced code put whitespaces after commas around arithmetic operators etc gt gt gt a 1 yes gt gt gt a 1 too cramped A certain number of rules for writing beautiful code and more importantly using the same conven tions as anybody else are given in the Style Guide for Python Code Quick read If you want to do a first quick pass through
247. is the sub dtype in the structured data corresponding to the name format e The second one is its offset in bytes from the beginning of the item 8 1 Life of ndarray 163 Scipy lecture notes Edition 2015 2 Exercise Mini exercise make a sparse dtype by using offsets and only some of the fields gt gt gt wav_header_dtype np dtype dict names format sample_rate data_id offsets Loffset_1 offset_2 offset_3 counted from start of structure in bytes formats list of dtypes for each of the fields and use that to read the sample rate and data_id as sub array gt gt gt f open data test wav r gt gt gt wav_header np fromfile f dtype wav_header_dtype count 1 gt gt gt f close gt gt gt print wav_header Pa RIEF AO WAVE fme e er dh de 16000 S2000Kk 2 16 Ii dese Varela lie av 366L gt gt gt wav_header sample_rate array 16000 dtype uint32 Let s try accessing the sub array gt gt gt wav_header data_id array IlL dar man ae at a dtype S1 gt gt gt wav_header shape CE gt gt gt wav_header data_id shape Cie When accessing sub arrays the dimensions get added to the end There are existing modules such as wavfile audiolab etc for loading sound data Casting and re interpretation views casting e on assignment e on array construction e on arithmetic e etc e and manually astype dtype data re interpretation
248. ite a comparison between IQ of male and female using a linear model gt gt gt model ols VIQ Gender 1 data fitQ gt gt gt print model summary Dep Variable Model Method OLS Regression Results VIQ OLS Least Squares R squared Adj R squared P statistic 0 015 0 010 0 5969 15 3 Linear models multiple factors and analysis of variance 293 Date Time No Observations 40 Df Residuals 38 Df Model 1 coef std err Intercept 109 4500 5 308 Gender T Male 5 8000 7 507 Omnibus 26 188 Prob 0mnibus 0 000 Skew 0 010 Kurtosis 1 510 Tips on specifying model Scipy lecture notes Edition 2015 2 Prob F statistic 0 445 Log Likelihood 182 42 ATG 368 8 BIC Sf2 2 t P gt t 95 0 Conf Int 20 619 0 000 98 704 120 196 0 773 0 445 9 397 20 997 Durbin Watson 1 709 Jarque Bera JB S OE Prob JB 0 157 Cond No 2262 Forcing categorical the Gender is automatical detected as a categorical variable and thus each of its different values are treated as different entities An integer column can be forced to be treated as categorical using gt gt gt model ols VIQ C Gender data fitQ Intercept We can remove the intercept using 1 in the formula or force the use of an intercept using 1 By default statsmodel treats a categorical variable with K possible values as K 1 dummy boolean variables the last level being absorbed into the intercept
249. ization opening a dialog Deferral Notification Some more advanced traits 322 Scipy lecture notes Edition 2015 2 18 1 Introduction The Enthought Tool Suite enable the construction of sophisticated application frameworks for data analysis 2D plotting and 3D visualization These powerful reusable components are released under liberal BSD style licenses Trarts Mayavi TVTK VTK Chaco Envisage Enable Kiva The main packages of the Enthought Tool Suite are e Traits component based approach to build our applications e Kiva 2D primitives supporting path based rendering affine transforms alpha blending and more e Enable object based 2D drawing canvas e Chaco plotting toolkit for building complex interactive 2D plots e Mayavi 3D visualization of scientific data based on VTK e Envisage application plugin framework for building scriptable and extensible applications In this tutorial we will focus on Traits 18 2 Example Throughout this tutorial we will use an example based on a water resource management simple case We will try to model a dam and reservoir system The reservoir and the dams do have a set of parameters e Name e Minimal and maximal capacity of the reservoir hm3 e Height and length of the dam m e Catchment area km2 e Hydraulic head m 18 1 Introduction 323 Scipy lecture notes Edition 2015 2 e Power of the turbines MW e Minimal and maximal release
250. k that the resulting histogram based segmentation is more accurate See also More advanced segmentation algorithms are found in the scikit image see Scikit image image processing See also Other Scientific Packages provide algorithms that can be useful for image processing In this example we use the spectral clustering function of the scikit learn in order to segment glued objects gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt from sklearn feature_extraction import image from sklearn cluster import spectral_clustering x y np indices 1 1 center1 28 24 center2 40 50 center3 67 58 center4 24 70 radiusi radius2 radius3 radius4 16 14 15 14 circlei x center1 0 2 y centeri 1 2 lt radius1 2 circle2 x center2 0 2 y center2 1 2 lt radius2 2 circle3 x center3 0 2 y center3 1 2 lt radius3 2 circle4 x center4 0 2 y center4 1 2 lt radius4 2 4 circles img circleil circle2 circle3 circle4 mask img astype bool1 img img astype float img 1 0 2 np random randn img shape
251. king with Numpy and Scipy have been introduced the interested user is invited to try these exercises 5 11 1 Maximum wind speed prediction at the Sprog station The exercise goal is to predict the maximum wind speed occurring every 50 years even if no measure exists for such a period The available data are only measured over 21 years at the Sprog meteorological station located in Denmark First the statistical steps will be given and then illustrated with functions from the scipy interpolate module At the end the interested readers are invited to compute results from raw data and in a slightly different approach Statistical approach The annual maxima are supposed to fit a normal probability density function However such function is not going to be estimated because it gives a probability from a wind speed maxima Finding the maximum wind speed occurring every 50 years requires the opposite approach the result needs to be found from a defined probability That is the quantile function role and the exercise goal will be to find it In the current model it is supposed that the maximum wind speed occurring every 50 years is defined as the upper 2 quantile By definition the quantile function is the inverse of the cumulative distribution function The latter describes the probability distribution of an annual maxima In the exercise the cumulative probability p_i for a given year i is defined as p_i i N 1 withN 21 the number of measur
252. laced at arbitrary positions and until now they were on the border of the axis We ll change that since we want to have them in the middle Since there are four of them top bottom left right we ll discard the top and right by setting their color to none and we ll move the bottom and left ones to coordinate 0 in data space coordinates ax plt gca gca stands for get current azis ax spines right set_color none ax spines top set_color none ax xaxis set_ticks_position bottom ax spines bottom set_position data 0 ax yaxis set_ticks_position left ax spines left set_position data 0 4 2 8 Adding a legend cosine sine Documentation e Legend guide e legend command e Legend API Let s add a legend in the upper left corner This only requires adding the keyword argument label that will be used in the legend box to the plot commands plt plot X C color blue linewidth 2 5 linestyle label cosine plt plot X S color red linewidth 2 5 linestyle label sine plt legend loc upper left 4 2 Simple plot 87 Scipy lecture notes Edition 2015 2 4 2 9 Annotate some points cosine sine Documentation e Annotating axis e annotate command Let s annotate some interesting points using the annotate command We chose the 27 3 value and we want to annotate both the sine and the cosine We ll first draw a marker on the curve as well as a straight dotted
253. le for example singular value decomposition SVD 5 2 Special functions scipy special 103 Scipy lecture notes Edition 2015 2 gt gt gt arr np arange 9 reshape 3 3 np diag 1 0 1 gt gt gt uarr spec vharr linalg svd arr The resulting array spectrum is gt gt gt spec array 14 88982544 0 45294236 0 29654967 The original matrix can be re composed by matrix multiplication of the outputs of svd with np dot gt gt gt sarr np diag spec gt gt gt svd_mat uarr dot sarr dot vharr gt gt gt np allclose svd_mat arr True SVD is commonly used in statistics and signal processing Many other standard decompositions QR LU Cholesky Schur as well as solvers for linear systems are available in scipy linalg 5 4 Fast Fourier transforms scipy fftpack The scipy fftpack module allows to compute fast Fourier transforms As an illustration a noisy input signal may look like gt gt gt time_step 0 02 gt gt gt period 5 gt gt gt time_vec np arange 0 20 time_step gt gt gt sig np sin 2 np pi period time_vec 0 5 np random randn time_vec size The observer doesn t know the signal frequency only the sampling time step of the signal sig The signal is supposed to come from a real function so the Fourier transform will be symmetric The scipy fftpack fftfreg function will generate the sampling frequencies and scipy fftpack fft will compute the
254. le Gouillart Ga l Varoquaux Python for scientific computing We introduce here the Python language Only the bare minimum necessary for getting started with Numpy and Scipy is addressed here To learn more about the language consider going through the excellent tutorial https docs python org tutorial Dedicated books are also available such as http www diveintopython net python Python is a programming language as are C Fortran BASIC PHB etc Some specific features of Python are as follows an interpreted as opposed to compiled language Contrary to e g C or Fortran one does not compile Python code before executing it In addition Python can be used interactively many Python inter preters are available from which commands and scripts can be executed a free software released under an open source license Python can be used and distributed free of charge even for building commercial software multi platform Python is available for all major operating systems Windows Linux Unix MacOS X most likely your mobile phone OS etc a very readable language with clear non verbose syntax a language for which a large variety of high quality packages are available for various applications from web frameworks to scientific computing a language very easy to interface with other languages in particular C and C Some other features of the language are illustrated just below For example Python is an object oriented lan
255. le to provide more values to finish immediately It allows the generator __del__ method to destroy objects holding the state of generator Let s define a generator which just prints what is passed in through send and throw gt gt gt import itertools gt gt gt def gQ Print ctert for i in itertools count print yielding 7i i bday ans yield i except GeneratorExit print gt closing raise except Exception as e print yield raised r e else print yield returned s ans gt gt gt it g gt gt gt next it z Start yielding 0 O gt gt gt it send 11 yield returned 11 yielding 1 1 gt gt gt it throw IndexError yield raised IndexError yielding 2 2 gt gt gt it close c losing next or__next__ In Python 2 x the iterator method to retrieve the next value is called next It is invoked implicitly through the global function next which means that it should be called __next__ Just like the global function iter calls __iter__ This inconsistency is corrected in Python 3 x where it next becomes it __next__ For other generator methods send and throw the situation is more complicated because they are not called implicitly by the interpreter Nevertheless there s a proposed syntax exten sion to allow continue to take an argument which will be passed to send of the loop s iterator If this extension is accept
256. lf function function return self _wrapper def _wrapper self args kwargs print in the wrapper 7s Zs args kwargs T return self function args xkwargs gt gt gt deco_instance replacing _decorator_class foo in decorator init foo gt gt gt deco_instance def function args xkwargs print in function 7s 7s args kwargs in decorator call foo gt gt gt function 1ii 12 in the wrapper 11 12 1 in f nction 11 12 A decorator like this can do pretty much anything since it can modify the original function object and mangle the arguments call the original function or not and afterwards mangle the return value 1 2 3 Copying the docstring and other attributes of the original function When a new function is returned by the decorator to replace the original function an unfortunate conse quence is that the original function name the original docstring the original argument list are lost Those attributes of the original function can partially be transplanted to the new function by setting __doc__ the docstring __module__and__name__ the fullname ofthe function and __annotations__ extra informa tion about arguments and the return value of the function available in Python 3 This can be done automati cally by using functools update_wrapper 7 2 Decorators 150 Scipy lecture notes Edition 2015 2 functools update_wrapper wrapper wrapped Update a wrapper function t
257. line Then we ll use the annotate command to display some text with an arrow t 2 np pi 3 plt plot t t LO np cos t color blue linewidth 2 5 linestyle plt scatter t mp cos t 50 color blue plt annotate r sin frac 2 pi 3 frac sqrt 3 2 xy t np sin t xycoords data xytext 10 30 textcoords offset points fontsize 16 arrowprops dict arrowstyle gt connectionstyle arc3 rad 2 plt plot t t 0 np sin t color red linewidth 2 5 linestyle plt scatter t np sin t 50 color red plt annotate r cos frac 2 pi 3 frac 1 2 xy t np cos t xycoords data xytext 90 50 textcoords offset points fontsize 16 arrowprops dict arrowstyle gt connectionstyle arc3 rad 2 4 2 10 Devil is in the details sin 2r _v3 Jay oo cosine sine Documentation e Artists e BBox The tick labels are now hardly visible because of the blue and red lines We can make them bigger and we can also adjust their properties such that they ll be rendered on a semi transparent white background This will allow us to see both the data and the labels for label in ax get_xticklabels ax get_yticklabels label set_fontsize 16 label set_bbox dict facecolor white edgecolor None alpha 0 65 4 2 Simple plot 38 Scipy lecture notes Edition 2015 2 4 3 Figures Subplots Axes and Ticks A figure in
258. ll manage the way the trait is rendered to the screen e g the Range trait is displayed as a slider etc In the very same vein as the Traits declarative way of creating classes TraitsUI provides a declarative interface to build user interfaces code from traits api import HasTraits Str Float Range from traitsui api import View class Reservoir HasTraits name Str max_storage Float 1e6 desc Maximal storage hm3 max_release Float 10 desc Maximal release m3 s head Float 10 desc Hydraulic head m efficiency Range 0O 1 traits_view View name max_storage max_release head efficiency title Reservoir resizable True def energy_production self release Returns the energy production Wh for the given release m3 s ord power 1000 9 81 self head release self efficiency return power 3600 if name _ main __ reservoir Reservoir name Project A max_storage 30 max release 100 0 head 60 efficiency 0 8 18 3 What are Traits 327 Scipy lecture notes Edition 2015 2 reservoir configure_traits Pe ie ie Reservoir Name Project A Max storage 30 0 Max release 100 0 Head 60 0 Efficiency 0 0 1 0 08 whan 18 3 5 Deferral Being able to defer the definition of a trait and its value to another object is a powerful feature of Traits from traits api import HasTraits Instance DelegatesTo Float Rang
259. llowing initial solution gt gt gt x0 np array 3 50 20 1 dtype float compare the result of scipy optimize leastsq and what you can get with scipy optimize fmin_slsqp when adding boundary constraints 5 11 Summary exercises on scientific computing 131 Scipy lecture notes Edition 2015 2 5 11 3 Image processing application counting bubbles and unmolten grains Det Spot Mag 10 4 mm 15 0 kY SSD 5 8 41x 3 25 mm MV 15 05 2009 1280 C Statement of the problem 1 Open the image file MV_HFV_012 jpg and display it Browse through the keyword arguments in the doc string of imshow to display the image with the right orientation origin in the bottom left corner and not the upper left corner as for standard arrays This Scanning Element Microscopy image shows a glass sample light gray matrix with some bubbles on black and unmolten sand grains dark gray We wish to determine the fraction of the sample covered by these three phases and to estimate the typical size of sand grains and bubbles their sizes etc 2 Crop the image to remove the lower panel with measure information 3 Slightly filter the image with a median filter in order to refine its histogram Check how the histogram changes 4 Using the histogram of the filtered image determine thresholds that allow to define masks for sand pixels glass pixels and bubble pixels Other option homework write a function that determines automatically
260. lookfor looks for keywords inside the docstrings of specified modules In 45 numpy lookfor convolution search results for convolution numpy convolve Returns the discrete linear convolution of two one dimensional sequences numpy bartlett Return the Bartlett window numpy correlate Discrete linear correlation of two 1 dimensional sequences In 46 numpy lookfor remove module os search results for remove os remove remove path os removedirs removedirs path os rmdir rmdir path os unlink unlink path os walk Directory tree generator e Ifeverything listed above fails and Google doesn t have the answer don t despair Write to the mailing list suited to your problem you should have a quick answer if you describe your problem well Experts on scientific python often give very enlightening explanations on the mailing list Numpy discussion numpy discussion scipy org all about numpy arrays manipulating them indexation questions etc 138 Scipy lecture notes Edition 2015 2 SciPy Users List scipy user scipy org scientific computing with Python high level data process ing in particular with the scipy package matplotlib users lists sourceforge net for plotting with matplotlib 139 Part II Advanced topics 140 Scipy lecture notes Edition 2015 2 This part of the Scipy lecture notes is dedicated to advanced usage It strives to educate the proficient Python coder t
261. lormaps All colormaps can be reversed by appending _r For instance gray_r is the reverse of gray If you want to know more about colormaps checks Documenting the matplotlib colormaps 4 6 Quick references 99 Scipy lecture notes Edition 2015 2 JUM ule EE Ee Jowwns 6uuds Eee jepads DIUUSIOS SS moquies BE O usid E a a a a a a ae yuid E ueg z0 jeuzdeds Adiu 0 TE SS 3 3x3 E ie zojdnu Jojdnub Die si M u1 4s_4Si NUS M LTT i I ou e e y ysi Kei6 3si6 yyee 4si6 by ESSE SSS SS SS Bi xijeyaqns 9000 Wwe WE O jooo M nq M uq 100 Quick references 4 6 CHAPTER 9 Scipy high level scientific computing Authors Adrien Chauve Andre Espaze Emmanuelle Gouillart Ga l Varoquaux Ralf Gommers Scipy The scipy package contains various toolboxes dedicated to common issues in scientific computing Its different submodules correspond to different applications such as interpolation integration optimiza tion image processing statistics special functions etc scipy can be compared to other standard scientific computing libraries such as the GSL GNU Scientific Library for C and C or Matlab s toolboxes scipy is the core package for scientific routines in Python it is meant to operate efficiently on numpy arrays so that numpy and scipy work hand in hand Before implementing a routine it is worth checking if the desired data processing is not already imple mented in Sci
262. m3 s e Efficiency of the turbines The reservoir has a known behaviour One part is related to the energy production based on the water released A simple formula for approximating electric power production at a hydroelectric plant is P phrgk where e P is Power in watts e pis the density of water 1000 kg m3 h is height in meters e ris flowrate in cubic meters per second e gis acceleration due to gravity of 9 8 m s2 k is a coefficient of efficiency ranging from 0 to 1 Annual electric energy production depends on the available water supply In some installations the water flow rate can vary by a factor of 10 1 over the course of a year The second part of the behaviour is the state of the storage that depends on controlled and uncontrolled pa rameters storager storage inflows release spillage irrigation A The data used in this tutorial are not real and might even not have sense in the reality 18 3 What are Traits A trait is a type definition that can be used for normal Python object attributes giving the attributes some additional characteristics e Standardization Initialization Validation Deferral e Notification e Visualization e Documentation A class can freely mix trait based attributes with normal Python attributes or can opt to allow the use of only a fixed or open set of trait attributes within the class Trait attributes defined by a class are automatically inherit
263. mages Exercise Compare histograms for the different filtered images 5 10 Image processing scipy ndimage 120 Scipy lecture notes Edition 2015 2 noisy Gaussian filter median filter Wiener filter 5 10 3 Mathematical morphology Mathematical morphology is a mathematical theory that stems from set theory It characterizes and trans forms geometrical structures Binary black and white images in particular can be transformed using this theory the sets to be transformed are the sets of neighboring non zero valued pixels The theory was also extended to gray valued images Erosion Dilation Opening Closing al jer Elementary mathematical morphology operations use a structuring element in order to modify other geomet rical structures Let us first generate a structuring element gt gt gt el ndimage generate_binary_structure 2 1 gt gt gt el array False True False True True Truel False True False dtype bo01 gt gt gt el astype np int array CLO 1 O i E OPE lt Olp e Erosion gt gt gt a np zeros 7 7 dtype np int gt gt gt all G gt gt gt a array LO LO LO LO LO LO gt gt gt ndimage binar array LLO O ol oi OF ol ole 0119 _erosion a astype a dtype o v v v v v v v LO LO LO LO 5 10 Image processing scipy O 1 1 1 1 1 0 y 0 0 1 1 1 O ol 0l ol ndima
264. max temperatures Hint this function has to have a period of 1 year Hint include a time offset 3 Fit this function to the data with scipy optimize curve_fit Plot the result Is the fit reasonable If not why 5 Is the time offset for min and max temperatures the same within the fit accuracy aN 5 5 Optimization and fit scipy optimize Scipy lecture notes Edition 2015 2 Exercise 2 D minimization Six hump Camelback function The six hump camelback function 4 f x y 4 2 1x7 x xy 4y 4y has multiple global and local minima Find the global minima of this function Hints e Variables can be restricted to 2 lt x lt 2and 1 lt y lt 1 e Use numpy meshgrid and pylab imshow to find visually the regions e Use scipy optimize fmin_bfgs or another multi dimensional minimizer How many global minima are there and what is the function value at those points What happens for an initial guess of x y 0 0 See the summary exercise on Non linear least squares curve fitting application to point extraction in topo graphical lidar data for another more advanced example 5 6 Statistics and random numbers scipy stats The module scipy stats contains statistical tools and probabilistic descriptions of random processes Ran dom number generators for various random process can be found in numpy random 5 6 1 Histogram and probability density function Given observations of a random process th
265. metric statistics to test the difference between VIQ in males and females Conclusion we find that the data does not support the hypothesis that males and females have different VIQ 15 2 Hypothesis testing comparing two groups 291 Scipy lecture notes Edition 2015 2 15 3 Linear models multiple factors and analysis of variance 15 3 1 formulas to specify statistical models in Python A simple linear regression 15 10 10 15 20 25 6 4 2 0 2 4 6 x Given two set of observations x and y we want to test the hypothesis that y is a linear function of x In other terms y xx coef intercept e where e is observation noise We will use the statmodels module to 1 Fit a linear model We will use the simplest strategy ordinary least squares OLS 2 Test that coef is non zero First we generate simulated data according to the model gt gt gt import numpy as np gt gt gt x np linspace 5 5 20 gt gt gt np random seed 1 gt gt gt normal distributed noise gt gt gt y 5 3 x 4 np random normal size x shape gt gt gt Create a data frame containing all the relevant variables gt gt gt data pandas DataFrame x x y y formulas for statistics in Python See the statsmodels documentation Then we specify an OLS model and fit it 15 3 Linear models multiple factors and analysis of variance 292 gt gt gt from statsmodels form
266. mine the provided image moonlanding png which is heavily contaminated with periodic noise In this exercise we aim to clean up the noise using the Fast Fourier Transform 2 Load the image using pylab imread 3 Find and use the 2 D FFT function in scipy fftpack and plot the spectrum Fourier transform of the image Do you have any trouble visualising the spectrum If so why 4 The spectrum consists of high and low frequency components The noise is contained in the high frequency part of the spectrum so set some of those components to zero use array slicing 5 Apply the inverse Fourier transform to see the resulting image 5 5 Optimization and fit scipy optimize Optimization is the problem of finding a numerical solution to a minimization or equality The scipy optimize module provides useful algorithms for function minimization scalar or multi dimensional curve fitting and root finding gt gt gt from scipy import optimize Finding the minimum of a scalar function Let s define the following function gt gt gt def f x return x 2 10 np sin x and plot it gt gt gt x np arange 10 10 0 1 gt gt gt plt plot x f x gt gt gt plt show 5 5 Optimization and fit scipy optimize 109 Scipy lecture notes Edition 2015 2 120 100 80 60 40 20 20 10 I O 5 10 This function has a global minimum around 1 3 and a local minimum around 3 8 The general and effic
267. move elements gt gt gt L red blue green black white gt gt gt L append pink gt gt gt L L red blue green black white pink gt gt gt L pop removes and returns the last item pink gt gt gt L red blue green black white gt gt gt L extend pink purple extend L in place gt gt gt L red blue green black white pink purple gt gt gt L L 2 gt gt gt L red blue green black white Reverse 2 2 Basic types 14 gt gt gt r L 1 gt gt gt r white black green blue red gt gt gt r2 list L gt gt gt r2 red blue green black white gt gt gt r2 reverse in place gt gt gt r2 white black green blue red Concatenate and repeat lists gt a rer E L white black green blue red gt gt gt r 2 white black green blue red Sort gt gt gt sorted r new object black blue green red white gt gt gt x L white black green blue red gt gt gt r sort in place gt gt gt r black blue green red white Methods and Object Oriented Programming red white plue green black green Scipy lecture notes Edition 2015 2 black white plue red The notation r method e g r append 3
268. mple quadratic function Optimize the following function using K 0 as a starting point np random seed 0 K np random normal size 100 100 def f x return np sum np dot K x 1 2 np sum x 2 2 Time your approach Find the fastest approach Why is BFGS not working well 13 3 Practical guide to optimization with scipy 259 Scipy lecture notes Edition 2015 2 Exercice A locally flat minimum Consider the function exp 1 1 x 2 y 2 This function admits a minimum in 0 0 Starting from an initialization at 1 1 try to get within le 8 of this minimum point 0 45 0 40 0 35 0 30 0 25 0 20 0 15 1 0 0 5 0 0 0 5 1 0 1 5 13 4 Special case non linear least squares 13 4 1 Minimizing the norm of a vector function Least square problems minimizing the norm of a vector function have a specific structure that can be used in the Levenberg Marquardt algorithm implemented in scipy optimize leastsq Lets try to minimize the norm of the following vectorial function gt gt gt def f x return np arctan x np arctan np linspace O 1 len x gt gt gt x0 np zeros 10 gt gt gt optimize leastsq f x0 array O 11111111 0 22222222 0 33333333 0 44444444 QO S0505a00 0 66666667 0O 77777776 0 3368393999 TI ee 2 This took 67 function evaluations check it with full_output 1 What if we compute the norm ourselves and use a good g
269. n array with more dimensions than input data Worked Example Broadcasting Let s construct an array of distances in miles between cities of Route 66 Chicago Springfield Saint Louis Tulsa Oklahoma City Amarillo Santa Fe Albuquerque Flagstaff and Los Angeles gt gt gt mileposts np array 0 198 303 736 871 1175 1475 1544 T 1913 2448 gt gt gt distance_array np abs mileposts mileposts np newaxis gt gt gt distance_array array 0 198 303 736 871 1175 1475 1544 1913 2448 198 0 105 538 6 3 ott 1277 1346 1715 22501 303 1105 Os 433 5568 872 1172 1241 1610 21451 736 1538 433 0 135 439 739 80s 1177 1712 L 8 1 6 3 569 135 0 304 604 1673 1042 1577 1175 977 872 439 304 O 3200 T 36S Teen 1273 1475 1277 1172 739 604 300 OF 69 438 973 1544 1346 1241 808 673 369 69 0 369 904 1913 1715 1610 1177 1042 738 438 369 oS S 2448 2250 2145 1712 1577 1273 973 904 535 0 A lot of grid based or network based problems can also use broadcasting For instance if we want to compute the distance from the origin of points on a 10x10 grid we can do 3 2 Numerical operations on arrays 63 Scipy lecture notes Edition 2015 2 gt gt gt x y np arange 5 np arange 5 np newaxis gt gt gt distance np sqrt x 2 y 2 gt gt gt distance array 0 alee ee eee ae Se aes Alls Peles
270. n building blocks 5 Scipy lecture notes Edition 2015 2 Python language data types string int flow control data collections lists dictionaries pat terns etc Modules of the standard library A large number of specialized modules or applications written in Python web protocols web framework etc and scientific computing Development tools automatic testing documentation generation Shell Konsole lt 2 gt soe e IPython an advanced Python shell http ipython org e Numpy provides powerful numerical arrays objects and routines to manipulate them http www numpy org e Scipy high level data processing routines Optimization regression interpolation etc http www scipy org 0 54 0 53 0 52 0 51 e Matplotlib 2 D visualization publication ready plots http matplotlib org e Mayavi 3 D visualization http code enthought com projects mayavi 1 3 The interactive workflow IPython and a text editor Interactive work to test and understand algorithms In this section we describe an interactive workflow with Python that is handy to explore and understand algorithms 1 3 The interactive workflow IPython and a text editor 6 Scipy lecture notes Edition 2015 2 Python is a general purpose language As such there is not one blessed environment to work in and not only one way of using it Although this makes it harder for beginners to find their way it make
271. n rational arithmetic 16 1 3 Symbols In contrast to other Computer Algebra Systems in SymPy you have to declare symbolic variables explicitly gt gt gt from sympy import gt gt gt x Symbol x gt gt gt y Symbol y Then you can manipulate them ee RN Ex Yy Q X gt gt gt x y 2 16 1 First Steps with SymPy 302 2 x y Scipy lecture notes Edition 2015 2 Symbols can now be manipulated using some of python operators arithmetic amp gt gt lt lt boolean Printing Here we use the following setting for printing gt gt gt sympy init_printing use_unicode False wrap_line True 16 2 Algebraic manipulations SymPy is capable of performing powerful algebraic manipulations We ll take a look into some of the most frequently used expand and simplify 16 2 1 Expand Use this to expand an algebraic expression It will try to denest powers and multiplications gt gt gt expand x y 3 3 2 2 3 x 3 x y 3 x y y gt gt gt S XKYKAD QeY KKXAD x 3 y 3 3 2 2 3 x 3 x y 3tx y y Further options can be given in form on keywords gt gt gt expand x y complex True re x re y I im x I im y gt gt gt I xim x I im y re x re y re x re y I im x I im y gt gt gt expand cos x y trig True sin x sin y cos x cos y gt gt gt cos x cos y sin x sin y sin x sin y cos
272. nalg eig reductions abs argmin comparisons all np linalg norm etc Solution Python source file 3 5 Some exercises 1 CHAPTER 4 Matplotlib plotting Thanks Many thanks to Bill Wing and Christoph Deil for review and corrections Authors Nicolas Rougier Mike Miiller Ga l Varoquaux Chapter contents e Introduction e Simple plot e Figures Subplots Axes and Ticks e Other Types of Plots examples and exercises e Beyond this tutorial e Quick references 4 1 Introduction Matplotlib is probably the single most used Python package for 2D graphics It provides both a very quick way to visualize data from Python and publication quality figures in many formats We are going to explore matplotlib in interactive mode covering most common cases 4 1 1 IPython and the matplotlib mode Python is an enhanced interactive Python shell that has lots of interesting features including named inputs and outputs access to shell commands improved debugging and many more It is central to the scientific computing workflow in Python for its use in combination with Matplotlib For interactive matplotlib sessions with Matlab Mathematica like functionality we use Python with it s spe cial Matplotlib mode that enables non blocking plotting IPython console When using the Python console we start it with the command line argument matplotlib pylab in very old versions IPython notebook In the Python notebook we insert
273. name St fan UnicodeDecodeError ascii codec can t decode byte Oxc3 in position 2 ordinal not in range 128 e Exceptions to pass messages between parts of the code def achilles _arrow x if abs x 1 lt 1e 3 raise StopIteration e return x while True try x achilles_arrow x except StopIteration break 2 8 Exception handling in Python 41 Scipy lecture notes Edition 2015 2 In 20 x Out 20 0 9990234375 Use exceptions to notify certain conditions are met e g StopIteration or not e g custom error raising 2 9 Object oriented programming OOP Python supports object oriented programming OOP The goals of OOP are e to organize the code and e to re use code in similar contexts Here is a small example we create a Student class which is an object gathering several custom functions methods and variables attributes we will be able to use gt gt gt class Student object def init__ self name self name name def set_age self age self age age def set_major self major self major major gt gt gt anna Student anna gt gt gt anna set_age 21 gt gt gt anna set_major physics In the previous example the Student class has __init__ set_age and set_major methods Its at tributes are name age and major We can call these methods and attributes with the following notation classinstance method or classinstance attribute The __init__ constr
274. nd meaning it can make an ndarray an instance of a specialized ndarray subclass 8 4 2 masked_array missing data Masked arrays are arrays that may have missing or invalid entries For example suppose we have an array where the fourth entry is invalid gt gt gt x np array 1 2 3 99 51 One way to describe this is to create a masked array gt gt gt mx np ma masked_array x mask 0 0 0 1 0 gt gt gt mx masked_array data 1 2 3 5 mask False False False True False fill_value 999999 Masked mean ignores masked data gt gt gt mx mean 2 15 gt gt gt np mean mx 2 1 5 A Notall Numpy functions respect masks for instance np dot so check the return types The masked_array returns a view to the original array gt gt gt mx 1 9 gt gt gt X array 1 9 3 99 5 8 4 Array siblings chararray maskedarray matrix 185 Scipy lecture notes Edition 2015 2 The mask You can modify the mask by assigning gt gt gt mx 1 np ma masked gt gt gt mx masked_array data i 3 5 mask False True False True False fill_value 999999 The mask is cleared on assignment gt gt gt mx 1 9 gt gt gt mx masked_array data 1 9 3 5 mask False False False True False fill_value 999999 The mask is also available directly gt gt gt mx mask array False False False True False dtype bool1 The masked entries can be
275. nd returns the parameters corresponding to the minimum value The parameters are specified with ranges given to numpy mgrid By default 20 steps are taken in each direction gt gt gt def f x The rosenbrock function j return 5 1 x 0 2 x 1 x 0 2 2 gt gt gt optimize brute f 1 2 1 2 array 1 00001462 1 00001547 13 3 Practical guide to optimization with scipy 13 3 1 Choosing a method 10 10 2 10 U c 2 2 y 10 Ill conditioned Gaussian 2 gt Ill conditioned quadratic 10 hi PREN eau e dim 2 dim 32 TOERE aaa o dim 8 dim 128 Well conditioned quadratic 4 a cASugate gradient Powell Nelder mead L BFGS BFGS Conjugate gradient BFGS L BFGS Newton wf wf wf w Hessian Without knowledge of the gradient e In general prefer BFGS scipy optimize fmin_bfgs or L BFGS scipy optimize fmin_l_bfgs_b even if you have to approximate numeri cally gradients e On well conditioned problems Powell scipy optimize fmin_powell and Nelder Mead scipy optimize fmin both gradient free methods work well in high dimension but they collapse for ill conditioned problems With knowledge of the gradient e BFGS scipy optimize fmin_bfgs or L BFGS scipy optimize fmin_l_bfgs_b e Computational overhead of BFGS is larger than that L BFGS itself larger than that of conjugate gradient On the other side BFGS usually needs less function evalua
276. ndimage measurements 246 Scipy lecture notes Edition 2015 2 e Other more powerful and complete modules OpenCV Python bindings CellProfiler ITK with Python bindings 12 6 Measuring objects properties ndimage measurements 247 CHAPTER 13 Mathematical optimization finding minima of functions Authors Ga l Varoquaux Mathematical optimization deals with the problem of finding numerically minimums or maximums or zeros of a function In this context the function is called cost function or objective function or energy Here we are interested in using scipy optimize for black box optimization we do not rely on the mathe matical expression of the function that we are optimizing Note that this expression can often be used for more efficient non black box optimization Prerequisites e Numpy Scipy e matplotlib See also References Mathematical optimization is very mathematical If you want performance it really pays to read the books e Convex Optimization by Boyd and Vandenberghe pdf available free online e Numerical Optimization by Nocedal and Wright Detailed reference on gradient descent methods e Practical Methods of Optimization by Fletcher good at hand waving explainations 248 Scipy lecture notes Edition 2015 2 Chapters contents e Knowing your problem Convex versus non convex optimization Smooth and non smooth problems Noisy versus exact cost functions Constraints e A rev
277. ndom_walker is similar to the Watershed but with a more probabilistic approach It is based on the idea of the diffusion of labels in the image gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt from skimage import segmentation Transform markers image so that O valued pixels are to be labelled and 1 valued pixels represent background markers image 1 labels_rw segmentation random_walker image markers image distance map watershed segmentation random walker segmentation Bh O bd Postprocessing label images S kimage provides several utility functions that can be used on label images ie images where different discrete values identify different regions Functions names are often self explaining S S kimage segmentation clear_border skimage segmentation relabel_from_one kimage morphology remove_small_objects etc 17 4 Image segmentation 317 Scipy lecture notes Edition 2015 2 Exercise e Load the coins image from the data submodule e Separate the coins from the background by testing several segmentation methods Otsu threshold ing adaptive thresholding and watershed or random walker segmentation e If necessary use a postprocessing function to improve the coins background segmentation 17 5 Measuring regions properties gt gt gt from skimage import measure Example compute the size and perimeter of the two segmented regions gt gt gt properti
278. ne2D object at gt gt gt gt plt xlabel size n lt matplotlib text Text object at gt gt gt gt plt ylabel memory MB lt matplotlib text Text object at gt 11 1 2 Sparse Matrices vs Sparse Matrix Storage Schemes e sparse matrix is a matrix which is almost empty storing all the zeros is wasteful gt store only nonzero items think compression e pros huge memory savings 210 Scipy lecture notes Edition 2015 2 e cons depends on actual storage scheme usually does not hold 11 1 3 Typical Applications e solution of partial differential equations PDEs the finite element method mechanical engineering electrotechnics physics e graph theory nonzero at i j means that node i is connected to node j 11 1 4 Prerequisites recent versions of numpy e scipy e matplotlib optional e ipython the enhancements come handy 11 1 5 Sparsity Structure Visualization e spy from matplotlib e example plots 11 1 Introduction 211 Scipy lecture notes Edition 2015 2 11 2 Storage Schemes e seven sparse matrix types in scipy sparse l 2 3 4 5 6 7 csc_matrix Compressed Sparse Column format csr_matrix Compressed Sparse Row format bsr_matrix Block Sparse Row format lil matrix List of Lists format dok_matrix Dictionary of Keys format coo_matrix COOrdinate format aka IJV triplet format dia_matrix DIAgonal format
279. ned very non quadratic function Methods based on conjugate gradient are named with cg in scipy The simple conjugate gradient method to minimize a function is scipy optimize fmin_cg gt gt gt def f x The rosenbrock function return 5 1 x 0 2 x 1 x 0 2 2 gt gt gt optimize fmin_cg f 2 2 Optimization terminated successfully Current function value 0 000000 Iterations 13 Function evaluations 120 Gradient evaluations 30 array 0 99998968 0 99997855 These methods need the gradient of the function They can compute it but will perform better if you can pass them the gradient gt gt gt def fprime x return np array 2 5 1 x 0 4 x 0 x 1 x 0 2 2 x 1 x 0 2 gt gt gt optimize fmin_cg f 2 2 fprime fprime Optimization terminated successfully Current function value 0 000000 Iterations 13 Function evaluations 30 Gradient evaluations 30 array 0 99999199 0 99998336 Note that the function has only been evaluated 30 times compared to 120 without the gradient 13 2 3 Newton and quasi newton methods Newton methods using the Hessian 2nd differential Newton methods use a local quadratic approximation to compute the jump direction For this purpose they rely on the 2 first derivative of the function the gradient and the Hessian 13 2 A review of the different optimizers 254 Scipy lecture notes Edition 2015 2 An ill c
280. ngs containing max 7 letters dtype S7 Much more e int32 e int64 e uint32 e uint64 3 1 4 Basic visualization Now that we have our first data arrays we are going to visualize them Start by launching IPython ipython Or the notebook ipython notebook Once IPython has started enable interactive plots 3 1 The Numpy array object 47 Scipy lecture notes Edition 2015 2 gt gt gt matplotlib Or from the notebook enable plots in the notebook gt gt gt jmatplotlib inline The inline is important for the notebook so that plots are displayed in the notebook and not in a new window Matplotlib is a 2D plotting package We can import its functions as below gt gt gt import matplotlib pyplot as plt the tidy way And then use note that you have to use show explicitly if you have not enabled interactive plots with yAMatplotlib gt gt gt plt plot x y line plot gt gt gt plt show lt shows the plot not needed with interactive plots Or if you have enabled interactive plots with matplotlib gt gt gt plot x y Line plot e 1D plotting gt gt gt x np linspace 0 3 20 gt gt gt y np linspace 0 9 20 gt gt gt plt plot x y line plot lt matplotlib lines Line2D object at gt gt gt gt plt plot x 7 o dot plot lt matplotlib lines Line2D object at gt O 0 0 0 5 1 0 1 5 2 0 2 5 3 0 e 2D arrays such as images 3 1 Th
281. nor ticks can be located and formatted independently from each other Per default minor ticks are not shown i e there is only an empty list for them because it is as NullLocator see below Tick Locators Tick locators control the positions of the ticks They are set as follows ax plt gca ax xaxis set_major_locator eval locator NullLocator MultipleLocator 1 0 ESP va eves rar a OV VY Yn CUS CF OV PO SO SP PES E Ur ar Vr CW CY UY ES CY OO OO SO CE 0 1 2 3 4 5 6 7 8 9 1C FixedLocator 0 2 8 9 10 0 2 8 9 1C IndexLocator 3 1 1 4 7 1C LinearLocator 5 0 2 5 5 0 7 5 10 LogLocator 2 1 0 0510 20 4 0 8 0 AutoLocator There are several locators for different kind of requirements n i j All of these locators derive from the base class matplotlib ticker Locator You can make your own lo cator deriving from it Handling dates as ticks can be especially tricky Therefore matplotlib provides special locators in matplotlib dates 4 4 Other Types of Plots examples and exercises 4 4 Other Types of Plots examples and exercises 90 Scipy lecture notes Edition 2015 2 4 4 1 Regular Plots You need to use the fill between command Starting from the code below try to reproduce the graphic on the right taking care of filled areas 256 np linspace np pi np pi n endpoint True np sin 2 X plt plot X Y 1 color blue alpha 1 00 plt plot X Y 1
282. not the amplitudes of the terms we can find those by least squares curve fitting First we have to define the function to fit gt gt gt def f2 x a b return a x 2 b np sin x Then we can use scipy optimize curve_fit to find a and b gt gt gt guess 2 2 gt gt gt params params_covariance optimize curve_fit f2 xdata ydata guess gt gt gt params array 0 99667386 10 17808313 Now we have found the minima and roots of f and used curve fitting on it we put all those resuls together in a single plot 5 5 Optimization and fit scipy optimize 111 Scipy lecture notes Edition 2015 2 120 f x 100 Curve fit result e e Minima 80 v v Roots 60 x 40 20 0 20 10 5 0 5 10 X In Scipy gt 0 11 unified interfaces to all minimization and root finding algorithms are available scipy optimize minimize scipy optimize minimize_scalar and scipy optimize root They allow comparing various algorithms easily through the method keyword You can find algorithms with the same functionalities for multi dimensional problems in scipy optimize Exercise Curve fitting of temperature data The temperature extremes in Alaska for each month starting in January are given by in degrees Celcius max 17 9 21 28 383 38 37 37 31 23 19 16 min O27 094 965 46 oo Slo 9 613 oS 2oy 46 hoe oo 1 Plot these temperature extremes 2 Define a function that can describe min and
283. np array 1 1 0 0 dtype bool gt gt gt b np array 1 0 1 0 dtype bool gt gt gt np logical_or a b array True True True False dtype bool gt gt gt np logical_and a b array True False False False dtype bool1 Transcendental functions gt gt gt a np arange 5 gt gt gt np sin a array lO 0 84147098 0 90929743 0 14112001 0 7568025 gt gt gt np log a array Sane 3 Ox 0 69314718 1 09861229 1 38629436 gt gt gt np exp a array 1 2 1220193 173890561 20 03553692 54 593150031 3 2 Numerical operations on arrays 56 Scipy lecture notes Edition 2015 2 Shape mismatches gt gt gt a np arange 4 gt gt gt a np array 1 2 Traceback most recent call last File lt stdin gt line 1 in lt module gt ValueError operands could not be broadcast together with shapes 4 2 Traceback most recent call last File lt stdin gt line 1 in lt module gt ValueError operands could not be broadcast together with shapes 4 2 Broadcasting We ll return to that later Transposition gt gt gt a np triu np ones 3 3 1 see help np triu gt gt gt a array l O0 dti Ll k Osr One Aala Os 40m OTO gt gt gt a T array CELOng yOL 0 ode Ora Oris Eaka a cO A The transposition is a view As a results the following code is wrong and will not make a matrix symmetric gt gt gt a a T It will work
284. ns in DLLs or shared libraries It can be used to wrap these libraries in pure Python Advantages e Part of the Python standard library e Does not need to be compiled e Wrapping code entirely in Python 14 3 Ctypes 268 Scipy lecture notes Edition 2015 2 Disadvantages e Requires code to be wrapped to be available as a shared library roughly speaking d11 in Windows so in Linux and dylib in Mac OSX e No good support for C 14 3 1 Example As advertised the wrapper code is in pure Python wu Example of wrapping cos function from math h using ctypes import ctypes from ctypes util import find_library find and load the library libm ctypes cdll LoadLibrary find_library m set the argument type libm cos argtypes ctypes c_double set the return type libm cos restype ctypes c_double def cos_func arg iT Wrapper for cos from math h 13 return libm cos arg e Finding and loading the library may vary depending on your operating system check the documentation for details e This may be somewhat deceptive since the math library exists in compiled form on the system already If you were to wrap a in house library you would have to compile it first which may or may not require some additional effort We may now use this as before In 1 import cos_module Ia 2 cos_module Type module string Form lt module cos_module from cos_module py gt File home esc git w
285. nts 1 astype np int 1 gt gt gt im ndimage gaussian_filter im sigma 1 4 n gt gt gt mask im gt im mean astype np float gt gt gt mask 0 1 im gt gt gt img mask 0 2 np random randn mask shape gt gt gt hist bin_edges np histogram img bins 60 gt gt gt bin_centers 0 5 bin_edges 1 bin_edges 1 gt gt gt binary_img img gt 0 5 histogram Use mathematical morphology to clean up the result gt gt gt Remove small white regions gt gt gt open_img ndimage binary_opening binary_img gt gt gt Remove small black hole gt gt gt close_img ndimage binary_closing open_img Exercise Check that reconstruction operations erosion propagation produce a better result than open ing closing gt gt gt eroded_img ndimage binary_erosion binary_img gt gt gt reconstruct_img ndimage binary_propagation eroded_img mask binary_img gt gt gt tmp np logical_not reconstruct_img gt gt gt eroded_tmp ndimage binary_erosion tmp gt gt gt reconstruct_final np logical_not ndimage binary_propagation eroded_tmp mask tmp gt gt gt np abs mask close_img mean O700T 27636 gt gt gt np abs mask reconstruct_final mean 0000595027 12 5 Feature extraction 241 Scipy lecture notes Edition 2015 2 Exercise Check how a first denoising step e g with a median filter modifies the histogram and chec
286. o be an expert and tackles various specific topics 141 CHAPTER Advanced Python Constructs Author Zbigniew Jedrzejewski Szmek This section covers some features of the Python language which can be considered advanced in the sense that not every language has them and also in the sense that they are more useful in more complicated pro grams or libraries but not in the sense of being particularly specialized or particularly complicated It is important to underline that this chapter is purely about the language itself about features supported through special syntax complemented by functionality of the Python stdlib which could not be implemented through clever external modules The process of developing the Python programming language its syntax is very transparent proposed changes are evaluated from various angles and discussed via Python Enhancement Proposals PEPs As a result features described in this chapter were added after it was shown that they indeed solve real problems and that their use is as simple as possible Chapter contents e Iterators generator expressions and generators Iterators Generator expressions Generators Bidirectional communication Chaining generators e Decorators Replacing or tweaking the original object Decorators implemented as classes and as functions Copying the docstring and other attributes of the original function Examples in the standard library
287. o debug for example SWIG Cython comes with an extension to the GNU debugger that helps debug Python Cython and C code The autogenerated C code uses the Python C Api Advantages e Python like language for writing C extensions e Autogenerated code e Supports incremental optimization e Includes a GNU debugger extension e Support for C Since version 0 13 Disadvantages e Must be compiled e Requires an additional library but only at build time at this problem can be overcome by shipping the generated C files 14 5 1 Example The main Cython code for our cos_module is contained in the file cos_module pyx nnm wu Example of wrapping cos Junction from math h using Cython cdef extern from math h double cos double arg def cos_func arg return cos arg Note the additional keywords such as cdef and extern Also the cos_func is then pure Python Again we can use the standard distutils module but this time we need some additional pieces from the Cython Distutils 14 5 Cython 276 Scipy lecture notes Edition 2015 2 from distutils core import setup Extension from Cython Distutils import build_ext setup cmdclass build_ext build_ext ext_modules Extension cos_module cos_module pyx Compiling this cd advanced interfacing with_c cython 1s cos_module pyx setup py python setup py build_ext inplace running build_ext cythoning cos_module pyx to cos_module c building co
288. o look like the wrapped function gt gt gt import functools gt gt gt def replacing decorator_with_args arg print defining the decorator def _decorator function print doing decoration r arg def _wrapper args xkwargs print inside wrapper r Zr args kwargs return function args xkwargs return functools update_wrapper _wrapper function return _decorator gt gt gt replacing decorator_with_args abc def function extensive documentation print inside function return 14 defining the decorator doing decoration abc gt gt gt function lt function function at Ox gt gt gt gt print function __doc__ extensive documentation One important thing is missing from the list of attributes which can be copied to the replacement func tion the argument list The default values for arguments can be modified through the __defaults__ __kwdefaults__ attributes but unfortunately the argument list itself cannot be set as an attribute This means that help function will display a useless argument list which will be confusing for the user of the function An effective but ugly way around this problem is to create the wrapper dynamically using eval This can be automated by using the external decorator module It provides support for the decorator decorator which takes a wrapper and turns it into a decorator which preserves the function signature To sum things up decorators should alwa
289. odules Extension _cos_doubles sources cos_doubles c cos_doubles i include_dirs numpy get_include As previously we need to use include_dirs to specify the location 1s cos_doubles c cos_doubles h cos_doubles i numpy i setup py test_cos_doubles py python setup py build_ext i running build_ext building _cos_doubles extension swigging cos_doubles i to cos_doubles_wrap c swig python o cos_doubles_wrap c cos_doubles i cos_doubles i 24 Warning 490 Fragment NumPy_Backward_Compatibility not found cos_doubles i 24 Warning 490 Fragment NumPy_Backward_Compatibility not found cos_doubles i 24 Warning 490 Fragment NumPy_Backward_Compatibility not found creating build creating build temp linux x86_64 2 7 gcc pthread fno strict aliasing g 02 DNDEBUG g fwrapv 03 Wall Wstrict prototypes fPIC I home esc an gcc pthread fno strict aliasing g 02 DNDEBUG g fwrapv 03 Wall Wstrict prototypes fPIC I home esc an In file included from home esc anaconda lib python2 7 site packages numpy core include numpy ndarraytypes h 17 from home esc anaconda lib python2 7 site packages numpy core include numpy ndarrayobject h 1 from home esc anaconda lib python2 7 site packages numpy core include numpy arrayobject h 15 from cos_doubles_wrap c 2706 _ home esc anaconda 1lib python2 7 site packages numpy core include numpy npy_deprecated_api h 11 2 warning wa gcc pthread shared bu
290. odules meaningful names not mathematics names 9 1 2 pyflakes fast static analysis They are several static analysis tools in Python to name a few e pylint e pychecker e pyflakes e pep8 e flake8 Here we focus on pyf lakes which is the simplest tool e Fast simple e Detects syntax errors missing imports typos on names Another good recommendation is the flake8 tool which is a combination of pyflakes and pep8 Thus in addition to the types of errors that pyflakes catches flake8 detects violations of the recommendation in PEP8 style guide Integrating pyflakes or flake8 in your editor or IDE is highly recommended it does yield productivity gains Running pyflakes on the current edited file You can bind a key to run pyflakes in the current buffer e In kate Menu settings gt configure kate In plugins enable external tools In external Tools add pyflakes kdialog title pyflakes filename msgbox pyflakes filename e In TextMate Menu TextMate gt Preferences gt Advanced gt Shell variables add a shell variable TM_PYCHECKER Library Frameworks Python framework Versions Current bin pyflakes Then Ctrl Shift V is binded to a pyflakes report e In vim In your vimrc binds F5 to pyf lakes autocmd FileType python let amp mp echo running pyflakes autocmd FileType tex mp rst python imap lt Esc gt 15 lt C 0 gt make M autocmd FileType tex mp rst python map lt
291. of the array if it s a sub array gt gt gt np dtype int type lt type numpy int64 gt gt gt gt np dtype int itemsize 8 gt gt gt np dtype int byteorder Example reading wav files The wav file header 8 1 Life of ndarray 162 Scipy lecture notes Edition 2015 2 4 byte unsigned little endian integer e 44 byte block of raw data in the beginning of the file e followed by data_size bytes of actual sound data The wav file header as a Numpy structured data type gt gt gt wav_header_dtype np dtype Chunk_id bytes 4 flexible sized scalar type item size 4 chunk size lt u4 little endian unsigned 32 bit integer format TSA 4 byte string imt 1da 1S4 Clint size uan audio fmt lt u2 num_channels lt u2 more of the same sample_rate lt u4 byte_rate lt u4 block_align lt u2 bits_per_sample lt u2 data _ id S1 2 2 sub array Just for fun data size Tuani the sound data itself cannot be represented here it does not have a fixed size See also wavreader py gt gt gt wav_header_dtype format dtype S4 gt gt gt wav_header_dtype fields dict _proxy block align dtrypel uint16 32 format diypet st Jo 5 data _id dtypel S51 2 2 gt gt gt wav_header_dtype fields format dtype S4 8 e The first element
292. of the linear system Has shape N or N 1 e optional x0 array matrix Starting guess for the solution tol float Relative tolerance to achieve before terminating maxiter integer Maximum number ofiterations Iteration will stop after maxiter steps even if the spec ified tolerance has not been achieved M sparse matrix dense matrix LinearOperator Preconditioner for A The preconditioner should ap proximate the inverse of A Effective preconditioning dramatically improves the rate of conver gence which implies that fewer iterations are needed to reach a given error tolerance callback function User supplied function to call after each iteration It is called as callback xk where xk is the current solution vector LinearOperator Class from scipy sparse linalg interface import LinearQperator e common interface for performing matrix vector products useful abstraction that enables using dense and sparse matrices within the solvers as well as matrix free solutions e has shape and matvec some optional parameters e example gt gt gt import numpy as np gt gt gt from scipy sparse linalg import LinearOperator gt gt gt def mv v return np array 2 vl0 3 v 1 gt gt gt A LinearOperator 2 2 matvec mv 11 3 Linear System Solvers 226 Scipy lecture notes Edition 2015 2 gt gt gt A lt 2x2 LinearOperator with unspecified dtype gt gt gt gt A matvec np ones 2 array l 2
293. omplete declarations for the C math library which simplifies the code above to become n Simpler example of wrapping cos function from math h using Cython from libc math cimport cos def cos_func arg return cos arg In this case the cimport statement is used to import the cos function 14 5 2 Numpy Support Cython has support for Numpy via the numpy pyx file which allows you to add the Numpy array type to your Cython code I e like specifying that variable i is of type int you can specify that variable a is of type numpy ndarray with a given dtype Also certain optimizations such as bounds checking are supported Look at the corresponding section in the Cython documentation In case you want to pass Numpy arrays as C arrays to your Cython wrapped C functions there is a section about this in the Cython wiki In the following example we will show how to wrap the familiar cos_doubles function using Cython void cos_doubles double in_array double out_array int size include lt math h gt Compute the cosine of each element in in_array storing the result in out array void cos_doubles double in_array double out_array int size int i for i 0 i lt size i out_array i cos in_arrayLi This is wrapped as cos_doubles_func using the following Cython code n Ezample of wrapping a C function that takes C double arrays as input using the Numpy declarations from Cython
294. on debugger pdb https docs python org library pdb html allows you to inspect your code inter actively Specifically it allows you to View the source code Walk up and down the call stack Inspect values of variables Modify values of variables Set breakpoints print Yes print statements do work as a debugging tool However to inspect runtime it is often more efficient to use the debugger 9 2 Debugging workflow 195 Scipy lecture notes Edition 2015 2 9 3 1 Invoking the debugger Ways to launch the debugger 1 Postmortem launch debugger after module errors 2 Launch the module with the debugger 3 Call the debugger inside the module Postmortem Situation You re working in Python and you get a traceback Here we debug the file index_error py When running it an IndexError is raised Type idebug and drop into the debugger In 1 run index_error py IndexError Traceback most recent call last home varoquau dev scipy lecture notes advanced debugging index_error py in lt module gt 6 T if _name _main_ gt 8 index_error 9 home varoquau dev scipy lecture notes advanced debugging index_error py in index_error 3 def index_error 4 tst List foobar gt 5 print lst len 1st 6 T if _name _main_ IndexError list index out of range In 2 debug gt home varoquau dev scipy lecture notes advanced debugging index_error py 5 index_error 4 lst list
295. onditionned quadratic function Note that as the quadratic approximation is exact the Newton method is blazing fast An ill conditionned non quadratic function Here we are optimizing a Gaussian which is always below its quadratic approximation As a result the Newton method overshoots and leads to oscillations An ill conditionned very non quadratic function In scipy the Newton method for optimization is implemented in scipy optimize fmin_ncg cg here refers to that fact that an inner operation the inversion of the Hessian is performed by conjugate gradient scipy optimize fmin_tnc can be use for constraint problems although it is less versatile gt gt gt def f x The rosenbrock function return 5 1 x 0 2 x 1 x 0 2 2 gt gt gt def fprime x return np array 2 5 1 x 0 4 xlO x 1 x 0 2 2 x 1 x l0 2 gt gt gt optimize fmin_ncg f 2 2 fprime fprime Optimization terminated successfully Current function value 0 000000 Iterations 9 Function evaluations 11 Gradient evaluations 51 Hessian evaluations 0 array ili ID Note that compared to a conjugate gradient above Newton s method has required less function evaluations but more gradient evaluations as it uses it to approximate the Hessian Let s compute the Hessian and pass it to the algorithm gt gt gt def hessian x Computed with sympy return np array
296. ones zeros eye and diag e Create different kinds of arrays with random numbers e Try setting the seed before creating an array with random values e Look at the function np empty What does it do When might this be useful 3 1 3 Basic data types You may have noticed that in some instances array elements are displayed with a trailing dot e g 2 vs 2 This is due to a difference in the data type used gt gt gt a np array 1 2 3 gt gt gt a dtype dtype int64 gt gt gt b np array 1 2 3 3 1 The Numpy array object 46 Scipy lecture notes Edition 2015 2 gt gt gt b dtype dtype float64 Different data types allow us to store data more compactly in memory but most of the time we simply work with floating point numbers Note that in the example above NumPy auto detects the data type from the input You can explicitly specify which data type you want gt gt gt c np array 1 2 3 dtype float gt gt gt c dtype dtype float64 The default data type is floating point gt gt gt a np ones 3 3 gt gt gt a dtype dtype float64 There are also other types Complex gt gt gt d np array 1 2j 3 4j 5 6 1j gt gt gt d dtype dtype complex128 Bool gt gt gt e np array True False False True gt gt gt e dtype dtype bool Strings gt gt gt f np array Bonjour Hello Hallo gt gt gt dtype lt stri
297. op e Zcpaste allows you to paste code especially code from websites which has been prefixed with the stan dard Python prompt e g gt gt gt or with an ipython prompt e g in 3 In 5 cpaste Pasting code enter alone on the line to stop or use Ctrl D In 3 timeit x 10 10000000 loops best of 3 85 9 ns per loop In 6 cpaste Pasting code enter alone on the line to stop or use Ctrl D gt gt gt timeit x 10 10000000 loops best of 3 86 ns per loop e debug allows you to enter post mortem debugging That is to say if the code you try to execute raises an exception using debug will enter the debugger at the point where the exception was thrown In 7 x 10 File lt ipython input 6 12fd421b5f28 gt line 1 SyntaxError invalid syntax 1 3 The interactive workflow IPython and a text editor 8 Scipy lecture notes Edition 2015 2 In 8 debug gt IPython core compilerop py 87 ast_parse 86 and are passed to the built in compile fumction or return compile source filename symbol self flags PyCF_ONLY_AST 1 88 ipdb gt locals source u x 10 n symbol exec self lt IPython core compilerop CachingCompiler instance at Ox2ad8ef0 gt filename lt ipython input 6 12fd421b5f28 gt IPython help e The built in Python cheat sheet is accessible via the quickref magic function e A list of all available magic functions is shown when
298. or this specific situation Hint you know the size of the output array so it may be possible to construct a Numpy array from the returned double 14 8 4 Cython l Look at the code that Cython autogenerates Take a closer look at some of the comments that Cython inserts What do you see Look at the section Working with Numpy from the Cython documentation to learn how to incrementally optimize a pure python script that uses Numpy Modify the Numpy example such that cos_doubles_func handles the preallocation for you thus mak ing it more like the Numpy C API example 14 8 Exercises 281 Part III Packages and applications 282 Scipy lecture notes Edition 2015 2 This part of the Scipy lecture notes is dedicated to various scientific packages useful for extended needs 283 CHAPTER 15 Statistics in Python Author Ga l Varoquaux Requirements e Standard scientific Python environment numpy scipy matplotlib e Pandas e Statsmodels e Seaborn To install Python and these dependencies we recommend that you download Anaconda Python or En thought Canopy or preferably use the package manager if you are under Ubuntu or other linux See also Bayesian statistics in Python This chapter does not cover tools for Bayesian statistics Of particular interest for Bayesian modelling is PyMC which implements a probabilistic programming language in Python Why Python for statistics R is a language dedicate
299. orage Property depends_on inflows release control attributes inflows Float desc Inflows hm3 release Range low min_release high max_release spillage Property desc Spillage hm3 depends_on storage inflows release Private traits HHHHHHHHH 18 3 What are Traits 335 Scipy lecture notes Edition 2015 2 _storage Float Traits property implementation def _get_storage self new_storage self _storage self release self inflows return min new_storage self max_storage def _set_storage self storage_value self _storage storage_value def _get_spillage self new_storage self _storage self release self inflows overflow new_storage self max_storage return max overflow 0 on_trait_change storage def print_state self print Storage tRelease tInflows tSpillage str_format t join 7 2f for i in range 4 print str_format format self storage self release self inflows self spillage print 79 __name__ main _ projectA Reservoir name Project A max_storage 30 max release 5 hydraulic_head 60 0 8 efficiency state ReservoirState reservoir projectA storage 25 state release 4 state inflows The patterns supported by the on_trait_change method and decorator are powerful The reader should look at the docstring of HasTraits on_trait_change for the details 18 3 7 Some more advanced traits Th
300. orking scipy lecture notes advanced interfacing with_c ctypes cos_module py Docstring lt no docstring gt In 3 dir cos_module Gurls e b iltins ek ge o eoe i __file__ __name__ __package__ cos_func ctypes find_library libm In 4 cos_module cos_func 1 0 Out 4 0 5403023058681398 In 5 cos_module cos_func 0 0 Dusel 120 14 3 Ctypes 269 Scipy lecture notes Edition 2015 2 In 6 cos_module cos_func 3 14159265359 Out 6 1 0 As with the previous example this code is somewhat robust although the error message is not quite as helpful since it does not tell us what the type should be In 7 cos_module cos_func foo ArgumentError Traceback most recent call last lt ipython input 7 11bee483665d gt in lt module gt gt 1 cos_module cos_func foo home esc git working scipy lecture notes advanced interfacing with_c ctypes cos_module py in cos_func arg 12 def cos_func arg 13 u1 Wrapper for cos from math bh gt 14 return libm cos arg ArgumentError argument 1 lt type exceptions TypeError gt wrong type 14 3 2 Numpy Support Numpy contains some support for interfacing with ctypes In particular there is support for exporting certain attributes of a Numpy array as ctypes data types and there are functions to convert from C arrays to Numpy arrays and back For more information consult the corresponding section in the Numpy Cookbook and the API
301. ort the importlib module first and then do In 10 importlib reload demo 2 5 4 __main__ and module loading Sometimes we want code to be executed when a module is run directly but not when it is imported by another module if __name__ __main__ allows us to check whether the module is being run directly File demo2 py def print_b Prints D print b def print_a Q Prints a print a print_b runs on import print_b if __name__ __main__ print_a is only executed when the module is run directly print_a Importing it In 11 import demo2 b In 12 import demo2 Running it In 13 run demo2 b a 2 5 5 Scripts or modules How to organize your code Rule of thumb e Sets of instructions that are called several times should be written inside functions for better code reusability e Functions or other bits of code that are called from several scripts should be written inside a module so that only the module is imported in the different scripts do not copy and paste your functions in the different scripts How modules are found and imported When the import mymodule statement is executed the module mymodule is searched in a given list of direc tories This list includes a list of installation dependent default path e g usr lib python as well as the 2 5 Reusing code scripts and modules 31 Scipy lecture notes Edition 2015 2 list of directories specif
302. ould cluster the observations into several groups by some criterion 20 3 Clustering grouping observations together 357 Scipy lecture notes Edition 2015 2 20 3 1 K means clustering The simplest clustering algorithm is k means This divides a set into k clusters assigning each observation to a cluster so as to minimize the distance of that observation in n dimensional space to the cluster s mean the means are then recomputed This operation is run iteratively until the clusters converge for a maximum for max_iter rounds An alternative implementation of k means is available in SciPy s cluster package The scikit learn im plementation differs from that by offering an object API and several additional features including smart ini tialization gt gt gt from sklearn import cluster datasets gt gt gt iris datasets load_iris gt gt gt k_means cluster KMeans n_clusters 3 gt gt gt k_means fit iris data KMeans gt gt gt print k_means labels_ 10 111110000022222 gt gt gt print iris target 10 O0 0 00A A2222 e e e o e e oth Virginica we Li Petal length Versicolour Petal length Petal length K means 3 clusters K means 8 clusters Application to Image Compression Clustering can be seen as a way of choosing a small number of information from the observations like a projection on a smaller space For instance t
303. pace 3 3 n Y np meshgrid x y plt contourf X Y f X Y 8 alpha 75 cmap jet C plt contour X Y f X Y 8 colors black linewidth 5 Click on figure for solution 4 4 Other Types of Plots examples and exercises 92 Scipy lecture notes Edition 2015 2 4 4 5 Imshow You need to take care of the origin of the image in the imshow command and use a col Starting from the code below try to reproduce the graphic on the right taking care of colormap image inter polation and origin f z y return 1 x 2 x 5 y 3 np exp x 2 y 2 10 np linspace 3 3 4 n np linspace 3 3 3 n X Y np meshgrid x y plt imshow f X Y Click on the figure for the solution 4 4 6 Pie Charts WD T Starting from the code below try to reproduce the graphic on the right taking care of colors and slices size You need to modify Z Z np random uniform 0 1 20 plt pie Z Click on the figure for the solution 4 4 Other Types of Plots examples and exercises 93 Scipy lecture notes Edition 2015 2 4 4 7 Quiver Plots wher 4 AA w bf A A ww AXA 444 amp E E FY NWA ee y YS F YY YN You need to draw arrows twice Starting from the code above try to reproduce the graphic on the right taking care of colors and orientations n 8 X Y np mgrid 0O n O n plt quiver X Y Click on figure for solution 4 4 8 Grids
304. place immutable objects cannot be modified once created See also A very good and detailed explanation of the above issues can be found in David M Beazley s article Types and Objects in Python 2 3 Control Flow Controls the order in which the code is executed 2 3 1 if elif else gt gt gt if 2 2 Princ O0bvious 2 3 Control Flow 18 Scipy lecture notes Edition 2015 2 Obvious Blocks are delimited by indentation Type the following lines in your Python interpreter and be careful to respect the indentation depth The Ipython shell automatically increases the indentation depth after a column sign to decrease the indentation depth go four spaces to the left with the Backspace key Press the Enter key twice to leave the logical block gt gt gt a 10 gt gt gt if a pring 1 elif a print 2 else print A lot A lot Indentation is compulsory in scripts as well As an exercise re type the previous lines with the same indenta tion in a script condition py and execute the script with run condition py in Ipython 2 3 2 for range Iterating with an index gt gt gt for i in range 4 plane a wW Ne Ore But most often it is more readable to iterate over values gt gt gt for word in cool powerful readable print Python is s word Python is cool Python is powerful Python is readable 2 3 3 while break continue Typical C style whil
305. preferred In 3 timeit np linalg svd data 1 loops best of 3 14 5 s per loop In 4 from scipy import linalg In 5 timeit linalg svd data 1 loops best of 3 14 2 s per loop In 6 timeit linalg svd data full_matrices False 1 loops best of 3 295 ms per loop 10 3 Making code go faster 206 Scipy lecture notes Edition 2015 2 In 7 timeit np linalg svd data full_matrices False 1 loops best of 3 293 ms per loop We can then use this insight to optimize the previous code def test data np random random 5000 100 u s v linalg svd data full_matrices False pea np dot ul 10 T data results fastica pca T whiten False In 1 import demo In 2 timeit demo demo fastica demo np demo prof pdf demo py demo pyc demo linalg demo prof demo prof png demo py lprof demo test In 2 timeit demo test ica py 65 RuntimeWarning invalid value encountered in sqrt W u np diag 1 0 np sqrt s u T W W W W T 1 2 W 1 loops best of 3 17 5 s per loop In 3 import demo_opt In 4 timeit demo_opt test 1 loops best of 3 206 ms per loop Real incomplete SVDs e g computing only the first 10 eigenvectors can be computed with arpack available in scipy sparse linalg eigsh Computational linear algebra For certain algorithms many of the bottlenecks will be linear algebra computations In this case using the right function to solve the right problem is k
306. preprocessing enhancement Goals denoising feature edges extraction 17 3 1 Local filters Local filters replace the value of pixels by a function of the values of neighboring pixels The function can be linear or non linear Neighbourhood square choose size disk or more complicated structuring element Example horizontal Sobel filter gt gt gt text data text gt gt gt hsobel_text filters hsobel text Uses the following linear kernel for computing horizontal gradients 17 3 Image preprocessing enhancement 312 Scipy lecture notes Edition 2015 2 17 3 2 Non local filters Non local filters use a large region of the image or all the image to transform the value of one pixel gt gt gt from skimage import exposure gt gt gt camera data camera gt gt gt camera_equalized exposure equalize_hist camera Enhances contrast in large almost uniform regions 17 3 3 Mathematical morphology See wikipedia for an introduction on mathematical morphology Probe an image with a simple shape a structuring element and modify this image according to how the shape locally fits or misses the image Default structuring element 4 connectivity of a pixel gt gt gt from skimage import morphology gt gt gt morphology diamond 1 array 0 1 01 Efe Aik 0 1 OJ dtype uint8 Erosion minimum filter Replace the value of a pixel by the minimal value covered
307. py Get this tutorial http www euroscipy org talk 882 8 6 1 Why e There s a bug e I don t understand what this is supposed to do e Ihave this fancy code Would you like to have it e Td like to help What can I do 8 6 2 Reporting bugs e Bug tracker prefer this http projects scipy org numpy http projects scipy org scipy Click the Register link to get an account e Mailing lists scipy org Mailing Lists If you re unsure 8 5 Summary 188 Scipy lecture notes Edition 2015 2 No replies in a week or so Just file a bug ticket Good bug report Title numpy random permutations fails for non integer arguments I m trying to generate random permutations using numpy random permutations When calling numpy random permutation with non integer arguments it fails with a cryptic error message gt gt gt np random permutation 12 array L 6 11 4 10 2 Sa 1 oe 3 0 5I gt gt gt np random permutation 12 Traceback most recent call last File lt stdin gt line 1 in lt module gt File mtrand pyx line 3311 in mtrand RandomState permutation File mtrand pyx line 3254 in mtrand RandomState shuffle TypeError len of unsized object This also happens with long arguments and so np random permutation X shape 0 where X is an array fails on 64 bit windows where shape is a tuple of longs It would be great if it could cast to integer or at least
308. py As non professional programmers scientists often tend to re invent the wheel which leads to buggy non optimal difficult to share and unmaintainable code By contrast Scipy s routines are optimized and tested and should therefore be used when possible Chapters contents e File input output scipy io e Special functions scipy special e Linear algebra operations scipy linalg e Fast Fourier transforms scipy fftpack e Optimization and fit scipy optimize e Statistics and random numbers scipy stats e Interpolation scipy interpolate e Numerical integration scipy integrate e Signal processing scipy signal e Image processing scipy ndimage e Summary exercises on scientific computing A This tutorial is far from an introduction to numerical computing As enumerating the different submodules and functions in scipy would be very boring we concentrate instead on a few examples to give a general idea of how to use scipy for scientific computing scipy is composed of task specific sub modules 101 Scipy lecture notes Edition 2015 2 Vector quantization Kmeans They all depend on numpy but are mostly independent of each other The standard way of importing Numpy and these Scipy modules is gt gt gt import numpy as np gt gt gt from scipy import stats same for other sub modules The main scipy namespace mostly contains functions that are really numpy functions try scipy cos is np cos Those are exposed
309. py gt 3 2 concurrent futures ProcessPoolExecutor invoke in parallel then kill process pool py gt 3 2 nogil solve the GIL problem temporarily cython only 7 3 1 Catching exceptions When an exception is thrown in the with block it is passed as arguments to __exit__ Three arguments are used the same as returned by sys exc_info type value traceback When no exception is thrown None is used for all three arguments The context manager can swallow the exception by returning a true value from 7 3 Context managers 156 Scipy lecture notes Edition 2015 2 __exit__ Exceptions can be easily ignored because if __exit__ doesn t use return and just falls of the end None is returned a false value and therefore the exception is rethrown after ___exit__ is finished The ability to catch exceptions opens interesting possibilities A classic example comes from unit tests we want to make sure that some code throws the right kind of exception class assert_raises object based on pytest and unittest TestCase def _init__ self type self type type def enter self pass def __exit__ self type value traceback if type is None raise AssertionError exception expected if issubclass type self type return True swallow the expected exception raise AssertionError wrong exception type with assert_raises KeyError foo 7 3 2 Using generators to define context managers
310. py lecture notes Edition 2015 2 Example docstring mlab mesh Plots a surface using grid spaced data supplied as 2D arrays Function signatures mesh x y z X y Zare 2D arrays all of the same shape giving the positions of the vertices of the surface The connec tivity between these points is implied by the connectivity on the arrays For simple structures such as orthogonal grids prefer the surf function as it will create more efficient data structures Keyword arguments color the color of the vtk object Overides the colormap if any when specified This is specified as a triplet of float ranging from 0 to 1 eg 1 1 1 for white colormap type of colormap to use extent xmin xmax ymin ymax zmin zmax Default is the x y z arrays extents Use this to change the extent of the object created figure Figure to populate line_width The with of the lines if any used Must be a float Default 2 0 mask boolean mask array to suppress some data points mask_points If supplied only one out of mask_points data point is displayed This option is usefull to reduce the number of points displayed on large datasets Must be an integer or None mode the mode of the glyphs Must be 2darrow or 2dcircle or 2dcross or 2ddash or 2ddiamond or 2dhooked_arrow or 2dsquare or 2dthick_arrow or 2dthick_cross or 2dtriangle or 2dvertex or arrow or cone or cube
311. py lecture notes Edition 2015 2 Fitting a waveform with a simple Gaussian model The signal is very simple and can be modeled as a single Gaussian function and an offset corresponding to the background noise To fit the signal with the function we must e define the model e propose an initial solution e callscipy optimize leastsq Model A Gaussian function defined by can be defined in python by gt gt gt def model t coeffs return coeffs 0 coeffsli np exp t coeffs 2 coeffs 3 2 where e coeffs 0 is B noise e coeffs 1i is A amplitude e coeffs 2 is u center e coeffs 3 is width Initial solution An approximative initial solution that we can find from looking at the graph is for instance gt gt gt x0 np array 3 30 15 1 dtype float Fit scipy optimize leastsq minimizes the sum of squares of the function given as an argument Basically the function to minimize is the residuals the difference between the data and the model gt gt gt def residuals coeffs y t return y model t coeffs So let s get our solution by calling scipy optimize leastsq with the following arguments e the function to minimize e an initial solution e the additional arguments to pass to the function gt gt gt from scipy optimize import leastsq gt gt gt x flag leastsq residuals x0 args waveform_1 t gt gt gt print x 2 70363341 27 82020742 15 47924562 305636228
312. quation gt gt gt dsolve sin x cos f x cos x sin f x f x diff x f x hint separable Sr ee ere ee X E E ead eee Re Cal Ci f x asin eea 1 pi f x asin M A ekp ist 37 2 2 Nowe cos x 1 NAZ cos x ee AER eee an Se eee Ci CA f x asin 11 x asin 11 l of 2 Il 2 wy cos x NW cos x Exercises 1 Solve the Bernoulli differential equation d f x ALO 5 Fy Fo 0 X 2 Solve the same equation using hint Bernoulli What do you observe 16 5 Linear Algebra 307 CHAPTER 17 Scikit image image processing Author Emmanuelle Gouillart scikit image is a Python package dedicated to image processing and using natively NumPy arrays as image objects This chapter describes how to use scikit image on various image processing tasks and insists on the link with other scientific Python modules such as NumPy and SciPy See also For basic image manipulation such as image cropping or simple filtering a large number of simple operations can be realized with NumPy and SciPy only See Image manipulation and processing using Numpy and Scipy Note that you should be familiar with the content of the previous chapter before reading the current one as basic operations such as masking and labeling are a prerequisite Chapters contents e Introduction and concepts scikit image and the SciPy ecosystem What
313. r Support vector machines SVMs for classification Clustering grouping observations together K means clustering Dimension Reduction with Principal Component Analysis Putting it all together face recognition Linear model from regression to sparsity Sparse models Model selection choosing estimators and their parameters Grid search and cross validated estimators 353 Scipy lecture notes Edition 2015 2 20 1 Loading an example dataset First we will load some data to play with The data we will use is a very simple flower database known as the Iris dataset We have 150 observations of the iris flower specifying some measurements sepal length sepal width petal length and petal width together with its subtype Iris setosa Iris versicolor Iris virginica To load the dataset into a Python object gt gt gt from sklearn import datasets gt gt gt iris datasets load_iris This data is stored in the data member which isa n_samples n_features array gt gt gt iris data shape 150 4 The class of each observation is stored in the target attribute of the dataset This is an integer 1D array of length n_samples gt gt gt iris target shape GEO gt gt gt import numpy as np gt gt gt np unique iris target array 0 1 21 20 1 Loading an example dataset 354 Scipy lecture notes Edition 2015 2 An example of reshaping data the digits dataset SN ODO WwW BP WN KF CO
314. raise a proper error for non integer types I m using Numpy 1 4 1 built from the official tarball on Windows 64 with Visual studio 2008 on Python org 64 bit Python 0 What are you trying to do 1 Small code snippet reproducing the bug if possible e What actually happens e What you d expect 2 Platform Windows Linux OSX 32 64 bits x86 PPC 3 Version of Numpy Scipy gt gt gt print np __version__ eh Check that the following is what you expect gt gt gt print np __file__ In case you have old broken Numpy installations lying around If unsure try to remove existing Numpy installations and reinstall 8 6 3 Contributing to documentation 1 Documentation editor e http docs scipy org doc numpy e Registration Register an account Subscribe to scipy dev mailing list subscribers only 8 6 Contributing to Numpy Scipy 189 Scipy lecture notes Edition 2015 2 Problem with mailing lists you get mail But you can turn mail delivery off change your subscription options at the bottom of http mail scipy org mailman listinfo scipy dev Send a mail scipy dev mailing list ask for activation To scipy dev scipy org Hi I d like to edit Numpy Scipy docstrings My account is XXXXX Cheers N N e Check the style guide http docs scipy org doc numpy Don t be intimidated to fix a small thing just fix it e Edit 2 Edit sources and send patches as for
315. rate 116 Scipy lecture notes Edition 2015 2 As an introduction let us solve the ODE dy dt 2y between t 0 4 with the initial condition y t 0 1 First the function computing the derivative of the position needs to be defined gt gt gt def calc_derivative ypos time counter_arr counter_arr 1 return 2 ypos An extra argument counter_arr has been added to illustrate that the function may be called several times for a single time step until solver convergence The counter array is defined as gt gt gt counter np zeros 1 dtype np uint16 The trajectory will now be computed gt gt gt from scipy integrate import odeint gt gt gt time_vec np linspace 0 4 40 gt gt gt yvec info odeint calc_derivative 1 time_vec args counter full_output True Thus the derivative function has been called more than 40 times which was the number of time steps gt gt gt counter array 129 dtype uint16 and the cumulative number of iterations for each of the 10 first time steps can be obtained by gt gt gt infol nfe 10 array clas 35 43 49 53 57 59 63 65 69 dtype int32 Note that the solver requires more iterations for the first time step The solution yvec for the trajectory can now be plotted 1 0 0 8 0 6 0 4 y position m 0 2 0 0 00 05 10 15 20 25 3 0 3 5 4 0 Time s Another example with scipy integrate odeint will be a damped spring mas
316. ray has the same shape than the array of integers gt gt gt a np arange 10 gt gt gt idx np array 3 4 9 7 gt gt gt idx shape OTa gt gt gt alidx array 3 4 9 7 The image below illustrates various fancy indexing applications gt gt gt al 0 1 2 3 4 1 2 3 4 5 array 1 12 23 34 45 gt gt gt al3 0 2 5 array 30 32 35 40 42 45 50 52 55 gt gt gt mask array 1 0 1 0 0 1 dtype bool gt gt gt a mask 2 array 2 22 52 3 1 The Numpy array object 54 Scipy lecture notes Edition 2015 2 Exercise Fancy indexing e Again reproduce the fancy indexing shown in the diagram above e Use fancy indexing on the left and array creation on the right to assign values into an array for instance by setting parts of the array in the diagram above to zero 3 2 Numerical operations on arrays Section contents e Elementwise operations e Basic reductions e Broadcasting e Array shape manipulation e Sorting data e Summary 3 2 1 Elementwise operations Basic operations With scalars gt gt gt a np array 1 2 3 4 PP arr 1 array 2 3 4 5J gt gt gt 2 a array 2 4 8 16 All arithmetic operates elementwise gt gt gt b np ones 4 1 gt gt gt a b array lar 0y ies 2l gt gt gt a b array Cl 2 4 6 38 41 gt gt gt j np arange 5 gt gt gt 2 j 1 j array L2 3
317. re notes Edition 2015 2 User 14 3929 s System 02256016 S and profile it In 2 run p demo py 916 function calls in 14 551 CPU seconds Ordered by internal time ncalls tottime percall cumtime percall filename lineno function 1 14 457 14 457 14 479 14 479 decomp py 849 svd 1 0 054 0 054 0 054 0 054 method random_sample of mtrand RandomState objects 1 0 017 0 017 0 021 0 021 function_base py 645 asarray_chkfinite 54 Oro 0 000 0 011 0 000 numpy core _dotblas dot 2 0 005 0 002 0 005 0 002 method any of numpy ndarray objects 6 0 001 0 000 0 001 0 000 ica py 195 gprime 6 0 001 0 000 0 001 0 000 ica py 192 g 14 0 001 0 000 0 001 0 000 numpy linalg lapack_lite dsyevd 19 0 001 0 000 0 001 0 000 twodim_base py 204 diag 1 0 001 0 001 0 008 0 008 ica py 69 _ica_par 1 0 001 0 001 14 551 14 551 execfile 107 0 000 0 000 0 001 0 000 defmatrix py 239 __array_finalize__ 7 0 000 0 000 0 004 0 001 ica py 58 _sym_decorrelation 7 0 000 0 000 0 002 0 000 linalg py 841 eigh 172 0 000 0 000 0 000 0 000 fisinstance 1 0 000 0 000 14 551 14 551 demo py 1 lt module gt 29 0 000 0 000 0 000 0 000 numeric py 180 asarray 35 0 000 0 000 0 000 0 000 defmatrix py 193 __new__ 35 0 000 0 000 0 001 0 000 defmatrix py 43 asmatrix 21 0 000 0 000 0 001 0 000 defmatrix py 287 __mul__ 41 0 000 0 000 0 000 0 000 numpy core multiarray zeros 28 0 000 0 000 0 000 0 000 method transpose of numpy ndarray obje
318. re raised In 2 run wiener_filtering py wiener_filtering py 40 RuntimeWarning divide by zero encountered in divide noise level 1 noise l_var We can turn these warnings in exception which enables us to do post mortem debugging on them and find our problem more quickly In 3 np seterr all raise Out 3l d divide print invalid print ever print under ignore In 4 run wiener_filtering py FloatingPointError Traceback most recent call last home esc anaconda 1lib python2 7 site packages IPython utils py3compat pyc in execfile fname where 176 else 177 filename fname gt 178 __builtin__ execfile filename where home esc physique cuso python 2013 scipy lecture notes advanced debugging wiener_filtering py in lt module gt 55 pl matshow noisy_face cut cmap pl cm gray 56 gt 57 denoised_face iterated_wiener noisy_face 58 pl matshow denoised_face cut cmap pl cm gray 59 home esc physique cuso python 2013 scipy lecture notes advanced debugging wiener_filtering py in iterated_wi 38 res noisy_img denoised_img 39 noise res 2 sum res size gt 40 noise_level 1 noise l_var 41 noise_level noise_level lt 0 0 42 denoised_img noise_level res FloatingPointError divide by zero encountered in divide Other ways of starting a debugger e Raising an exception as a poor man break point If you find it tedious to note the line number to
319. read of sequential access is allowed 7 1 Iterators generator expressions and generators 143 Scipy lecture notes Edition 2015 2 7 1 2 Generator expressions A second way in which iterator objects are created is through generator expressions the basis for list compre hensions To increase clarity a generator expression must always be enclosed in parentheses or an expression If round parentheses are used then a generator iterator is created If rectangular parentheses are used the pro cess is short circuited and we get a list gt gt gt i for i in nums lt generator object lt genexpr gt at Ox gt gt gt gt i for i in nums ie es gt gt gt list i for i in nums OO Si The list comprehension syntax also extends to dictionary and set comprehensions A set is created when the generator expression is enclosed in curly braces A dict is created when the generator expression contains pairs of the form key value gt gt gt i for i in range 3 set 0 1 21 gt gt gt i i 2 for i in range 3 t0 0 to AL 2 4 One gotcha should be mentioned in old Pythons the index variable i would leak and in versions gt 3 this is fixed 7 1 3 Generators Generators A generator is a function that produces a sequence of results instead of a single value David Beazley A Curious Course on Coroutines and Concurrency A third way to create iterator objects is to call a generator function A generator
320. red fash blue p fish In 104 slicer rhyme step 2 O t 104 one two gt red plue l In 105 slicer rhyme 1 step 2 Qut 1o05 d fish 2 fish fish fish In 106 slicer rhyme start 1 stop 4 step 2 Qut LOG fish fish The order of the keyword arguments does not matter In 107 slicer rhyme step 2 start 1 stop 4 Qutlilo l fish fash but it is good practice to use the same ordering as the function s definition Keyword arguments are a very convenient feature for defining functions with a variable number of arguments especially when default values are to be used in most calls to the function 2 4 4 Passing by value Can you modify the value of a variable inside a function Most languages C Java distinguish passing by value and passing by reference In Python such a distinction is somewhat artificial and it is a bit subtle whether your variables are going to be modified or not Fortunately there exist clear rules Parameters to functions are references to objects which are passed by value When you pass a variable to a function python passes the reference to the object to which the variable refers the value Not the variable itself If the value passed in a function is immutable the function does not modify the caller s variable If the value is mutable the function may modify the caller s variable in place gt gt gt def try_to_modify x
321. resize 8 gt gt gt a array l0 1 2 3 0 0 0 0 3 2 Numerical operations on arrays 66 Scipy lecture notes Edition 2015 2 However it must not be referred to somewhere else gt gt gt b a gt gt gt a resize 4 Traceback most recent call last File lt stdin gt line 1 in lt module gt ValueError cannot resize an array that has been referenced or is referencing another array in this way Use the resize function Traceback most recent call last File lt stdin gt line 1 in lt module gt ValueError cannot resize an array that has been referenced or is Exercise Shape manipulations e Look at the docstring for reshape especially the notes section which has some more information about copies and views e Use flatten as an alternative to ravel What is the difference Hint check which one returns a view and which a copy e Experiment with transpose for dimension shuffling 3 2 5 Sorting data Sorting along an axis gt gt gt a np array 14 3 5 1 2 1 gt gt gt b np sort a axis 1 gt gt gt b array 3 4 5 ED Sorts each row separately In place sort gt gt gt a sort axis 1 gt gt a array 3 4 5 se als 2a Sorting with fancy indexing gt gt gt a np array 4 3 1 2 gt gt gt j np argsort a gt gt gt j array I2 3 1 019 gt gt gt aljl array Cll 2 3 4J Finding minima and maxima gt gt gt a np
322. rn the answer for this point z_out 0 z Boilerplate Cython definitions You don t really need to read this part tt just pulls in stuff from the Numpy C headers 8 2 Universal functions 178 Scipy lecture notes Edition 2015 2 cdef extern from numpy arrayobject h void import_array ctypedef int npy_intp cdef enum NPY_TYPES NPY_CDOUBLE cdef extern from numpy ufuncobject h void import_ufunc ctypedef void PyUFuncGenericFunction char npy_intp npy_intp void object PyUFunc_FromFuncAndData PyUFuncGenericFunction func void data char types int ntypes int nin int nout int identity char name char doc int c void PyUFunc_DD_D char npy_intp npy_intp void Required module initialization import_array import_ufunc The actual ufunc declaration cdef PyUFuncGenericFunction loop_funcl i cdef char input_output_types 3 cdef void elementwise_funcs 1 loop_func 0 PyUFunc_DD_D input _output_types 0 NPY_CDOUBLE input _output_types 1i NPY_CDOUBLE input_output_types 2 NPY_CDOUBLE elementwise_funcs 0 lt void gt mandel_single_point mandel PyUFunc_FromFuncAndData loop_func elementwise_funcs input _output_types 1 number of supported input types 2 number of wnpult args 1 number of output args O tdentity element never mind this mandel function name mandel z c gt computes iterated z z c docstring O unused
323. rt vectors which are the observations closest to the separating hyperplane 20 2 Classification 350 Scipy lecture notes Edition 2015 2 gt gt gt from sklearn import svm gt gt gt svc svm SVC kernel linear gt gt gt svc fit iris data iris target SVC There are several support vector machine implementations in scikit learn The most commonly used ones are svm SVC svm NuSVC and svm LinearSVC SVC stands for Support Vector Classifier there also exist SVMs for regression which are called SVR in scikit learn Exercise Train an svm SVC on the digits dataset Leave out the last 10 and test prediction performance on these observations Using kernels Classes are not always separable by a hyperplane so it would be desirable to have a decision function that is not linear but that may be for instance polynomial or exponential Polynomial kernel RBF kernel Radial Basis Func tion gt gt gt svm SVC kernel lineax svc svm SVC kernel poly gt gt gt ae degree 3 gt gt gt gamma inverse of size of gt gt gt degree polynomial degr e gt radial kernel Exercise Which of the kernels noted above has a better prediction performance on the digits dataset 20 3 Clustering grouping observations together Given the iris dataset if we knew that there were 3 types of iris but did not have access to their labels we could try unsupervised learning we c
324. rvoir name DelegatesTo reservoir max_storage DelegatesTo reservoir max_release DelegatesTo reservoir min release Float state attributes storage Property depends_on inflows release control attributes inflows Float desc Inflows hm3 release Range low min_release high max_release spillage Property desc Spillage hm3 depends_on storage inflows release Traits view traits_view View Group VGroup Item name Item storage Item spillage label State style readonly ye VGroup Item inflows Item release label Control Private traits _storage Float Traits property implementation def _get_storage self new_storage self _storage self release self inflows return min new_storage self max_storage 18 3 What are Traits 331 Scipy lecture notes Edition 2015 2 def _set_storage self storage_value self _storage storage_value _get_spillage self new_storage self _storage self release self inflows overflow new_storage self max_storage return max overflow 0 print_state self print Storage tRelease tInflows tSpillage str_format t join 7 2f for i in range 4 print str_format format self storage self release self inflows self spillage print 79 __name__ main _ projectA Reservoir name Project A max_storage 30 max release Bs hydraulic_head 6
325. rvoir projectA storage 10 register the dynamic listener state on_trait_change wake_up_watchman_if_spillage name spillage state release 90 state inflows O state print_state print Forcing spillage state inflows 100 state release 0 print Why do we have two executions of the callback The dynamic trait notification signatures are not the same as the static ones e def wake_up_watchman pass def wake_up_watchman new pass def wake_up_watchman name new pass def wake_up_watchman object name new pass def wake_up_watchman object name old new pass Removing a dynamic listener can be done by e calling the remove_trait_listener method on the trait with the listener method as argument e calling the on_trait_change method with listener method and the keyword remove True e deleting the instance that holds the listener Listeners can also be added to classes using the on_trait_change decorator from traits api import HasTraits Instance DelegatesTo Float Range from traits api import Property on_trait_change from reservoir import Reservoir class ReservoirState HasTraits Keeps track of the reservoir state given the initial storage For the simplicity of the example the release is considered in hm3 timestep and not in m3 s reservoir Instance Reservoir max_storage DelegatesTo reservoir min release Float max_release DelegatesTo reservoir state attributes st
326. s File demo py Function test at line 5 Total time 14 2793 s Line Hits Time Per Hit A Time Line Contents 5 profile 6 def test 7 1 19015 1901520 omi data np random random 5000 100 8 1 14242163 14242163 0 99 u s v linalg svd data 9 1 10282 10282 0 01 pca np dot ul 10 data 10 1 7799 7799 0 Oral results fastica pca T whiten False The SVD is taking all the time We need to optimise this line 10 3 Making code go faster Once we have identified the bottlenecks we need to make the corresponding code go faster 10 3 1 Algorithmic optimization The first thing to look for is algorithmic optimization are there ways to compute less or better For a high level view of the problem a good understanding of the maths behind the algorithm helps However it is not uncommon to find simple changes like moving computation or memory allocation outside a for loop that bring in big gains Example of the SVD In both examples above the SVD Singular Value Decomposition is what takes most of the time Indeed the computational cost of this algorithm is roughly n in the size of the input matrix However in both of these example we are not using all the output of the SVD but only the first few rows of its first return argument If we use the svd implementation of scipy we can ask for an incomplete version of the SVD Note that implementations of linear algebra in scipy are richer then those in numpy and should be
327. s print args is args print kwargs is kwargs In 36 variable_args one two x 1 y 2 z 3 args is one two kwyares is ay 22 a te eZ tee 2 4 7 Docstrings Documentation about what the function does and its parameters General convention In 67 def funcname params Concise one line sentence describing the function Extended summary which can contain multiple paragraphs function body pass In 68 funcname Type function Base Class type function gt String Form lt function funcname at Oxeaa0f0 gt Namespace Interactive Fille lt ipython console gt Definition funcname params Docstring Concise one line sentence describing the function Extended summary which can contain multiple paragraphs Docstring guidelines For the sake of standardization the Docstring Conventions webpage documents the semantics and conven tions associated with Python docstrings Also the Numpy and Scipy modules have defined a precise standard for documenting scientific func tions that you may want to follow for your own functions with a Parameters section an Examples section etc See http projects scipy org numpy wiki CodingStyleGuidelines docstring standard and http projects scipy org numpy browser trunk doc example py L37 2 4 8 Functions are objects Functions are first class objects which means they can be e assigned to a variable e an item in a list or any collection e passed as
328. s advantages A function can be much shorter than the definition of a class with the required next and __iter__ methods What is more important it is easier for the author of the generator to understand the state which is kept in local variables as opposed to instance attributes which have to be used to pass data between consecutive invocations of next on an iterator object A broader question is why are iterators useful When an iterator is used to power a loop the loop becomes very simple The code to initialise the state to decide if the loop is finished and to find the next value is extracted into a separate place This highlights the body of the loop the interesting part In addition it is possible to reuse the iterator code in other places 7 1 4 Bidirectional communication Each yield statement causes a value to be passed to the caller This is the reason for the introduction of generators by PEP 255 implemented in Python 2 2 But communication in the reverse direction is also useful One obvious way would be some external state either a global variable or a shared mutable object Direct communication is possible thanks to PEP 342 implemented in 2 5 It is achieved by turning the previously boring yield statement into an expression When the generator resumes execution after a yield statement the caller can call a method on the generator object to either pass a value into the generator which then is returned by the yield statement
329. s interpreted by taking all elements in the unspecified dimensions Slicing Arrays like other Python sequences can also be sliced gt gt gt a np arange 10 gt gt gt a array 0 dls 2 3 4 5 6 7 Ose GND gt gt gt a 2 9 3 start end step array I2 5 81 Note that the last index is not included gt gt gt al 4 arrayclO da 2 3J All three slice components are not required by default start is 0 end is the last and step is 1 Peery weet array 1 2 oor al 2 array C O 2 4 6 ol gt gt gt al 3 arrayC 3 4 5 6 7 8 9 A small illustrated summary of Numpy indexing and slicing 3 1 The Numpy array object 50 Scipy lecture notes Edition 2015 2 gt gt gt al0 3 5 array 3 4 gt gt gt al4 4 array 44 45 54 55 gt gt gt al 2 array 2 12 22 32 42 52 gt gt gt a 2 2 2 array 20 22 24 40 42 44 You can also combine assignment and slicing gt gt gt a np arange 10 gt gt gt al5 10 gt gt gt a array CO b e2e08 Se 4 o lo tO dO Oi gt gt gt b np arange 5 gt gt gt ald b 1 gt gt gt a array O T 2a A Aa eg ao Exercise Indexing and slicing e Try the different flavours of slicing using start end and step starting from a linspace try to obtain odd numbers counting backwards and even numbers counting forwards e Reproduce the slices in the diagram above You may
330. s it possible for Python to be used to write programs in web servers or embedded devices Reference document for this section IPython user manual http ipython org ipython doc dev index html 1 3 1 Command line interaction Start ipython In 1 pring Hello world Hello world Getting help by using the operator after an object In l2 print Type builtin_function_or_method Base Class lt type builtin_function_or_method gt String Form lt built in function print gt Namespace Python builtin Docstring print value sep end n file sys stdout Prints the values to a stream or to sys stdout by default Optional keyword arguments file a file like object stream defaults to the current sys stdout sep string inserted between values default a space end string appended after the last value default a newline 1 3 2 Elaboration of the algorithm in an editor Create a file my_file py in a text editor Under EPD Enthought Python Distribution you can use Scite available from the start menu Under Python x y you can use Spyder Under Ubuntu if you don t already have your favorite editor we would advise installing Stani s Python editor In the file add the following lines s Hello world print s Now you can run it in Python and explore the resulting variables In 1 run my_file py Hello world In 2 s Out 2 Helle world In 3 whos Variable Type Data Info
331. s oscillator 2nd order os cillator The position of a mass attached to a spring obeys the 2nd order ODEy 2 eps wo y wo72 y O with wo 2 k m with k the spring constant m the mass and eps c 2 m wo with c the damping coefficient For this example we choose the parameters as gt gt gt mass 0 5 kg gt gt gt kspring 4 N m gt gt gt cviscous 0 4 N s m so the system will be underdamped because 5 8 Numerical integration scipy integrate 117 Scipy lecture notes Edition 2015 2 gt gt gt eps cviscous 2 mass np sqrt kspring mass gt gt gt eps lt 1 True For the scipy integrate odeint solver the 2nd order equation needs to be transformed in a system of two first order equations for the vector Y y y It will be convenient to define nu 2 eps wo c mandom wo 2 k m gt gt gt nu_coef cviscous mass gt gt gt om_coef kspring mass Thus the function will calculate the velocity and acceleration by gt gt gt def calc_deri yvec time nuc omc return yvec 1 nuc yvec 1 omc yvec 0 gt gt gt time_vec np linspace 0O 10 100 gt gt gt yarr odeint calc_deri 1 0 time_vec args nu_coef om_coef The final position and velocity are shown on the following Matplotlib figure 0 2 4 6 8 10 There is no Partial Differential Equations PDE solver in Scipy Some Python packages for solving PDE s are available such as fipy
332. s_module extension creating build creating build temp linux x86_64 2 7 gcc pthread fno strict aliasing g 02 DNDEBUG g fwrapv 03 Wall Wstrict prototypes fPIC I home esc an gcc pthread shared build temp linux x86_64 2 7 cos_module o L home esc anaconda lib lpython2 7 o home esc 1s build cos_module c cos_module pyx cos_module so setup py And running it In 1 import cos_module In 2 cos_module Type module String Form lt module cos_module from cos_module so gt File home esc git working scipy lecture notes advanced interfacing with_c cython cos_module so Docstring lt no docstring gt In 3 dir cos_module Gutisi oS builtins l Pe BOG 3 Ye Tenn __name__ __package__ itest cog_func In 4 cos_module cos_func 1 0 Out 4 0 5403023058681398 In 5 cos_module cos_func 0 0 fie a 1 0 In 6 cos_module cos_func 3 14159265359 Cie es 10 And testing a little for robustness we can see that we get good error messages In 7 cos_module cos_func foo TypeError Traceback most recent call last lt ipython input 7 11bee483665d gt in lt module gt gt 1 cos_module cos_func foo home esc git working scipy lecture notes advanced interfacing with_c cython cos_module so in cos_module cos_fu TypeError a float is required 14 5 Cython 277 Scipy lecture notes Edition 2015 2 Additionally it is worth noting that Cython ships with c
333. set a break point you can simply raise an exception at the point that you want to inspect and use Python s debug Note that in this case you cannot step or continue the execution e Debugging test failures using nosetests You can run nosetests pdb to drop in post mortem debugging on exceptions and nosetests pdb failure to inspect test failures using the debugger In addition you can use the Python interface for the debugger in nose by installing the nose plugin ipdbplugin You can than pass ipdb and ipdb failure options to nosetests e Calling the debugger explicitly Insert the following line where you want to drop in the debugger import pdb pdb set_trace A When running nosetests the output is captured and thus it seems that the debugger does not work Simply run the nosetests with the s flag 9 3 Using the Python debugger 199 Scipy lecture notes Edition 2015 2 Graphical debuggers and alternatives e For stepping through code and inspecting variables you might find it more convenient to use a graphical debugger such as winpdb e Alternatively pudb is a good semi graphical debugger with a text user interface in the console e Also the pydbgr project is probably worth looking at 9 3 2 Debugger commands and interaction a Printthelocalvariables SSS A Debugger commands are not Python code You cannot name the variables the way you want For instance if in you cannot override the variables in the curr
334. sposes never make copies it just swaps strides gt gt gt x np zeros 10 10 10 dtype np float gt gt gt x strides 800 80 8 gt gt gt x T strides 8 80 800 But not all reshaping operations can be represented by playing with strides gt gt gt a np arange 6 dtype np int8 reshape 3 2 gt gt b a T gt gt gt b strides 12 So far so good However gt gt gt str a data x00 x01 x02 x03 x04 x05 gt gt gt D array ClO 2 4 1 3 5 dtype int8 gt gt gt c b reshape 3 2 gt gt gt c array 0 2 4 1 3 5 dtype int8 Here there is no way to represent the array c given one stride and the block of memory for a Therefore the reshape operation needs to make a copy here Example fake dimensions with strides Stride manipulation gt gt gt from numpy lib stride_tricks import as_strided gt gt gt help as_strided as_strided x shape None strides None Make an ndarray from the given array with the given shape and strides A as_strided does not check that you stay inside the memory block bounds gt gt gt x np array 1 2 3 4 dtype np int16 gt gt gt as_strided x strides 2 2 shape 2 array 1 3 dtype int16 gt gt gt x 2 array 1 3 dtype int16 See also stride fakedims py Exercise 8 1 Life of ndarray 169 Scipy lecture notes Edition 2015 2 array 1 2 3 4 dtype np int8 gt array Cluills 2
335. surf and mayavi mlab mesh the connectivity is implicity given by the layout of the arrays See also mayavi mlab triangular_mesh Our data is often more than points and values it needs some connectivity information Volumetric data contour3d __ Lt If your data is dense in 3D it is more difficult to display One option is to take iso contours of the data mlab clf x y 2 np mgrid 5 5 64 5 5 64 5 5 64j values x x 0 5 y y z z 2 0 mlab contour3d values 19 1 Mlab the scripting interface 342 Scipy lecture notes Edition 2015 2 This function works with a regular orthogonal grid the value array is a 3D array that gives the shape of the grid 19 1 2 Figures and decorations Figure management Here is a list of functions useful to control the current figure Get the current figure mlab gcf Clear the current figure mlab clf Set the current figure mlab figure 1 bgcolor 1 1 1 fgcolor 0 5 0 5 0 5 Save figure to image file mlab savefig foo png size 300 300 Change the view mlab view azimuth 45 elevation 54 distance 1 Changing plot properties In general many properties of the various objects on the figure can be changed If these visualization are created via mlab functions the easiest way to change them is to use the keyword arguments of these functions as described in the docstrings 19 1 Mlab the scripting interface 343 Sci
336. sx sy f shape X Y np ogrid 0 sx O sy r np hypot X sx 2 Y sy 2 rbin 20 r r max astype np int radial_mean ndimage mean f labels rbin index np arange 1 rbin max 1 e Other measures Correlation function Fourier wavelet spectrum etc One example with mathematical morphology granulometry gt gt gt def disk_structure n struct np zeros 2 n 1 2 n 1 x y np indices 2 n 1 2 n 1 mask x n 2 y n 2 lt n 2 struct mask 1 return struct astype np bool gt gt gt gt gt gt def granulometry data sizes None s max data shape if sizes None sizes range 1 s 2 2 granulo ndimage binary_opening data structure disk_ structure n sum for n in sizes return granulo 12 6 Measuring objects properties ndimage measurements 245 Scipy lecture notes Edition 2015 2 gt gt gt np random seed 1 gt gt gt n 10 gt gt gt l1 256 gt gt gt im np zeros 1 1 gt gt gt points 1 np random random 2 n 2 gt gt gt imL points 0 astype np int points 1 astype np int 1 gt gt gt im ndimage gaussian_filter im sigma 1 4 n gt gt gt gt gt gt mask im gt im mean gt gt gt gt gt gt granulo granulometry mask sizes np arange 2 19 4 4 6 8 10 12 14 16 18 See also More on image processing e The chapter on Scikit image 12 6 Measuring objects properties
337. system you can directly skip to the next chapter Matplotlib plotting The remainder of this chapter is not necessary to follow the rest of the intro part But be sure to come back and finish this chapter as well as to do some more exercices 3 3 More elaborate arrays Section contents e More data types e Structured data types e maskedarray dealing with propagation of missing data 3 3 1 More data types Casting Bigger type wins in mixed type operations gt gt gt op array l1 2 3 1 5 armay le2 25505 yee Assignment never changes the type 3 3 More elaborate arrays 68 gt gt gt a np array 1 2 3 gt gt gt a dtype dtype int64 gt gt gt alO 1 9 lt float is truncated to integer gt gt gt a array li 2 319 Forced casts gt gt gt a np array 1 7 1 2 1 61 gt gt gt b a astype int lt truncates to integer gt gt gt b array ILe abl Rounding gt gt gt a np array 1 2 1 5 1 6 2 5 3 5 4 5J gt gt gt b np around a gt gt gt b Still floating pornit array l Irse Qi see Beg Ary Aal gt gt gt c np around a astype int gt gt gt C array 1 2 2 2 4 4 Different data type sizes Integers signed T6 bits 32 bits same as int on 32 bit platform 64 bits same as int on 64 bit platform gt gt gt np array 1 dtype int dtype dtype int64 gt gt gt np iinfo np int32 max 2 31
338. t cm gray lt matplotlib image AxesImage object at 0x gt Increase contrast by setting min and max values gt gt gt plt imshow f cmap plt cm gray vmin 30 vmax 200 lt matplotlib image AxesImage object at 0x gt gt gt gt Remove azes and ticks gt gt gt plt axis off C05 1023 5 767 5 0 5 Draw contour lines gt gt gt plt contour f 50 200 lt matplotlib contour QuadContourSet gt 0 200 400 600 800 1000 For fine inspection of intensity variations use interpolation nearest 12 2 Displaying images 232 Scipy lecture notes Edition 2015 2 gt gt gt plt imshow f 320 340 510 530 cmap plt cm gray lt matplotlib image AxesImage object at Ox gt gt gt gt plt imshow f 320 340 510 530 cmap plt cm gray interpolation nearest lt matplotlib image AxesImage object at Ox gt 3 D visualization Mayavi See also See 3D plotting with Mayavi e Image plane widgets e Isosurfaces 1 60e 16 fae O 267 0 Qd 0 192 12 3 Basic manipulations Images are arrays use the whole numpy machinery 12 3 Basic manipulations 233 gt gt gt gt gt gt 127 gt gt gt gt gt gt face misc face gray True face 0 40 Slicing facel10 13 20 23 array 141 153 145 133 134 rie 96 92 94 dtype uints face l100 120 255 lx ly face shape X Y np ogrid 0 1lx 0 1ly mask X lx 2 2
339. t learn provides an object that given data computes the score during the fit of an estimator on a parameter grid and chooses the parameters to maximize the cross validation score This object takes an esti mator during the construction and exposes an estimator API gt gt gt from sklearn import svm grid_search gt gt gt gammas np logspace 6 1 10 gt gt gt svc svm SVC gt gt gt clf grid_search GridSearchCV estimator svc param_grid dict gamma gammas cast n_jobs 1 gt gt gt clf fit digits data 1000 digits target 1000 GridSearchCV cv None gt gt gt clf best_score_ Oro TE gt gt gt clf best_estimator_ gamma 0 00059948425031894088 By default the GridSearchCV uses a 3 fold cross validation However if it detects that a classifier is passed rather than a regressor it uses a stratified 3 fold Cross validated estimators Cross validation to set a parameter can be done more efficiently on an algorithm by algorithm basis This is why for certain estimators the scikit learn exposes CV estimators that set their parameter automatically by cross validation gt gt gt from sklearn import linear_model datasets gt gt gt lasso linear_model LassoCV gt gt gt diabetes datasets load_diabetes gt gt gt X_diabetes diabetes data gt gt gt y_diabetes diabetes target gt gt gt lasso fit X_diabetes y_diabetes LassoCV alphas None gt gt gt The
340. t order gt gt gt p 0 O gt gt gt p roots array l 1 3 0 33333333 gt gt gt p degree In general polynomials do not always expose order 2 Example using polynomials in Chebyshev basis for polynomials in range 1 1 gt gt gt x np linspace 1 1 2000 gt gt gt y np cos x 0 3 np random rand 2000 gt gt gt p np polynomial Chebyshev fit x y 90 gt gt gt t np linspace 1 1 200 oe pi pilot Goa eer lt matplotlib lines Line2D object at gt gt gt gt plt plot t p t k lv 3 lt matplotlib lines Line2D object at gt 3 4 Advanced operations 173 Scipy lecture notes Edition 2015 2 1 0 0 5 0 0 0 5 The Chebyshev polynomials have some advantages in interpolation 3 4 2 Loading data files Text files Example populations txt year hare lynx carrot 1900 30e3 4e3 48300 1901 47 2e3 6 1e3 48200 1902 70 2e3 9 8e3 41500 1903 7 4e3 35 2e3 38200 gt gt gt data np loadtxt data populations txt gt gt gt data array 1900 30000 4000 48300 1901 47200 6100 48200 1902 70200 9800 41500 gt gt gt np savetxt pop2 txt data gt gt gt data2 np loadtxt pop2 txt If you have a complicated text file what you can try are e np genfromtxt e Using Python s I O functions and e g regexps for parsing Python is quite well suited for this 3 4 Advanced operations 14 Reminder Navigating t
341. t prototypes fPIC I home esc an In file included from home esc anaconda lib python2 7 site packages numpy core include numpy ndarraytypes h 17 from home esc anaconda lib python2 7 site packages numpy core include numpy ndarrayobject h 1 from home esc anaconda 1lib python2 7 site packages numpy core include numpy arrayobject h 15 from _cos_doubles c 253 home esc anaconda lib python2 7 site packages numpy core include numpy npy_deprecated_api h 11 2 warning wa home esc anaconda lib python2 7 site packages numpy core include numpy __ufunc_api h 236 warning _import_um gcc pthread fno strict aliasing g 02 DNDEBUG g fwrapv 03 Wall Wstrict prototypes fPIC I home esc an gcc pthread shared build temp linux x86_64 2 7 _cos_doubles o build temp linux x86_64 2 7 cos_doubles o L ho 1s build _cos_doubles c cos_doubles c cos_doubles h _cos_doubles pyx cos_doubles so setup py test_cos_dou And as before we convince ourselves that it worked import numpy as np import pylab import cos_doubles lx np arange O 2 np pi 0 1 y np empty_like x cos_doubles cos_doubles_func x y pylab plot x y pylab show L 14 6 Summary In this section four different techniques for interfacing with native code have been presented The table below roughly summarizes some of the aspects of the techniques Part of CPython Compiled Numpy Support Pon CAPI Ctypes 14 6 Summ
342. te entries allowed values are corresponding non zero values e efficient for constructing sparse matrices incrementally e constructor accepts dense matrix array sparse matrix 11 2 Storage Schemes 216 Scipy lecture notes Edition 2015 2 shape tuple create empty matrix efficient O 1 access to individual elements flexible slicing changing sparsity structure is efficient can be efficiently converted to a coo_matrix once constructed e slow arithmetics for loops with dict iteritems e use when sparsity pattern is not known apriori or changes Examples e create a DOK matrix element by element gt gt gt mtx sparse dok_matrix 5 5 dtype np float64 gt gt gt mtx lt 5x5 sparse matrix of type lt numpy float64 gt with O stored elements in Dictionary Of Keys format gt gt gt gt for ir in range 5 for ic in range 5 TF mtx ir ic 1 0 ir ic gt gt gt mtx lt 5x5 sparse matrix of type lt numpy float64 gt with 20 stored elements in Dictionary Of Keys format gt gt gt gt mtx todense matrix L 0 Sle Slee tiles v a Abc Oa s a e eee e Ore Eae iols Ease Slee lee O e bra e aa le aD e slicing and indexing gt gt gt mtx 1 1 0 0 gt gt gt mtx 1 1 3 lt 1x2 sparse matrix of type lt numpy float64 gt with 1 stored elements in Dictionary Of Keys format gt gt gt gt mtx 1 1 3 todense matrix CLO ale iD gt gt
343. te scipy special gammaln which will give the log of Gamma to a higher numerical precision e Erf the area under a Gaussian curve scipy special erf 5 3 Linear algebra operations scipy linalg The scipy linalg module provides standard linear algebra operations relying on an underlying efficient implementation BLAS LAPACK e The scipy linalg det function computes the determinant of a square matrix gt gt gt from scipy import linalg gt gt gt arr np array 1 2 ee 3 4 gt gt gt linalg det arr a2 20 gt gt gt arr np array L3 2 6 4 gt gt gt linalg det arr 0 0 gt gt gt linalg det np ones 3 4 Traceback most recent call last ValueError expected square matrix Traceback most recent call last ValueError expected square matrix e The scipy linalg inv function computes the inverse of a square matrix gt gt gt arr np array 1 2 ue 3 4 gt gt gt iarr linalg inv arr gt gt gt larr array l 2 3 Glue 1 5 0 5 gt gt gt np allclose np dot arr iarr np eye 2 True Finally computing the inverse of a singular matrix its determinant is zero will raise LinAlgError gt gt gt arr np array 3 2 sen 6 4 gt gt gt linalg inv arr Traceback most recent call last LinAlgError singular matrix Traceback most recent call last LinAlgError singular matrix e More advanced operations are availab
344. ter items one by one 3 1 e Evenly spaced gt gt gt a np arange 10 0 n 1 gt gt gt a array l0 lg 2 tsa wid 40h wlan Ou Oil gt gt gt b np arange 1 9 2 start end exclusive step gt gt gt b array lis 5 5 7J The Numpy array object 45 Scipy lecture notes Edition 2015 2 e or by number of points gt gt gt c np linspace 0O 1 6 start end num points gt gt gt C array L0 o O82 O45 8065 0S 1 19 gt gt gt d np linspace 0 1 5 endpoint False gt gt gt d array LO n 02r 0a 0o 0D e Common arrays gt gt gt a np ones 3 3 reminder 3 3 is a tuple gt gt gt a array Ikl iee eal Ea ay dels Ete Aen A gt gt gt b np zeros 2 2 gt gt gt b array LIL 0 0 EO ol gt gt gt c np eye 3 gt gt gt C array Il dg 0 0T POr iea TOs Foa O tea gt gt gt d np diag np array 1 2 3 4 gt gt gt d array Ii 0 0 70l OR 0 Or LO oa O I0 0 aCe aA e np random random numbers Mersenne Twister PRNG gt gt gt a np random rand 4 uniform in 0 1 gt gt gt a array 0 95799151 0 14222247 0 08777354 0 51887998 gt gt gt b np random randn 4 Gaussian gt gt gt b array 0 37544699 0 11425369 0 47616538 1 79664113 gt gt gt np random seed 1234 Setting the random seed Exercise Creating arrays using functions e Experiment with arange linspace
345. that opening amounts to eroding then dilating An opening operation removes small structures while a closing operation fills small holes Such operations 5 10 Image processing scipy ndimage 122 Scipy lecture notes Edition 2015 2 can therefore be used to clean an image gt gt gt a np zeros 50 50 gt gt gt a 10 10 10 10 1 gt gt gt a 0 25 np random standard_normal a shape gt gt gt mask a gt 0 5 gt gt gt opened_mask ndimage binary_opening mask gt gt gt closed_mask ndimage binary_closing opened_mask opened_mask closed_mask Check that the area of the reconstructed square is smaller than the area of the initial square The opposite would occur if the closing step was performed before the opening Exercise For gray valued images eroding resp dilating amounts to replacing a pixel by the minimal resp maximal value among pixels covered by the structuring element centered on the pixel of interest gt gt gt a np zeros 7 7 dtype np int gt gt gt all 6 1 6 3 gt gt gt al4 4 2 a 2 3 1 gt gt gt a array 10 0 0 0 0 0 0 LOS oe ao Ole Oa Sl LS aao Cece ce e e RE Ro ee Ee ee Oe O a a a a a O lO On Oe 0 0 0 0 gt gt gt ndimage grey_erosion a size 3 3 array CLIO 0 Oe 0 70 0 0 iO 0 0 0 05 0 Ol RO s O e es XO O TOE Seal Se ee oul Gi O35 Oe sO OMe Os 0 Or Oo 10s 80 O Os OF SO Om 10
346. the thresholds from the minima of the histogram 5 Display an image in which the three phases are colored with three different colors 6 Use mathematical morphology to clean the different phases 7 Attribute labels to all bubbles and sand grains and remove from the sand mask grains that are smaller than 10 pixels To do so use ndimage sum or np bincount to compute the grain sizes 8 Compute the mean size of bubbles Proposed solution gt gt gt import numpy as np gt gt gt import pylab as pl gt gt gt from scipy import ndimage 5 11 Summary exercises on scientific computing 132 Scipy lecture notes Edition 2015 2 5 11 4 Example of solution for the image processing exercise unmolten grains in glass MV 15 05 2009 1280 C 1 Open the image file MV_HFV_012 jpg and display it Browse through the keyword arguments in the docstring of imshow to display the image with the right orientation origin in the bottom left corner and not the upper left corner as for standard arrays gt gt gt dat pl imread data MV_HFV_012 jpg 2 Crop the image to remove the lower panel with measure information gt gt gt dat dat 60 3 Slightly filter the image with a median filter in order to refine its histogram Check how the histogram changes gt gt gt filtdat ndimage median_filter dat size 7 7 gt gt gt hi_dat np histogram dat bins np arange 256 gt gt gt hi_filtdat np histogr
347. the cos_func to Python To achieve this with SWIG we must write an interface file which contains the instructions for SWIG Example of wrapping cos function from math h using SWIG module cos_module At the resulting C file should be built as a python extension define SWIG_FILE_WITH_ INIT Includes the header in the wrapper code include cos_module h ht Parse the header file to generate wrappers Ainclude cos_module h As you can see not too much code is needed here For this simple example it is enough to simply include the header file in the interface file to expose the function to Python However SWIG does allow for more fine grained inclusion exclusion of functions found in header files check the documentation for details Generating the compiled wrappers is a two stage process 1 Run the swig executable on the interface file to generate the files cos_module_wrap c which is the source file for the autogenerated Python C extension and cos_module py which is the autogenerated pure python module 14 4 SWIG 212 Scipy lecture notes Edition 2015 2 2 Compile the cos_module_wrap c into the _cos_module so Luckily distutils knows how to handle SWIG interface files so that our setup py is simply from distutils core import setup Extension setup ext_modules Extension _cos_module sources cos_module c cos_module i cd advanced interfacing with_c swig 1s cos_module c
348. the cutoff for z real 2 zZ imag 2 TODO mandelbrot iteration should go here Return the answer for this point z_out O z Boilerplate Cython definitions The litany below is particularly long but you don t really need to read this part it just pulls in stuff from the Numpy C headers cdef extern from numpy arrayobject h void import_array ctypedef int npy_intp cdef enum NPY_TYPES NPY_DOUBLE NPY_CDOUBLE NPY_LONG cdef extern from numpy ufuncobject h void import_ufunc ctypedef void PyUFuncGenericFunction char npy_intp npy_intp void object PyUFunc_FromFuncAndData PyUFuncGenericFunction func void data char types int ntypes int nin int nout int identity char name char doc int c List of pre defined loop functions void PyUFunc_f_f_As_d_d char args npy_intp dimensions npy_intp steps void func void PyUFunc_d_d char x args npy_intp dimensions npy_intp steps void func void PyUFunc_f_f char args npy_intp dimensions npy_intp steps void func void PyUFunc_g_ g char x args npy_intp dimensions npy_intp steps void func void PyUFunc_F_F_As_D_D char args npy_intp dimensions npy_intp steps void func void PyUFunc_F_F char x args npy_intp dimensions npy_intp steps void func void PyUFunc_D_D char x args npy_intp dimensions npy_intp steps void func void PyUFunc_G_G char x args npy_intp dimensions npy_intp steps void func voi
349. the next statement in the current execution context while step will go across execution contexts i e enable exploring inside function calls ipdb gt s gt home varoquau dev scipy lecture notes advanced optimizing wiener_filtering py 35 iterated_wiener 2 34 noisy_img noisy_img gt 35 denoised_img local_mean noisy_img size size 36 l_var local_var noisy_img size size ipdb gt n gt home varoquau dev scipy lecture notes advanced optimizing wiener_filtering py 36 iterated_wiener 35 denoised_img local_mean noisy_img size size gt 36 l_var local_var noisy_img size size 37 for i in range 3 e Step a few lines and explore the local variables ipdb gt n gt home varoquau dev scipy lecture notes advanced optimizing wiener_filtering py 37 iterated_wiener 36 l_var local_var noisy_img size size saa Ef for i in range 3 38 res noisy_img denoised_img ipdb gt print 1 yar 5868 5379 5316 5071 4799 5149 5013 363 437 346 262 4355 5379 410 344 392 604 3377 435 362 308 275 198 1632 548 392 290 248 263 1653 466 789 736 1835 1725 1940 ipdb gt print l_var min 0 Oh dear nothing but integers and 0 variation Here is our bug we are doing integer arithmetic 9 3 Using the Python debugger 198 Scipy lecture notes Edition 2015 2 Raising exception on numerical errors When we run the wiener_filtering py file the following warnings a
350. the p value the probability of both processes being identical If it is close to 1 the two process are almost certainly identical The closer it is to zero the more likely it is that the processes have different means See also The chapter on statistics introduces much more elaborate tools for statistical testing and statistical data load ing and visualization outside of scipy 5 Interpolation scipy interpolate The scipy interpolate is useful for fitting a function from experimental data and thus evaluating points where no measure exists The module is based on the FITPACK Fortran subroutines from the netlib project By imagining experimental data close to a sine function gt gt gt measured_time np linspace 0 1 10 gt gt gt noise np random random 10 2 1 1e 1 gt gt gt measures np sin 2 np pi measured_time noise The scipy interpolate interpid class can build a linear interpolation function gt gt gt from scipy interpolate import interplid gt gt gt linear_interp interpid measured_time measures Then the scipy interpolate linear_interp instance needs to be evaluated at the time of interest gt gt gt computed_time np linspace 0 1 50 gt gt gt linear_results linear_interp computed_time A cubic interpolation can also be selected by providing the kind optional keyword argument 5 7 Interpolation scipy interpolate 115 Scipy lecture notes Edition 2015 2 gt gt gt
351. thon it is easy to transform exiting Python code in compiled code and with a good use of the numpy support yields efficient code on numpy arrays for instance by unrolling loops For all the above profile and time your choices Don t base your optimization on theoretical considerations 10 4 Writing faster numerical code 208 Scipy lecture notes Edition 2015 2 10 4 1 Additional Links e If you need to profile memory usage you could try the memory_profiler e If you need to profile down into C extensions you could try using gperftools from Python with yep e If you would like to track performace of your code across time i e as you make new commits to your repository you could try vbench e If you need some interactive visualization why not try RunSnakeRun 10 4 Writing faster numerical code 209 CHAPTER 11 Sparse Matrices in SciPy Author Robert Cimrman 11 1 Introduction dense matrix is e mathematical object e data structure for storing a 2D array of values important features e memory allocated once for all items usually a contiguous chunk think NumPy ndarray e fast access to individual items 11 1 1 Why Sparse Matrices e the memory that grows like n 2 e small example double precision matrix gt gt gt import numpy as np gt gt gt import matplotlib pyplot as plt gt gt gt x np linspace 0 1e6 10 gt gt gt plt plot x 8 0 x 2 1e6 lw 5 lt matplotlib lines Li
352. tion terminated successfully Current function value 0 000000 Iterations 16 Function evaluations 24 Gradient evaluations 24 array 1 00000017 1 00000026 L BFGS Limited memory BFGS Sits between BFGS and conjugate gradient in very high dimensions gt 250 the Hessian matrix is too costly to compute and invert L BFGS keeps a low rank version In addition the scipy version scipy optimize fmin_1_bfgs_b includes box bounds gt gt gt def f x The rosenbrock function return 5 1 x 0 2 x 1 x 0 2 2 gt gt gt def fprime x return np array 2 5 1 x 0 4 x 0 x 1 x 0 2 2 x 1 x 0 2 gt gt gt optimize fmin_1l_bfgs_b f 2 2 fprime fprime array 1 00000005 1 00000009 1 4417677473011859e 15 If you do not specify the gradient to the L BFGS solver you need to add approx_grad 1 13 2 4 Gradient less methods A shooting method the Powell algorithm Almost a gradient approach 13 2 A review of the different optimizers 256 Scipy lecture notes Edition 2015 2 An ill conditionned quadratic function Powell s method isn t too sensitive to local ill conditionning in low dimensions Error on f x An ill conditionned very non quadratic function Simplex method the Nelder Mead The Nelder Mead algorithms is a generalization of dichotomy approaches to high dimensional spaces The algorithm works by refining a simplex the generalization of int
353. tions than CG Thus conjugate gradient method is better than BFGS at optimizing computationally cheap functions With the Hessian e If you can compute the Hessian prefer the Newton method scipy optimize fmin_ncg 13 3 Practical guide to optimization with scipy 258 Scipy lecture notes Edition 2015 2 If you have noisy measurements e Use Nelder Mead scipy optimize fmin or Powell scipy optimize fmin_powell 13 3 2 Making your optimizer faster Choose the right method see above do compute analytically the gradient and Hessian if you can e Use preconditionning when possible e Choose your initialization points wisely For instance if you are running many similar optimizations warm restart one with the results of another Relax the tolerance if you don t need precision 13 3 3 Computing gradients Computing gradients and even more Hessians is very tedious but worth the effort Symbolic computation with Sympy may come in handy Avery common source of optimization not converging well is human error in the computation of the gradient You can use scipy optimize check_grad to check that your gradient is correct It returns the norm of the different between the gradient given and a gradient computed numerically gt gt gt optimize check _grad f fprime 2 2 2 384185791015625e 07 See also scipy optimize approx_fprime to find your errors 13 3 4 Synthetic exercices Exercice A si
354. to fit a model to some data The data used in this tutorial are lidar data and are described in details in the following introductory paragraph If you re impatient and want to practice now please skip it and go directly to Loading and visualization Introduction Lidars systems are optical rangefinders that analyze property of scattered light to measure distances Most of them emit a short light impulsion towards a target and record the reflected signal This signal is then processed to extract the distance between the lidar system and the target Topographical lidar systems are such systems embedded in airborne platforms They measure distances be tween the platform and the Earth so as to deliver information on the Earth s topography see for more de l Mallet C and Bretar F Full Waveform Topographic Lidar State of the Art ISPRS Journal of Photogrammetry and Remote Sensing 64 1 pp 1 16 January 2009 http dx doi org 10 1016 j isprsjprs 2008 09 007 5 11 Summary exercises on scientific computing 127 Scipy lecture notes Edition 2015 2 Gumbell cumulative probability Vso 34 25 m s r 30 25 30 35 40 45 Annual wind speed maxima m s Figure 5 3 Solution Python source file 5 11 Summary exercises on scientific computing 128 Scipy lecture notes Edition 2015 2 tails In this tutorial the goal is to analyze the waveform recorded by the lidar system Such a signal contains peaks whose center and amplitude
355. tor Let s say we have function which returns a lists of things and this list created by running a loop If we don t know how many objects will be needed the standard way to do this is something like def find_answers answers while True ans look _for_next_answer if ans is None break answers append ans return answers This is fine as long as the body of the loop is fairly compact Once it becomes more complicated as often happens in real code this becomes pretty unreadable We could simplify this by using yield statements but then the user would have to explicitly call List find_answers We can define a decorator which constructs the list for us def vectorized generator_func def wrapper args kwargs return list generator_func args kwargs return functools update_wrapper wrapper generator_func Our function then becomes vectorized def find_answers while True ans look _for_next_answer if ans is None break yield ans 7 2 7 A plugin registration system This is a class decorator which doesn t modify the class but just puts it in a global registry It falls into the category of decorators returning the original object class WordProcessor object PLUGINS def process self text for plugin in self PLUGINS text plugin cleanup text return text 7 2 Decorators 154 Scipy lecture notes Edition 2015 2 classmethod def plugin cls plugin cls PLUGIN
356. transparent True contours 0 5 for i in range 1 20 scalars np sin i x y Z x y Z iso mlab_source scalars scalars 19 4 Animating the data 349 Scipy lecture notes Edition 2015 2 See also More details in the Mayavi documentation Event loops For the interaction with the user for instance changing the view with the mouse Mayavi needs some time to process these events The for loop above prevents this The Mayavi documentation details a workaround 19 5 Making interactive dialogs It is very simple to make interactive dialogs with Mayavi using the Traits library see the dedicated chapter Traits building interactive dialogs 19 5 1 A simple dialog from traits api import HasTraits Instance from traitsui api import View Item HGroup from mayavi core ui api import SceneEditor MlabSceneModel def curve n_turns The function creating the x y z coordinates needed to plot phi np linspace 0 2 np pi 2000 return np cos phi 1 0 5 np cos n_turns phi np sin phi 1 0 5 np cos n_turns phi 0 5 np sin n_turns phi class Visualization HasTraits The class that contains the dialog scene Instance MlabSceneModel def init__ self HasTraits init__ self X y Z curve n_turns 2 Populating our plot self plot self scene mlab plot3d x y z Describe the dialog view View Item scene height 300 show_label False editor SceneEditor HGroup n_turns resiza
357. tring Form lt module demo from demo py gt Namespace Interactive File home varoquau Projects Python_talks scipy_2009_tutorial source demo py Docstring A demo module In 5 who demo In 6 whos Variable Type Data Info demo module lt module demo from demo py gt In 7 dir demo Out 7 m e builtins a 1 OC _file__ name_ __package__ tol Edi print a print_b In 8 demo demo __builtins__ demo __init__ demo __str__ demo __class__ demo __name__ demo __subclasshook__ demo __delattr__ demo __new__ demo c demo __dict__ demo __package__ demo d demo __doc__ demo __reduce__ demo print_a demo __file__ demo __reduce_ex__ demo print_b demo __format__ demo __repr__ demo py demo __getattribute__ demo __setattr__ demo pyc demo hash _ demo sizeof __ Importing objects from modules into the main namespace In 9 from demo import print_a print_b In 10 whos Variable Type Data Info demo module lt module demo from demo py gt print_a function lt function print_a at 0xb7421534 gt print_b function lt function print_b at 0xb74214c4 gt In 11 print_a a 2 5 Reusing code scripts and modules 30 Scipy lecture notes Edition 2015 2 A Module caching Modules are cached if you modify demo py and re import it in the old session you will get the old one Solution In 10 reload demo In Python3 instead reload is not builtin so you have to imp
358. turn the result NpyIter_Deallocate in_iter NpyIter_Deallocate out_iter Py_INCREF out_array return out_array in case bad things happen fail Py_XDECREF out_array return NULL 14 2 Python C Api 267 Scipy lecture notes Edition 2015 2 define functions in module static PyMethodDef CosMethods cos_func_np cos_func_np METH_VARARGS evaluate the cosine on a numpy array NULL NULL O NULL module initialization PyMODINIT_FUNC initcos_module_np void void Py_InitModule cos_module_np CosMethods IMPORTANT this must be called import_array To compile this we can use distutils again However we need to be sure to include the Numpy headers by using func numpy get_include from distutils core import setup Extension import numpy define the extension module cos_module_np Extension cos_module_np sources cos_module_np c include_dirs numpy get_include run the setup setup ext_modules cos_module_np To convince ourselves if this does actually works we run the following test script import cos_module_np import numpy as np import pylab x np arange O 2 np pi 0 1 y cos_module_np cos_func_np x pylab plot x y pylab show And this should result in the following figure 14 3 Ctypes Ctypes is a foreign function library for Python It provides C compatible data types and allows calling functio
359. u_cache memoizes an arbitrary function maintaining a limited cache of argu ments answer pairs Python 3 2 e functools total_ordering is a class decorator which fills in missing ordering methods __1t__ __gt__ __le__ based on a single available one Python 2 7 7 2 5 Deprecation of functions Let s say we want to print a deprecation warning on stderr on the first invocation of a function we don t like anymore If we don t want to modify the function we can use a decorator class deprecated object Print a deprecation warning once on first use of the function gt gt gt deprecated doctest SKIP def fO aes pass gt gt gt JO doctest SKIP f ts deprecated def _call_ self func self func func self count 0 return self _wrapper def _wrapper self xargs kwargs self count 1 if self count print self func __name__ is deprecated return self func args xkwargs It can also be implemented as a function def deprecated func Print a deprecation warning once on first use of the function gt gt gt deprecated doctest SKIP 7 2 Decorators 153 Scipy lecture notes Edition 2015 2 def fO TAT pass gt gt gt fO doctest SKIP f ts deprecated count 0 def wrapper args kwargs count 0 1 if count 0 1 print func name__ is deprecated return func args xkwargs return wrapper 7 2 6 A while loop removing decora
360. uctor is a special method we call with MyClass init parameters if any Now suppose we want to create a new class MasterStudent with the same methods and attributes as the pre vious one but with an additional internship attribute We won t copy the previous class but inherit from it gt gt gt class MasterStudent Student internship mandatory from March to June gt gt gt james MasterStudent james gt gt gt james internship mandatory from March to June gt gt gt james set_age 23 gt gt gt james age 23 The MasterStudent class inherited from the Student attributes and methods Thanks to classes and object oriented programming we can organize code with different classes correspond ing to different objects we encounter an Experiment class an Image class a Flow class etc with their own methods and attributes Then we can use inheritance to consider variations around a base class and re use code Ex from a Flow base class we can create derived StokesFlow TurbulentFlow PotentialFlow etc 2 9 Object oriented programming OOP 42 CHAPTER 3 NumPy creating and manipulating numerical data Authors Emmanuelle Gouillart Didrik Pinte Ga l Varoquaux and Pauli Virtanen This chapter gives an overview of Numpy the core tool for performant numerical computing with Python 3 1 The Numpy array object Section contents What are Numpy and Numppy arrays Creating arrays Basic data types
361. ue pea fit X_train X_train_pca pca transform X_train X_test_pca pca transform X_test classification clf svm SVC C 5 gamma 0 001 clf fit X_train_pca y_train 20 5 Putting it all together face recognition 360 Scipy lecture notes Edition 2015 2 predict on new tmages for i in range 10 print 1lfw_people target_names clf predict X_test_pcali 0 pl imshow X_test i reshape 50 37 cmap pl cm gray raw_input 20 6 Linear model from regression to sparsity Diabetes dataset The diabetes dataset consists of 10 physiological variables age sex weight blood pressure measure on 442 patients and an indication of disease progression after one year gt gt gt diabetes datasets load_diabetes gt gt gt diabetes_X_train diabetes data 20 gt gt gt diabetes_X_test diabetes data 20 gt gt gt diabetes_y_train diabetes target 20 gt gt gt diabetes_y_test diabetes target 20 The task at hand is to predict disease prediction from physiological variables 20 6 1 Sparse models To improve the conditioning of the problem uninformative variables mitigate the curse of dimensionality as a feature selection preprocessing etc it would be interesting to select only the informative features and set non informative ones to 0 This penalization approach called Lasso can set some coefficients to zero Such methods are called sparse method and sparsit
362. ugh to discuss class based decorators where arguments are given in the decorator expression and the decorator __init__ method is used for decorator construction gt gt gt class decorator_class object def _init__ self arg this method is called in the decorator expression print in decorator init 7s arg self arg arg def _call__ self function this method ts called to do the job print in decorator call s self arg eee return function gt gt gt deco_instance decorator_class foo in decorator init foo gt gt gt deco_instance def function args xkwargs print in function 7s 4s args kwargs in decorator call foo gt gt gt function in function O IT 7 2 Decorators 149 Scipy lecture notes Edition 2015 2 Contrary to normal rules PEP 8 decorators written as classes behave more like functions and therefore their name often starts with a lowercase letter In reality it doesn t make much sense to create a new class just to have a decorator which returns the original function Objects are supposed to hold state and such decorators are more useful when the decorator returns a new object gt gt gt class replacing decorator_class object def _init__ self arg this method ts called in the decorator expression print in decorator init 7s arg self arg arg def _call__ self function this method ts called to do the job print in decorator call 7s self arg se
363. ula api import ols gt gt gt model ols y x data fitQ We can inspect the various statistics derived from the fit gt gt gt print model summary Dep Variable Model Method Date Time No Observations Df Residuals Df Model OLS Regression Results y OLS Least Squares 20 18 R squared Adj R squared F statistic Prob F statistic Log Likelihood AIC X 2 9369 Omnibus Prob Omnibus Skew Kurtosis Terminology 0 100 0 951 0 058 2 390 Bice if P gt t 342 0 000 604 0 000 Durbin Watson Jarque Bera JB Prob JB Cond No Scipy lecture notes Edition 2015 2 0 804 0 794 74 03 8 56e 08 57 988 120 0 1220 95 0 Conf Int Selo 2 220 Soe oor 3 654 2 956 0 322 0 851 3 03 Statsmodel uses a statistical terminology the y variable in statsmodel is called endogenous while the x variable is called exogenous This is discussed in more detail here To simplify y endogenous is the value you are trying to predict while x exogenous represents the features you are using to make the prediction Exercise Retrieve the estimated parameters from the model above Hint use tab completion to find the relevent attribute Categorical variables comparing groups or multiple categories Let us go back the data on brain size gt gt gt data pandas read_csv examples brain_size csv sep na_values We can wr
364. urces modules and filters 348 Scipy lecture notes Edition 2015 2 Excercice Using the GUI add the ExtractVectorNorm filter to display iso contours of the field magnitude mlab pipeline the scripting layer The mlab scripting layer builds pipelines for you You can reproduce these pipelines programmatically with the mlab pipeline interface each step has a corresponding mlab pipeline function simply convert the name of the step to lower case underscore separated ExtractVectorNorm gives extract_vector_norm This function takes as an argument the node that it applies to as well as optional parameters and returns the new node For example iso contours of the magnitude are coded as mlab pipeline iso_surface mlab pipeline extract_vector_norm field contours 0 1 Bmax 0 4 Bmax opacity 0 5 Excercice Using the mlab pipeline interface generate a complete visualization with iso contours of the field mag nitude and a vector cut plane click on the figure for a solution 19 4 Animating the data To make movies or interactive application you may want to change the data represented on a given visualiza tion If you have built a visualization using the mlab plotting functions or the mlab pipeline function we can update the data by assigning new values to the mlab_source attributes Sey 3 2 p ogridli 5 5 100 5 5 100 gt 5 5 100 scalars np sin x y z x y Z iso mlab contour3d scalars
365. utions have a number of useful methods Explore them by reading the docstring or by using Python tab completion Can you find the shape parameter of 1 back by using the fit method on your random variates 5 6 2 Percentiles The median is the value with half of the observations below and half above 114 5 6 Statistics and random numbers scipy stats Scipy lecture notes Edition 2015 2 gt gt gt np median a 0 04041769593 It is also called the percentile 50 because 50 of the observation are below it gt gt gt stats scoreatpercentile a 50 0 0404176959 Similarly we can calculate the percentile 90 gt gt gt stats scoreatpercentile a 90 1 318353699120 The percentile is an estimator of the CDF cumulative distribution function 5 6 3 Statistical tests A Statistical test is a decision indicator For instance if we have two sets of observations that we assume are generated from Gaussian processes we can use a T test to decide whether the two sets of observations are significantly different gt gt gt a np random normal 0 1 size 100 gt gt gt b np random normal 1 1 size 10 gt gt gt stats ttest_ind a b array 3 177574054 00019370689 The resulting output is composed of e The T statistic value it is a number the sign of which is proportional to the difference between the two random processes and the magnitude is related to the significance of this difference e
366. with block is exited Since closing files is such a common operation the support for this is already present in the file class It has an __exit__ method which calls close and can be used as a context manager itself gt gt gt with open tmp file a as f f write more contents n The common use for try finally is releasing resources Various different cases are implemented similarly in the __enter__ phase the resource is acquired in the __exit__ phase it is released and the exception if thrown is propagated As with files there s often a natural operation to perform after the object has been used and it is most convenient to have the support built in With each release Python provides support in more places e all file like objects file automatically closed fileinput tempfile py gt 3 2 b22 BZ2File ezip Gziprkile tarfile larrile zipfile Ziprile ftplib nntplib close connection py gt 3 2 or 3 3 locks multiprocessing RLock lock and unlock multiprocessing Semaphore memoryview automatically release py gt 3 2 and 2 7 e decimal localcontext modify precision of computations temporarily e winreg PyHKEY open and close hive key e warnings catch_warnings kill warnings temporarily contextlib closing the same as the example above call close parallel programming concurrent futures ThreadPoolExecutor invoke in parallel then kill thread pool
367. wn anyone really needing to use the Python C API should read the section about objects types and reference counts from the official python doc umentation Additionally there is a tool by the name of cpychecker which can help discover common errors with reference counting 14 2 Python C Api 264 Scipy lecture notes Edition 2015 2 14 2 1 Example The following C extension module make the cos function from the standard math library available to Python Example of wrapping cos function from math h with the Python C API include lt Python h gt include lt math h gt wrapped cosine function static PyObject cos_func Py0bject self PyObject args double value double answer parse the input from python float to c double if PyArg ParseTuple args d amp value return NULL if the above function returns 1 an appropriate Python exception will have been set and the function simply returns NULL t call cos from libm answer cos value construct the output from cos from c double to python float return Py_BuildValue f answer define functions in module static PyMethodDef CosMethods cos_func cos_func METH VARARGS evaluate the cosine NULL NULL 0 NULL ie module initialization PyMODINIT_FUNC initcos_module void void Py_InitModule cos_module CosMethods As you can see there is much boilerplate both to massag
368. x cos y 16 2 2 Simplify Use simplify if you would like to transform an expression into a simpler form gt gt gt simplify x x y x y Simplification is a somewhat vague term and more precises alternatives to simplify exists powsimp simplifi cation of exponents trigsimp for trigonometric expressions logcombine radsimp together Exercises 1 Calculate the expanded form of x y 2 Simplify the trigonometric expression sin x cos x 16 2 Algebraic manipulations 303 Scipy lecture notes Edition 2015 2 16 3 Calculus 16 3 1 Limits Limits are easy to use in SymPy they follow the syntax limit function variable point so to compute the limit of f x as x 0 you would issue limit f x 0 gt gt gt limit sin x x x 0 you can also calculate the limit at infinity gt gt gt limit x x 00 OO gt gt gt limit 1 x x 00 0 gt gt gt limit x x x 0 1 16 3 2 Differentiation You can differentiate any SymPy expression using diff func var Examples gt gt gt diff sin z x cos x gt gt gt diff sin 2 x x 2 cos 2 x gt gt gt diff tan x x 2 tan x 1 You can check that it is correct by gt gt gt limit tan x y tan x y y 0 2 tan x 1 Higher derivatives can be calculated using the diff func var n method gt gt gt diff sin 2 x x 1 2 cos 2 x gt gt gt diff sin 2 x x 2 4 sin 2 x
369. x4 sparse matrix of type lt numpy int64 gt with 4 stored elements in COOrdinate format gt gt gt gt mtx todense matrix 4 0 9 0l O k On Ol HOR O 0 Ol Or 0 0 51 e duplicates entries are summed together gt gt gt row np array L0 0 1 3 1 0 0J gt gt gt col np array l0 2 1 3 1 0 0 gt gt gt data np array 1 1 1 1 1 1 11 gt gt gt mtx sparse coo_matrix data row col shape 4 4 gt gt gt mtx todense matrix ES 0 1 Ol O2 One Ole HO Os Or 0l 0 0 0 111 e no slicing gt gt gt mtx 2 3 Traceback most recent call last TypeError coo_matrix object 11 2 Storage Schemes 218 Scipy lecture notes Edition 2015 2 Traceback most recent call last TypeError coo_matrix object Compressed Sparse Row Format CSR e row oriented three NumPy arrays indices indptr data indices is array of column indices data is array of corresponding nonzero values indptr points to row starts in indices and data lengthisn_row 1 last item number of values length of both indices and data nonzero values of the i th row are data indptr Li indptr i 1 with column in dices indices Lindptr i indptr i 1 item i j can be accessed as dataLlindptr li k where k is position of j in indices indptr i indptr i 1 subclass of _cs_matrix common CSR CSC functionality subclass of _data_matrix sp
370. xt Text object at Ox gt gt gt gt plt ylabel Amplitude lt matplovtlib text Text object at 0x gt 5 4 Fast Fourier transforms scipy fftpack 105 Scipy lecture notes Edition 2015 2 Amplitude 0 5 10 15 20 Time s numpy fft Numpy also has an implementation of FFT numpy fft However in general the scipy one should be preferred as it uses more efficient underlying implementations 5 4 Fast Fourier transforms scipy fftpack 106 Scipy lecture notes Edition 2015 2 Worked example Crude periodicity finding 80 70 60 50 40 30 20 Population number 10 10 O 1900 1905 1910 1915 1920 Year 300 250 200 150 Power 10 100 50 O Ul 10 15 20 Period 5 4 Fast Fourier transforms scipy fftpack 107 Scipy lecture notes Edition 2015 2 Worked example Gaussian image blur Convolution file de KE DA fiw Koho 50 100 150 5 4 Fast Fourier transforms scipy fftpack 108 Scipy lecture notes Edition 2015 2 Exercise Denoise moon landing image oer hi bateaehae A ry gars ry seis 55 sone Bee Gusts os LADD OD Oe ey y 900 Fare P G ha hae Oy er D nah aaa APs het OSOON ri ae ROLIO rS i x i 5i g L Ho DEJ o bi 0 A 4 POIS O ROO ere A ome g Tis 0 Y gt Q Ah Yy 4 5 0 ong gored 005050 Mote enemies 1 Exa
371. y Zz x 23 y append 42 z 99 new reference Prine x print y print z gt gt gt a 77 immutable variable gt gt gt b 99 mutable variable gt gt gt c 28 gt gt gt try_to_modify a b c 23 99 42 99 gt gt gt print a I 2 4 Defining functions 24 Scipy lecture notes Edition 2015 2 gt gt gt print b 99 42 gt gt gt print c 28 Functions have a local variable table called a local namespace The variable x only exists within the function try_to_modify 2 4 5 Global variables Variables declared outside the function can be referenced within the function In 114 x 5 In 115 def addx y TPT return x y In 116 addx 10 Outlii6 15 But these global variables cannot be modified within the function unless declared global in the function This doesn t work In 117 def setx y TS x y ee PrineC sx is 70 x In 118 setx 10 x is 10 In 120 x Out 120 5 This works In 121 def setx y erts global x vee x y eee Prine x is 40 x In 122 setx 10 x is 10 In 123 x Out 123 10 2 4 0 Variable number of parameters Special forms of parameters e xargs any number of positional arguments packed into a tuple e xxkwargs any number of keyword arguments packed into a dictionary 2 4 Defining functions 25 Scipy lecture notes Edition 2015 2 In 35 def variable_args args kwarg
372. y can be seen as an application of Occam s razor prefer simpler models to complex ones gt gt gt from sklearn import linear_model gt gt gt regr linear_model Lasso alpha 3 gt gt gt regr fit diabetes_X_train diabetes_y_train Lasso gt gt gt regr coef_ very sparse coefficients array 0 5 z0 497 34075682 199 17441034 O zO 7 lS 269291545 OF 430 9379595 0 JD gt gt gt regr score diabetes_X_test diabetes_y_test 075510835453 m being the score very similar to linear regression Least Squares gt gt gt lin linear_model LinearRegression gt gt gt lin fit diabetes_X_train diabetes_y_train LinearRegression gt gt gt lin score diabetes_X_test diabetes_y_test 0 583850753022 Different algorithms for a same problem Different algorithms can be used to solve the same mathematical problem For instance the Lasso ob ject in the sklearn solves the lasso regression using a coordinate descent method that is efficient on large datasets However the sklearn also provides the LassoLARS object using the LARS which is very effi cient for problems in which the weight vector estimated is very sparse that is problems with very few observations 20 6 Linear model from regression to sparsity 361 Scipy lecture notes Edition 2015 2 20 7 Model selection choosing estimators and their parameters 20 7 1 Grid search and cross validated estimators Grid search The sciki
373. y example such that the function takes two input arguments where the second is the preallocated output array making it similar to the other Numpy examples Modify the example such that the function only takes a single input array and modifies this in place Try to fix the example to use the new Numpy iterator protocol If you manage to obtain a working solu tion please submit a pull request on github You may have noticed that the Numpy C API example is the only Numpy example that does not wrap cos_doubles but instead applies the cos function directly to the elements of the Numpy array Does this have any advantages over the other techniques Can you wrap cos_doubles using only the Numpy C API You may need to ensure that the arrays have the correct type are one dimensional and contiguous in memory 14 8 2 Ctypes l Modify the Numpy example such that cos_doubles_func handles the preallocation for you thus mak ing it more like the Numpy C API example 14 7 Further Reading and References 280 Scipy lecture notes Edition 2015 2 14 8 3 SWIG Look at the code that SWIG autogenerates how much of it do you understand Modify the Numpy example such that cos_doubles_func handles the preallocation for you thus mak ing it more like the Numpy C API example Modify the cos_doubles C function so that it returns an allocated array Can you wrap this using SWIG typemaps If not why not Is there a workaround f
374. yUfunc_d_d double elementwise_func double input_1 PyUfunc_dd_d double elementwise_func double input_1 double input_2 PyUfunc_D_D elementwise_func complex_double input complex_double output PyUfunc_DD_D elementwise_func complex_double in1 complex_double in2 complex_double out Type codes NPY_BOOL NPY_BYTE NPY_UBYTE NPY_SHORT NPY_USHORT NPY_INT NPY_UINT NPY_LONG NPY_ULONG NPY_LONGLONG NPY_ULONGLONG NPY_FLOAT NPY_DOUBLE NPY_LONGDOUBLE NPY_CFLOAT NPY_CDOUBLE NPY_CLONGDOUBLE NPY_DATETIME NPY_TIMEDELTA NPY_OBJECT NPY_STRING NPY_UNICODE NPY_VOID eS S 8 2 3 Solution building an ufunc from scratch The elementwise function cdef void mandel_single _point double complex z_in double complex c_in double complex z_out nogil The Mandelbrot iteration Some points of note It s NOT allowed to call any Python functions here The Ufunc loop runs with the Python Global Interpreter Lock released Hence the nogil gt And so all local variables must be declared with ede Note also that this function receives pointers to the data the traditional solution to passing complex variables around cdef double complex z z_in 0 cdef double complex c c_in 0 cdef int k the integer we use in the for loop Straightforward iteration for k in range 100 Zz Z Z Cc if z real 2 z imag 2 gt 1000 break Retu
375. ymPy is able to solve algebraic equations in one and several variables 16 4 Equation solving 305 Scipy lecture notes Edition 2015 2 In 7 solve x 4 1 x Outl7 LI 1 1 I As you Can see it takes as first argument an expression that is supposed to be equaled to 0 It is able to solve a large part of polynomial equations and is also capable of solving multiple equations with respect to multiple variables giving a tuple as second argument In 8 solve x S4y 2 3 x 6 y 15 x yl Out Seay i k er It also has limited support for trascendental equations In 9 solve exp x 1 x Out 9 pi I Another alternative in the case of polynomial equations is factor factor returns the polynomial factorized into irreducible terms and is capable of computing the factorization over various domains In 10 f x 4 3 x 2 1 In 11 factor f Out 11 1 x x 2 1 x x 2 In 12 factor f modulus 5 Out i2 2 x 2 2 x 2 SymPy is also able to solve boolean equations that is to decide if a certain boolean expression is satisfiable or not For this we use the function satisfiable In 13 satisfiable x amp y Out Lisi x True y True This tells us that x amp y is True whenever x and y are both True If an expression cannot be true i e no values of its arguments can make the expression True it will return False In 14 satisfiable x amp x Out 14 False
376. ype int32 gt gt gt print mtx 0 0 1 1 2 2 35 3 1 0 2 1 3 2 T 0 2 ia E 12 gt gt gt mtx todense matrix L1 0 114 Ol OnrF WN Fe 5 2 0 12 05 6 73 0 LAO 0 Ty Adal e explanation with a scheme 11 2 Storage Schemes 214 Scipy lecture notes Edition 2015 2 offset row 2 9 1 a lO gt Sess O O G a e i B32 vs 22 2 6 3 3 T 4 Drao aaa 8 e matrix vector multiplication gt gt gt vec np ones 4 gt gt gt vec array FA a a ld gt gt gt mtx vec array 12 19 Gi TID gt gt gt mtx toarray vec array Cll 1 Ore Sui 0 l Sins P Oa 12 i Oss Ons Sas CRIE Ore O ee A List of Lists Format LIL e row based linked list each row is a Python list sorted of column indices of non zero elements rows stored in a NumPy array dtype np object non zero values data stored analogously efficient for constructing sparse matrices incrementally e constructor accepts dense matrix array sparse matrix shape tuple create empty matrix flexible slicing changing sparsity structure is efficient e slow arithmetics slow column slicing due to being row based e use when sparsity pattern is not known apriori or changes example reading a sparse matrix from a text file Examples e create an empty LIL matrix gt gt gt mtx sparse 1lil_matrix 4 5 e prepare random data gt gt gt
377. ypes 8 2 Universal functions 180 Scipy lecture notes Edition 2015 2 2 number of supported input types Scl o SSS See 2 number of input args 1 number of output args O identity element never mind this mandel function name mandel z c gt computes iterated z z c docstring O unused 8 2 4 Generalized ufuncs ufunc output elementwise_function input Both output and input can be a single array element only generalized ufunc output and input can be arrays with a fixed number of dimensions For example matrix trace sum of diag elements input shape n n output shape i e scalar Cala ea Matrix product input_1 shape m n input_2 shape n p output shape m p m n n p gt m p e This is called the signature of the generalized ufunc e The dimensions on which the g ufunc acts are core dimensions Status in Numpy e g ufuncs are in Numpy already e new ones can be created with PyUFunc_FromFuncAndDataAndSignature e but we don t ship with public g ufuncs except for testing ATM gt gt gt import numpy core umath_tests as ut gt gt gt ut matrix_multiply signature mn ap mh p gt gt gt x np ones 10 2 4 gt gt gt y np ones 10 4 5 gt gt gt ut matrix_multiply x y shape COR aS e the last two dimensions became core dimensions and are modified as per the signature e otherwise
378. ys use functools update_wrapper or some other means of copy ing function attributes 7 2 4 Examples in the standard library First it should be mentioned that there s a number of useful decorators available in the standard library There are three decorators which really form a part of the language e classmethod causes a method to become a class method which means that it can be invoked without creating an instance of the class When a normal method is invoked the interpreter inserts the instance object as the first positional parameter self When a class method is invoked the class itself is given as the first parameter often called cls Class methods are still accessible through the class namespace so they don t pollute the module s namespace Class methods can be used to provide alternative constructors class Array object def init self data self data data classmethod def fromfile cls file data numpy load file return cls data This is cleaner then using a multitude of flags to ___init__ 7 2 Decorators 151 Scipy lecture notes Edition 2015 2 e staticmethod is applied to methods to make them static i e basically a normal function but acces sible through the class namespace This can be useful when the function is only needed inside this class its name would then be prefixed with _ or when we want the user to think of the method as connected to the class despite an implementation

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