Extrema User's Guide
Contents
1. first 10 rows from the first column of y Such expressions can result in scalars arrays vectors or matrices depending on the number of dimensions of the result The special characters and are also available for use in indices For example x all values from vector x x the last value from vector x x 1 the next to last value from vector x m all rows and all columns of matrix m m all rows and the last column of matrix m m 1 1 all rows and all but last column of matrix m Constants You can type numeric values or constants anywhere a scalar variable or value is expected Constant arrays are expressed as a list of values inside square brackets When typing out vector of matrix values separate successive indexes with a comma and successive values within an index with a semicolon Ja USE 8 scalar value 1 2 4 8 vector with 4 values OG GFir0 GAUL 3 by 3 identity matrix You can also use the start stop step notation to specify regular sequences of values with which to fill the variable 0 27 pl 01 vector from 0 to 27 in steps of 0 1 10 10 2 descending sequence from 10 to 10 in steps of 2 Expressions Extrema allows you to use mathematical expressions anywhere it would expect a variable or value provided the expression evaluates to the expected type Simple expressions involving dimensioned variables generally return a value of the same dimension Thus if x has 102. graph the fit function as a line SET CURVECOLOR BLUE GRAPH X A B X Free parameters should be initialized to an appropriate guess value from which the fit will begin In simple cases the actual value of the guess is not terribly important Extrema will find the correct value regardless In more complex cases the initial guess will affect how the fit progresses and could affect the final result In other words in some cases different fits can be found depending on where you start so choosing a reasonable guess to initialize the 39 free parameters can be important Once the fit is complete the free parameters will have their fitted values If Extrema failed to find a good fit the free parameters will have the last values Extrema tried to fit with or optionally they can be reset to their initial values upon failure Normally fitting results in multiple lines of text output describing the fit The values of the free parameters and various other values describing the accuracy of the fit are all contained in this output Extrema can optionally write some of this information into variables for access by scripts and expressions later in the analysis process For more detailed information please refer to the FIT command in the Extrema Command Reference Fitting different data segments to different functions In some cases you will want to divide the data into segments or groups and fit each group separately For example
3. 1 i WHERE ydiff lt spike min GRAPH x i yl i _ graph the selected data 46 Output Printing graphs Printing your graphs is very easy simply select Print and proceed as you would for any other Windows printing job Exporting graphs for inclusion in other documents Researchers commonly need to include their graphs in other documents such as research papers written reports or web pages For these purposes Extrema can export to several industry standard graphics formats PostScript EPS Portable Network Graphics PNG and Joint Photographic Experts Group JPEG The HARDCOPY command is used for saving the graphics to a file in one of the supported formats Encapsulated PostScript is the default format if no qualifier is entered with the HARDCOPY command PostScript amp EPS PostScript EPS is the industry standard for printed documents it provides excellent publication quality output that is completely scalable and is compatible with documents conforming to the Portable Document Format PDF or to the TeX and LaTeX systems that are common in scientific publishing PNG Portable Network Graphic PNG is a bitmap image format that is supported by most major web browsers including Explorer and Netscape As a bitmap format it is inferior for regular publication purposes but it is convenient for in lined tmage display in web pages It gives high rates of compression for conventional drawings and plots and is the re
4. DO IOOPS uius nu mu man aan mama nn 53 GETTING STARTED WITN EXTREMA CHAPTERS Getting started Data input Drawing graphs Customizing graphs Data analysis Output q M 38 8 Ge Scripting Getting Started An introduction to Extrema About This Guide T his guide is an introduction to using Extrema for data analysis and visualization It is not a comprehensive manual detailing all Extrema features but rather a concise guide to accomplishing the types of tasks most commonly performed by researchers It intended to be useful to new users and for experienced users who would like a quick introduction to features or methods they have not used before In most cases this guide instructs by example Typical data analysis and visualization tasks are described and then performed both using the GUI and using the scripting language The tasks are grouped into categories that are treated in subsequent sections of this guide The tasks described are generally simple but practical Extending them to cases that are more complex is generally straightforward although you may need to consult the Extrema Command Reference for further details Documentation that is more comprehensive is available in the following references 1 Extrema Command Reference a comprehensive guide to the Extrema command language 2 Extrema Online Help detailed online help is available through the program itself In many of th
5. GRAPH command for example GRAPH legend entry x y If LEGENDON 0 a string entered as a first parameter with the GRAPH command is ignored Following is an example script using a graph legend and the picture that it produces X S so T EGENDON 1 EGENDTITLECOLOR 16 EGENDTITLEFONT impact EGENDTITLE The Legend Title AGENDENTRYLINEON 1 SLEGENDFRAME 20 60 60 90 T symbolSizes 2 2 5 3 2 symbols 15 16 17 18 colors 1 red colors 2 blue colors 3 ecral colors 4 cyan DO i 1 4 SET PLOTSYMBOL symbols i PLOTSYMBOLCOLOR colors i SPLOTSYMBOLSIZE symbolSizes i EGENDSYMBOLS i CURVECOLOR colors 5 i CURVELINETYPE i 2 GRAPH legend entry lt gt RCHAR i x i x 2 NDDO E PLOT m Zo 30 400 The Legend Title legend entry legend entry 300 legend entry legend entry 200 100 Graph Axes To graph only the axes for a particular set of data use GRAPH AXESONLY x y To graph a set of data with no axes use GRAPH OVERLAY x y These options are handy if you make multiple drawing passes over the same graph In the GUI you can simply select the appropriate checkboxes to get the same behaviour Scaling Axes can be manually or automatically scaled Auto scaling is the default in which the
6. also specify the number of rows in the matrix READ matrix file3 dat m nrows read file into matrix m There are also options to read matrices in other ways for example by specifying the number of columns in the matrix See the READ command for more information Binary files Binary files are not human readable if you attempt to view or edit them you will see a lot of garbage They contain sequences of numbers or other values in raw machine format To read this data you need to know the details of the file format that is the sequence and types of data written to the file Generating Data Commonly you will need to create data spontaneously In simple cases you can type in the data directly Usually however you will be working with data sizes that make this approach too tedious There are numerous methods you can use for bulk data generation Sequences Simple sequences can be generated using the start stop step array notation pi ACOS 1 define scalar pi X 0 pi 01 make a sequence of values from 0 to T in increments of 0 01 You can create a regular sequence of values using the GENERATE facility The generated data can be specified using any of the following methods minimum value maximum value number of values minimum value maximum value step size minimum value step size number of values You can also request random values instead of a regular step size Functions By applying an expression to
7. an already existing variable you can generate a new variable in which every element of the input variable has been modified by the expression Capture this data in a new variable by simply setting the new variable to equal the expression y 10 SIN x x is a vector of values If your source data is a monotonically increasing sequence see above that serves as the dependent variable then you will get a fair representation of the function itself over that range For instance to produce data representing the function SIN x over the range 0 to 2T pi ACOS 1 x Os2 p170 01 y SIN x Interpolating Data You may have a sparse sampling of data that you wish to fill in by interpolation These techniques are described later in Chapter 5 Data Analysis Bamples Drawing Graphs Extrema can produc a wide variety of graph types not all of which are described here This section reviews the types of graphs that are most commonly used and how to make them For complete details on the variations that are possible consult the Exttema Command Reference Manual for the appropriate commands Graphics sub windows Use the WINDOW command to choose and or define a graphics sub window Graphics sub windows ate an easy way to subdivide the graphics output page into rectangular regions allowing multiple graphs and or multiple figures and or multiple text regions A window is a subset of the page A window other than the default zero level wind
8. axis will stretch or shrink to accommodate the full range of the plotted data This is convenient for well behaved data sets but maybe not for data with spikes infinities or related problems Autoscaling is also inconvenient when one is overlaying numerous similar graphs where one requires that the scale be fixed 31 Manual axis scaling is done using the SCALES command SCALES x min x max y min y max SCALES x min x max n x tics y min y max n y tics SCALES The first form simply sets axis ranges The second form also sets the number of large numbered tic marks that should be shown for each axis The last form freezes the axis scales at whatever is their current value Tics The parameters controlling x axis tic marks are XTICSON XTICSBOTHSIDES XTICANGLE XNLINCS XNSINCS XLARGETICLENGTH XSMALLTICLENGTH XIMAGTICANGLE XIMAGTICLENGTH controls whether or not tic marks both large and small are drawn on the x axis controls whether or not tic marks both large and small are drawn on both sides of the x axis controls the angle of the tic marks both large and small on the x axis controls the number of large labelled tic marks to be displayed on the x axis conttols the number of small unlabeled tic marks to be displayed between the large labelled tic marks on the x axis controls the length of the large labelled tic marks on the x axis controls the length of the optional sm
9. suppose you want to fit two line segments to the data such that they join at one end point Below on the left is an example where the two segments are forced to join and on the right an example where they are allowed to float X 1 19 La leepsrpeaeapesr 7 Ss 971079 Oy io oy 45 sy ey Loren K WINDOW 5 SET PLOTSYMBOL 1 GRAPH x y SCALAR FIT abcd X0 10 FIT y atb x x lt x0 ctd x x gt x0 atb x c d x 1000 x x0 SET PLOTSYMBOL 0 11 WHERE x lt x0 I2 WHERE x gt x0 Y1 atb x 2 p3 ET CURVECOLOR red RAPH OVERLAY x il yl il RAPH OVERLAY x i2 y2 i2 S G G WINDOW 7 SET PLOTSYMBOL 1 GRAPH x y FIT y atb x x lt x0 c d x x gt x0 S Y x S G G R ET PLOTSYMBOL 0 1 atb x 2 DE ET CURVECOLOR red RAPH OVERLAY x il yl il RAPH OVERLAY x i2 y2 i2 EPLOT ALL 40 Binning data has already been mentioned a few times as a means of converting one dimensional data into two dimensional data BIN command or two dimensional into three dimensional BIN2D command Simply put binning counts the data points falling into a certain range This results in a vector or vectors in the 2 D case describing the ranges the bins and a second vector or matrix describing the counts Simple binning is straightforward An input vector of values is taken as input a
10. the data be monotonically increasing They will not however return an actual algebraic function describing the shape of your data For this you need to do a proper fit see below There are also interpolation functions that will X in missing data using similar smoothing techniques INTERP and SPLINTERP functions Please refer to SMOOTH SPLSMOOTH SAVGOL INTERP and SPLINTERP functions Fitting to a function To describe your data as a function you ll need to know in advance what function you will be fitting to This function will be expressed with a number of free parameters whose precise values are unknown The purpose of the fit is to determine what values of those free parameters best match the data 38 Note Fitting is an uncertain process by its very nature There is no guarantee that an appropriate fit will be found in all cases and there is no guarantee that there is only one such fit that describes the data A free parameter is like a scalar variable except that instead of being set by you or your data analysis operations it is set by Extrema in the course of making the fit This difference in behaviour means that free parameters are declared differently so that Extrema knows it can vary the parameter instead of treating it as a fixed constant in the fitting expression SET PLOTSYMBOLCOLOR RED GRAPH X Y graph the raw data SCALAR FIT A B declare free parameters FIT Y A B X perform the fit SET PLOTSYMBOL 0
11. values then the expression Sin x 1 also has 10 values Other examples m x 2 the rows denoted in x and the last 3 columns of m x m n x times the nth row of m SIN atb the sines of the sums of respective values in a and b X 2 SIN x 1 a non linear function of the values in x There is no limit to the length or complexity of a mathematical expression in Extrema You can also index the results of an expression e g SIN x 1 4 8 selects 4th through 8th values of the expression Reading Data From Files In most cases your data will be contained in files You will need to read these files into Extrema variables before you can operate on the data Extrema is quite flexible in allowing you to read files of different formats although in more complex cases you will need to know the details of the file s data format Text files Text files are human readable that is they contain data written in ASCII format and they can be viewed or edited with a simple text editor such as Notepad Many spreadsheets can export their data into such a format If your text file contains data arranged in rows and columns with columns delimited by commas or white space then reading the file is simple To read each column into its own vector READ filel dat READ file2 dat X y read 2 columns into vectors x and y aa srr b b err read data and errors into vectors To read all columns into a single matrix you must
12. z 6 i 10 100 ig ao ME a 05 a0 D 20 oO OZ oO z 5 2 10 oO z 5 10 The WINDOW command can also be used to define your own custom set of drawing windows Draw two sets of data on the same graph at the same scale Method 1 Manually set the axis scales to values that are appropriate for both graphs see section 4 5 1 Then draw your first graph Without clearing the graph draw the second graph without axes SCALE 0 2 pi 1 J xmin xmax ymin ymax GRAPH x y SET CURVECOLOR RED change color for overlayed curve GRAPH OVERLAY x z 19 0 5 0 0 0 5 Method 2 Draw the graph that should be used to autoscale the axes first Then freeze the axes at those values before drawing the second graph without axes GRAPH x y SCALE freezes the current scale GRAPH OVERLAY x z Method 3 Graph either of the two data curves first then overlay the second data curve Then use the REPLOT command to redraw both curves on a common scale GRAPH x y GRAPH OVERLAY x z REPLOT Draw two sets of data on the same graph but at different Y scales Method 1 If you only need a labelled y axis for one data set the task is easy GRAPH x y y axis is for this graph GRAPH OVERLAY x z no y scale shown for this graph Method 2 By default the y axis is drawn at the left hand end of the x axis The GRAPH YONRIGHT command draws the y axis on the right For example the following commands produce the figure below X 1 100 GRAP
13. 1 0 then the numbers labelling the large tic marks on the y axis are displayed in exponential format 10000 1000 YLOGBASE 10 YLOGSTYLE 1 YLOGBASE 10 100 YLOGSTYLE 0 10 Axis placement The placement of the axes can be precisely controlled by manipulating the axis location parameters SXLOWERAXIS 6XUPPERAXIS 6YLOWERAXIS 34 6YUPPERAXIS The percentage versions specify positions as percentages of the current drawing window otherwise the positions are in the drawing coordinates By careful manipulation of these values you can place one graph at any point on the drawing with respect to another For instance to adjoin two graphs along the x axis so that there is an upper graph and a lower graph with a common edge Set SYUPPERAXIS to a reduced value e g 50 Plot the first graph Set 5YLOWERAXTIS to the value of SYUPPERAXIS Set SYUPPERAXIS to 85 Turn off drawing of the x axis labels with SET XAXIS 0 Plot the second graph oY ARS NE 10 8 100 80 60 40 20 0 0 20 40 60 80 100 In practice there are some other parameters you may need to play with to keep the y axis labelling clean but the above will suffice in simple cases 35 Data Analysis Examples Extrema provides numerous tools for data analysis induding data transformation tools filtering tools cutting and selection tools Elementary data manipulation is done using Extrema s built in expression evaluation capabilities A
14. 8 In addition to the plotting symbol you can also specify the size color and angle in degrees If scalar values are used for these the value will apply to every data point If vector values are used the vectors should be the length as the data vectors The corresponding values for each point are used to set the plotting style for that point In the GUI you can simply enter the size color and angle vectors or scalars or expressions in the appropriate fields In the command language use SET PLOTSYMBOL symbol SET SPLOTSYMBOLSIZE size SET PLOTSYMBOLCOLOR color SET PLOTSYMBOLANGLE angle For example to plot a vector field we could select an arrow symbol where the arrow is centred at on the data value 13 and then set the sizes and angles according to two vectors magnitude and direction SET PLOTSYMBOL 13 SET SPLOTSYMBOLSIZE magnitude SET PLOTSYMBOLCOLOR black SET PLOTSYMBOLANGLE direction GRAPH x y 26 draw vector field Line type The line type or style used for drawing lines on your graphs can be selected using the SET LINETYPE command The line style is used only for graphed lines it is not used for other lines such as graph axes Line width SET LINEWIDTH n controls the line width of the axes the data curve and the plotting symbols drawn when the GRAPH command is entered The units of 27 LINEWIDTH are pts where a pt is 1 72 of an inch For example a line width of 36 gives 1 2 inch w
15. Extrema s scripting capabilities include looping and decision making features so a fair amount of intelligence can be built into your scripts The examples in this guide include instructions for performing operations interactively using the GUI or using the command language Command language examples can be used interactively in the command window or in scripts Data Input Before you can analyze or visnakze your data you need to get your data into Extrema mn a form that can be manipulated Data Representation Data is stored internally in variables which have names that you use to reference the data they contain Except for a few automatically generated variables these names are chosen by the user The first character of a variable name must be an alphabetic character that is A to Z and the maximum number of characters in a name is thirty two 32 Except for these resttictions variable names can be any combination of alphabetic characters ABC XYZ digits 0123456789 underscore _ and dollar sign Variable names are case insensitive e g variable mydata is the same as MyData Function names are reserved names and cannot be used as variable names Variables can contain character data or numeric data Numeric data are always stored as double precision real values Character or string variables can be one of the following types string scalar a simple string of text string array an array of text strings N
16. H YONRIGHT X SIN X 20 20 05 0 0 0 5 If you need a labelled y axis for both data sets just graph the first data curve with the y axis on the left the default and then graph the second data curve with the y axis on the right You might want to change the color for the second y axis and curve to distinguish it from the first For example x L 20 5 yl x 2 y2 EXP SIN x 5 SET XLABEL This is the x axis label XLABELON 1 SXLABELHEIGHT 5 YLABEL x lt gt 2 YLABELON 1 SYLABELHEIGHT 5 GRAPH x yl SET YAXISCOLOR blue YNUMBERSCOLOR blue CURVECOLOR blue 21 XAXIS Q YLABELCOLOR blue YLABEL e lt gt sin x 5 GRAPH YONRIGHT x y2 400 3 0 25 300 2 0 s 200 15 gxuis 100 05 0 5 10 15 20 This is the x axis label The GRAPH XONTOP command draws the x axis on the top More information on customizing axes and graph placement is provided in the next chapter 22 Customizing Graph Presentation Once you can draw your data you will want to customize the details of the drawing to improve is presentation Extrema has a large number of internal parameters used to control the drawing details By altering these parameters you can vaty the appearance of your drawing in a great variety of ways The most commonly used parameters can be easily set from the GUI simply by checking off the desired options from those that are presented The more obscure parameters may not ha
17. H x i yli graph the selected data Example 2 select only the data points collected within a time window We have an unordered scattered set of data points in the vectors x and y and the times of each in a vector t Say our time window is defined by tmin and tmax i WHERE t gt tmin amp t lt tmax select data in time window GRAPH x i yli graph the selected data Example 3 select only the data points whose error is below a threshold We have a set of data points in the vectors x and y with errors denoted by vectors xerr and yerr We want to reject any data point with an x error exceeding xthresh or y error exceeding ythresh i WHERE xerr lt xthresh yerr lt ythresh select good data 45 GRAPH x i yli xerr i yerrli graph the selected data Example 4 eliminate spikes from the data We have a set of data points in the vectors x and y with occasional anomalous single point spikes where the y value goes very high In the simple case we can simply filter out any data over a certain y value say ymax i WHERE y lt ymax GRAPH x i yli graph the selected data This won t work if the good data occasionally can rise above ymax In this case you might only want to filter out spikes with a certain minimum height say spike min relative to adjacent good points Here is a simple way to accomplish that ydiff 1 0 get y differences between each point and the previous point ydiff 2 LEN y y 2 y 1
18. PLAY myfile dat updated with LEN x points There is no limit to how many times you call a subroutine how many different subroutines you call or how many levels deep subroutines may be nested inside each other Branching and looping Branches are places where the execution of a script can fork This is usually controlled using the IF THEN construct There are two ways this statement can be used Method 1 IF condition THEN command Method 2 IF condition THEN command command Cll ENDIF In both cases the commands are executed only if the condition evaluates to true Le 1 In method 1 only the single command is executed whereas in method 2 an entire sequence of commands is executed If the condition evaluates to false then the commands ate ignored and execution jumps to the next non conditional statement You can also jump non conditionally using the GOTO statement The command GOTO bel command transfers script execution to the given label A label is a special statement that does nothing it is simply a place marker in the script It consists of just a name no spaces followed by a colon The colon is not used in the GOTO statement start IF a gt b THEN RETURN GOTO start In the above example the GOTO statement restarts the script from the top this could repeat indefinitely if we didn t have the conditional RETURN statement to exit the script at some point This is a simple example of a l
19. all tic marks on the x axis These ate the unlabeled tic marks between the large numbered tic marks controls the angle in degrees measured counter clockwise between the x axis and a line joining the base of each large tic mark on the x axis to the centre of the number labelling that tic mark controls the distance measured from the base of each large tic mark on the x axis to the centre of the number labelling that tic mark 32 The parameters controlling y axis tic marks are YTICSON YTICSBOTHSIDES YTICANGLE YNLINCS YNSINCS YLARGETICLENGTH YSMALLTICLENGTH YIMAGTICANGLE YIMAGTICLENGTH controls whether or not tic marks both large and small are drawn on the y axis controls whether or not tic marks both large and small are drawn on both sides of the y axis controls the angle of the tic marks both large and small on the y axis controls the number of large labelled tic marks to be displayed on the y axis conttols the number of small unlabeled tic marks to be displayed between the large labelled tic marks on the y axis controls the length of the large labelled tic marks on the y axis controls the length of the optional small tic marks on the y axis These ate the unlabeled tic marks between the large numbered tic marks controls the angle in degrees measured counter clockwise between the y axis and a line joining the base of each large tic mark on the
20. commended graphics format for most drawings JPEG Joint Photographic Experts Group JPEG images are also stored in a bitmap format and suffer from all the drawbacks of PNG images They are also optimized for displaying photographic images and do not generally give good compression for conventional drawings and plots Some complex drawings that involve smoothly varying gradients of tone or color may benefit from being exported to JPEG format however Saving Data If you have done much data processing you may want to save your modified data in a file so that you can come back to it in a future Extrema session and analyze it further The simplest way to write an output file is using the WRITE command which takes your variables and writes them in columns to the designated output file WRITE mydata dat x y z write x y and z vectors to 3 columns This sort of data file is also easy to read back in to Extrema see section 2 It is fairly portable in general and could also be imported into most spreadsheet programs for instance There are many options and other uses for the WRITE command including writing text strings scalars matrixes and tensors specifying precise formats appending to files writing individual vectors in multiple columns 48 Scripting Interactive use of Extrema is adequate for one off data analysis and visualization jobs or for exploratory data analysis in which you are trying to gain an understa
21. ctors however Two vectors which contain a series of xy data pairs Normally these would be plotted using a two dimensional graph type but they can also be binned to form a 2 D histogram which is represented as a matrix and plotted as in the first case There are a number of different graph types that can be used to represent a surface function z x y These are summarized below 15 Contour Plots In contour plots the x value is interpreted as an elevation that is indicated using a contour map CONTOUR m 15 mis a matrix 30 25 20 15 10 Density plots In density plots the zx value is interpreted as an intensity that is indicated using color default tone dithering or scaled boxes DENSITY m mis a matrix 16 Surface plots In surface plots the x value is interpreted as a 3 spatial dimension that is drawn in perspective SURFACE m mis a matrix Fourt dimensional graphs There are limits to how well four or more dimensional data can be represented on a two dimensional surface but there are some tricks you can use Here are a few ideas to help you get started Assume we have a tensor of data called t and we wish to graph x 7 We can slice t at different values of z and graph these slices using the 3 D graphing techniques above Section 3 5 below explains how to tile multiple graphs on the same drawing You could also build a script to plot each slice in sequence automatically cleari
22. d These are checked off by name in the GUI or by selecting a color name or a negative color number in the command language purple brown 8 mamon 15 24 Default Drawing Color The characteristic COLOR sets the current drawing color If set this color is used for everything placed onto the drawing By changing this parameter in the course of adding things to the drawing you can easily control the plotting colors SET COLOR BLUE draw the curve in blue GRAPH OVERLAY x y SET COLOR BLACK draw the axes in black GRAPH AXESONLY x y There are also more specific drawing color parameters used to set the colors of certain graph items such as labels axes and so on If set these will override the global default color If not set the global color will be used SET CURVECOLOR BLUE draw the curve in blue SET XAXISCOLOR BLACK draw the x axis in black GRAPH x y Plotting Symbols The plotting symbol can be manually selected in the GRAPH window In the command language use SET PLOTSYMBOL n Where n is the symbol number taken from Ifnis positive successive points are connected by lines If n is negative the corresponding positive value is used but points will not be connected If n is zero no plotting symbol is used but the points are connected the data is drawn as a simple curve in this case 25 1 0 7 2 X 8 3 9 4 10 5 11 6 gt 12 13 1 14 E 15 16 17 1
23. e brief procedures described in this guide the reader will be referred to commands and functions for more information These commands and functions may be looked up in the above references Conventions used in this guide Examples of messages and prompts written by the program as well as examples of user typed input are displayed in typewriter type style Curly brackets enclose parameters that are optional and or have default values and indicate that it is not necessary to enter these parameters Parentheses besides being used in mathematical expressions also enclose formats The backslash separates a command from a command qualifier or a parameter from its qualifier Literal quote strings can be delimited by the opening quote and the single quote or by the single quote at the beginning and the end or by the double quote at the beginning and the end For example ABC ABC and ABC are all valid literal quote strings Parentheses the back slash and quotes must be included where indicated Extrema is case insensitive so input may be provided in upper or lower case or with a mix of cases In the examples in this guide Extrema keywords are given in UPPER CASE while variables and user defined words are given in lower case but this is simply for clarity you do not have to follow this convention Words that you should replace with your own variable names are given in alics Installing E
24. ed Data Analysis amp Visualization Extrema User s Guide TRIUMF 4004 Wesbrook Mall Vancouver British Columbia V6T 2A3 Phone 604 222 1047 Fax 604 222 1074 Table of Contents GETTING STARTED 1 About This Guide 1 Conventions used in this guide 2 Installing Extrema 2 Running Extrema 2 DATA INPUT een Data Representation 4 Addressing parts of arrays 5 Constant 6 Expressions 6 Reading Data From Files 6 Textiles Ghee eh thie 7 Binary files 7 Generating Data 7 Sequences 7 FUNGCUONS 2 44 22842 wack 8 Interpolating Data 8 DRAWING GRAPHS 9 Graphics sub windows 9 Pre defined windows 9 One dimensional graphs 10 Two dimensional graphs ji Line graphs Scatterplots Histograms and bar charts 14 Three dimensional graphs ER 15 Contour Plots 16 Density plots 16 Surface plots 17 Four dimensional graphs RE TE NE ES 17 Graphing two 3 D functions on the same drawing 17 Multiple plots on the same drawing Tile numerous graphs on the same drawing 18 Draw two sets of data on the same graph at the same scale TS M NS em be 19 Draw
25. er script and is executed the same way The RETURN command when encountered stops execution of the current script and returns to the next script above This is a common way of ending a subroutine although it is not strictly necessary at the very end RETURN is implied at the end of every script so it is only needed for abnormal returns in the middle of the script RETURN ing from a top level script will return the user to interactive mode Subroutine scripts will commonly require input parameters by means of which the calling script passes information for the subroutine to operate on Parameters are typically variables that are given after the subroutine name for example analyze x y z pass x y and z vectors to the analyze subroutine Parameters can also include text strings for instance file names savedata myfile dat x y save the x amp y vectors tomyfile dat Subroutines that require parameters can also be executed interactively without specifying the parameters In this case Extrema will interactively prompt you for the missing information To use a parameter inside a subroutine use the notation where n is the number of the parameter to substitute into that spot For instance in the previous example the script savedata pcm might contain the following lines WRITE APPEND 1 2 23 DISPLAY 1 updated with LEN 2 points which would be translated to 51 WRITE APPEND myfile dat x y DIS
26. es are assumed to be bins and y values are assumed to be counts X 1 19 1 0 Y SIN x 2 pi GRAPH HISTOGRAM x y 0 6 0 6 0 4 0 2 0 0 0 5 10 15 20 The other types of histogram are shown below each with the appropriate value of HTSTOGRAMTYPE 14 HISTOGRAMTYPE 1 HISTOSRAMTYPE 3 40 40 a 8 6 5 4 4 2 2 D 2 4 6 B 40 7 2 4 6 8 i0 HISTOGRAMTYPE 2 HISTOGRAMTYPE 4 40 40 a 8 6 5 4 4 2 2 D a 6 B 40 9 2 4 6 8 i0 Three dimensional graphs Three dimensional graphs represent data triplets These are typically interpreted as 3 D functions z x 1 and plotted accordingly 3 D data typically comes in three forms A matrix whose indices represent the x and y dimensions and whose values represent the z dimension Variables of this type are representable as surface functions and surface plots are usually generated from data of this type Three vectors which can represent the x y and z dimensions in a rectangular coordinate system Corresponding indexes in these vectors are your data coordinates so by analogy with 2 D graphs above the data can be plotted as a scatter plot or a line by connecting successive points If it is reasonable to interpret the data as sampling a function xy then it is also possible to interpolate a regular matrix from it using the GRID operation and then proceed as in the previous case Certain types of 3 D plots e g contour plots can be generated directly from the original data ve
27. ide lines The parameter n must be a scalar SET LINEWIDTH is a shorthand way to set CURVELINEWIDTH and PLOTSYMBOLLINEWIDTH 30 29 20 Text Graph titles axis labels and other drawn text strings have several characteristics that can be altered in particular the font size color placement and angle These can be specified for specific types of text labels e g x axis labels or for text labels in general These parameters are all easily controlled in the GUI by setting the appropriate fields in the TEXT window or the FONT sub window You can explicitly enter the location on the graph where the label should be placed or you can manually place it using the mouse The placement of the label is with respect to a particular point in the box that encloses the text string In command mode the various text drawing parameters are controlled using a number of settings SET SET SET SET SET XTEXTLOCATION x position YTEXTLOCATION y position SXTEXTLOCATION xp position SYTEXTLOCATION yp position FONT font name 28 expressed as a percentage l expressed as a percentage Axis Labels Axis labels are a special case of text strings since they have a standard placement and orientation The x axis text label is drawn centred below the x axis The y axis text label is drawn centred to the left of the y axis The axis text labels are drawn only when the axes are drawn The character string may contain format com
28. it in plain text txt format By default Extrema scripts are assumed to have the extension pcm Any other extension can be used however if it is specified when you run the script Every line of the script is an Extrema command By default the commands are executed in order unless you have loops or branches see below Comments Script lines that begin with the exclamation point character are comments These lines are ignored by Extrema and are used to add commentary to your scripts to help document what they are doing Many of the scripting examples in this guide include comments Comments can be placed on their own line or they can be appended to the end of the line Everything from the exclamation point character to the end of the line is ignored Comment characters are also sometimes used to disable lines in a script Running scripts To run a script use the syntax script By default this will execute the script in the file script pcm If you saved your script in a file with a different extension say txt you could say script txt instead By default scripts run silently with no output unless they perform actions that generate some kind of output e g draw graphs do a fit In some cases you may prefer that your script be chatty so that you can follow its progress You could insert commands to force some output at particular moments or you could just include the command ENABLE ECHO at or near the top of
29. mands The SET XLABEL command sets the automatic x axis text label Use the SET XLABELON command to toggle off on drawing the x axis text label Change the sizes of the text label with SET XLABELHEIGHT or SET SXLABELHEIGHT Change the font of the x axis text label with the SET XLABELFONT command and change the color of the x axis text label with the SET XLABELCOLOR command The SET YLABEL command sets the automatic y axis text label Use the SET YLABELON command to toggle off on drawing the y axis text label Change the sizes of the text label with SET YLABELHEIGHT or SET YLABELHEIGHT Change the font of the y axis text label with the SET YLABELFONT command and change the color of the y axis text label with the SET YLABELCOLOR command the y axis label 16 14 12 10 8 6 the x axis label Graph Legend Legends are boxes of descriptive text that describe certain details of the graph Typically they are used to label different point types different line types or colors contour elevations fit parameters and so on The LEGENDON characteristic is changed with the SET command and the current value is obtained with the GET command If LEGENDON 0 a legend entry is drawn into a legend frame box A legend entry consists of a short line segment with optional plotting symbol s and a text string The legend entry is drawn when the GRAPH command is entered The string portion of the legend entry is expected as the first parameter of the
30. nate interpolation method LINEAR simple linear interpolation LAGRANGE general Lagrange interpolation m FC Fritsch and Carlson method of monotone piecewise cubic interpolation If one s starting data is not monotonically increasing then one can use the SPLINTERP x y n function instead It accepts an arbitrary set of x and y values and a number of points to interpolate The output is a 2 column matrix the first column of which gives the interpolated points 1e x values and the second of which gives the interpolated values 1 e y values 2 D interpolation Beginning with a scattered set of 3 D data points in three vectors say x y and z you can interpolate a regular matrix using the GRID command The three vectors are assumed to represent scattered points where z i is the altitude corresponding to the coordinates x i y i The set of scattered data points is used to construct a Thiessen triangulation of the plane and a regular matrix m is interpolated For example the following script produces the pictures below ae Oy ink bes prose oe 2 10 10 10 10 100 10 100 500 GRID XYOUT X Y Z M XOUT YOUT SET PLOTSYMBOL 14 GRAPH X Y produce the graph on the left SET PLOTSYMBOL 0 DENSITY DITHER XOUT YOUT M produce the density plot 42 o P a a o N o a o D r a Integration Integration is the summing of areas and volumes under cu
31. nd two output vectors containing the bins and the counts are returned BIN x xbin xcount bin the values in x GRAPH HISTOGRAM xbin xcount There are many binning options among them various options for defining the bin boundaries the averages of the values in each bin can be returned values can be counted conditionally counts can be weighted lagrange binning Please refer to the BIN command for more information Interpolation There are many cases where one needs to interpolate data for instance estimating missing data values converting an irregular data sample to a monotonically increasing data sample 41 representing a set of data points as a smooth function Interpolation presumes the data can be represented as a smooth function and that this function passes through all of the data points Interpolation therefore consists of looking up the y values of this function for any x that is not represented in the original data This is normally done by means of the INTERP function which returns a data vector containing the interpolated values The INTERP function accepts three arguments vector a monotonically increasing set of x values y vector the values of y at each of the above x values interpolation points a set of x values at which to interpolate new y values The method of interpolation is normally interpolating splines but an optional fourth argument can be used to select an alter
32. nding of your data There are other cases where completely interactive control over the job are not desirable for instance favourite configurations or preferred defaults that you d like to load instantly routine or repetitive tasks that always involve the same set of steps production analysis in which established data analysis routines are used repeatedly on similar data sets long or intensive analysis jobs that don t need baby sitting In these cases you will use scripts to fully or partially automate the process Examples of Extrema scripting commands have been provided throughout this guide in fact every Extrema operation has both an interactive GUI method and a corresponding command driven method to accomplish it In addition there are special tools that are only available when in script mode such as branches loops subroutines If you have computer programming experience you ll recognize these as the essential elements of a programming language Using these elements you can make your Extrema scripts perform arbitrarily complicated tasks So long as the analysis procedure is quantifiable in some way Extrema can be configured to make all the necessary decisions and take the appropriate steps in the handling of your data Creating and editing scripts Extrema scripts ate simple text files with one command per line You can use any editor or word processor to cteate and or edit your script provided you save
33. ng and plotting the next slice as it went This would give you a simple animation as you moved up or down through the slices Graphing two 3 D functions on the same drawing Say we have two matrixes f and g which represent surface functions xy and g x y over the same X and Y ranges We would like to plot them together on the same drawing for easy comparison This is most effectively done by plotting f using some kind of density plot and then drawing g as a contour plot over top of it It may help to draw g ina different color to ensute it is clearly visible 17 eS 2e SE re PS K pe 2 be ae deel Gi g Pl ERE Multiple plots on the same drawing Researchers commonly need to combine graphs into the same drawing plot multiple data sets on the same graph draw different graphs with a common axis and so on There are many ways Extrema can be used to get these effects A few are mentioned here see also section 4 5 4 Tile numerous graphs on the same drawing Extrema divides the drawing area into windows which can be selected by their number to confine a graph to a particular section of the drawing Window number 0 is the default For example to tile four graphs on the same drawing simply select windows 5 6 7 and 8 in order and issue an appropriate GRAPH command for each WINDOW 5 GRAPH x yl WINDOW 6 GRAPH x y2 WINDOW 7 G W 18 a ao 5 25 20 z 15 oO ig o z 5 a 10 oO
34. ny expression involving a variable will return a similar variable each element of which has been modified by the expression the return value of the expression can be saved to another variable or operated on directly Examples y SIN Sie COs x 72 save expression results in variable y GRAPH x 3 x 2 6x4 2 graph expression directly Expressions are built up of constants variables operators and functions which can be combined in any algebraic syntax as in the examples above Operators In addition to the simple arithmetic operators exponentiation and grouping there are also special vector and matrix operators gt lt outer product lt gt inner product lt matrix transpose matrix reflect i vector union amp vector intersection and a set of Boolean operators that return true 1 or false 0 values Of exclusive or amp and not equal to not equal to gt greater than x less than gt greater than or equal to lt less than or equal to Functions Extrema has over 200 built in functions that can perform a wide range of other operations on your data Examples include conventional mathematical functions such as the trigonometric functions logarithms roots and exponentials and rounding functions advanced functions such as Bessel Clebsch Gordan etc calculus functions such as integral and derivative probability functi
35. ons programmers functions such as random number generation variable tests looping functions array and matrix functions such as where eigenvectors and eigenvalues etc string functions such as case date time etc In all cases these functions accept data of a certain type and return data of a certain type they may be freely used in any expression so long as the types they return make sense in the expression context For further information see Operators and Functions in the Extrema Command Reference Fitting Fitting data that is describing a set of data points as some sort of function is one of the most important forms of data analysis Extrema s data fitting capabilities are sophisticated and flexible complete details are provided in the Extrema Command Reference but some simple examples are given here 37 Smoothing Smoothing is a simple way of fitting a set of data points to a smooth curve There are several methods of calculating these smooth curves notably cubic splines under tension SMOOTH and SPLSMOOTH functions and Saviztky Golay filters SAVGOL function 250 200 150 100 50 Smoothing functions return a smoothed set of data that is they accept your data as input and output a new set of values that fall on a smooth curve of the appropriate type They can operate on any shape of data without any prior knowledge of the data s shape In some cases there is a requirement that
36. oop 52 Here is another way to use GOTO statements to continue executing a block of code until some condition is met begin loop to execute a block of code until a equals b IF a b THEN GOTO begin If you cannot get Extrema to make the key decisions on its own you could always prompt for a human decision on whether to repeat the loop begin loop to execute a block of code until user is happy INQUIRE Redo Y N answer IF EQS answer Y THEN GOTO begin You could use similar conditional branching statements to control the number of times a section of code gets executed nloop 0 ugly loop to execute a block of code 10 times do Loop nloop nlooptl IF nloop lt 10 THEN GOTO do loop DO loops The previous example can be written more concisely as DO amp 1 10 better loop to execute a block of code 10 times ENDDO The general form of the DO loop is DO scalar vector ENDDO The scalar is set to each element of the vector in turn with the loop code being executed once for each such setting Thus the following are all legal DO loops DO i f 10 10 21 loop from 10 to 10 in steps of 2 ENDDO DO i BAe loop over the squares of all values of x 53 ENDDO DO x 1 LEN al matrix c is the difference between matrixes a and b DO y 1 LEN a c x y afx y bfx y ENDDO ENDDO This last example is illustrative of the looping mechanism using nested lo
37. ops but otherwise it is a bit artificial This is how you might handle the operation of subtracting two matrices in a conventional programming language but loops of this nature are implicit in Extrema s variable handling and expression evaluation The same operation can be accomplished more efficiently with c a b implicit loop over every element 54
38. ow has a smaller plotting unit range than the full page Commensurateness is never lost in a sub window Pre defined windows Some of the initial pre defined windows in PORTRAIT orientation are displayed below WINDOW 3 WINDOW 5 WINDOW 7 WINDOW 1 WINDOW 2 pce WINDOW 4 WINDOW 6 WINDOW 8 WINDOW 10 WINDOW 9 WINDOW 11 One dimensional graphs One dimensional graphs are created from a single data vector If you do not provide an independent variable to graph against Extrema will use the vector index as the independent vatiable GRAPH lels0 1 1 0 0 5 0 0 0 5 1 0 0 5 10 15 20 Alternatively you can bin the values in the vector to turn it into data pairs i e bins and counts The resulting vectors can be plotted using any of the two dimensional graph types below Binning data is explained in Chapter 5 Data Analysis Examples 10 Two dimensional graphs Two dimensional graphs represent data pairs Typically you will have two vectors of the same size which should be plotted against each other in some way Line graphs Line graphs connect each subsequent x y data point with a line This presumes that the points are ordered so that they are connected in sequence GRAPH x y draw y x as a line graph A parametric line graph is also easy to make If our parametric independent variable is T then we generate X and Y vectors by passing T through app
39. ropriate parametric functions t O 2 p12 1 x t SIN t y C COs CI GRAPH x y 5 4 3 2 1 0 1 2 If your data is already in polar coordinates you can graph it directly using the POLAR option e g GRAPH POLAR radius_vector angle_vector Angles are presumed to be in degrees Note The plotting symbol characteristic PLOTSYMBOL must be set to 0 to get a line graph This is the default so no special action needs to be taken unless the plotting symbol has been otherwise set see Scatterplots Scatterplots Scatterplots take the corresponding elements of each vector and plot them as x y data points using whatever plot symbol has been selected There is no requirement that the vectors be ordered in any particular way To plot scattered points that are not joined by a line select a negative symbol type SET PLOTSYMBOL 1 GRAPH x y If we have errors in the data also stored in a matching vector then we can add that information to the plot by specifying the error vector s GRAPH x y yerr 12 If your data is ordered and you would like the data points to be joined with a line then simply use a positive plotting symbol number SET PLOTSYMBOL 1 GRAPH x y 13 For a detailed listing of plotting symbols see Chapter 4 Customizing Graph Presentation Histograms and bar charts Histograms bar charts with tails going to y 0 are drawn by using the HISTOGRAM qualifier with the GRAPH command x valu
40. rves and surfaces Extrema provides you with several tools to accomplish this The INTEGRAL function is the simplest method it accepts two vectors representing the x values monotonically increasing and_y values of the function to be integrated The return value is the integrated function 1 e the integral at each x value there is one additional value appended to the end of this output vector and that is the integral over the full range of x For example to find the area under cos x sin x for 0 lt x lt T pi ACOS 1 x O pi 1 yi INTEGRAL x COS x 3 SIN x 4 value yil 43 4 ficos sin x dx 1 5 1 0 oe 3 4 cos x sin x 0 0 0 5 u 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 Other functions Please refer to the DERIV function derivative of a function and the AREA function area within a polygon and the VOLUME function volume under a surface There are also numerous special integration functions such as elliptic integral Fresnel integral exponential integral sine integral S ININT and cosine integral COS INT Two dimensional integration is typically done using the VOLUME function which can operate on a variety of data types vectors containing scattered xy points vectors containing scattered polar coordinate points angle radius regular matrix Data Selection Filtering cutting and other forms of conditional data selection are a big part of many analysi
41. s tasks There are many ways this can be accomplished in Extrema 44 Many of these techniques involve selecting subsets of vectors matrices or tensors according to some arbitrary condition A trivial form of data selection simply consists of selecting the desired indexes for example good data m only the last column of the matrix is good If the good data is scattered throughout a vector say data and you have the indexes of the good values in another vector good then you can select the good data using the notation good data data good Determining which indexes are good and which are bad is the tricky part The WHERE function is invaluable for this It accepts a vector as input and returns the indexes where the input vector was not equal to zero The input vector is usually some kind of Boolean operation on the actual data vector such that a vector of true false 1 0 values is actually passed to the WHERE function The return vector of indexes is then used to select the values from the original data vectors The power of this function is best illustrated with a few simple examples Example 1 select the data points within 1 unit of the origin We have a scattered set of data points in the vectors x and y but we want only the ones that lie within the unit circle i e the points that satisfy SORT x 2 y 2 lt 1 i WHERE SQRT x 2 y 2 lt 1 select data in unit circle i is out list of selected indexes GRAP
42. the script which causes Extrema to echo each command as it is performed Use DISABLE ECHO to turn this behaviour off Interacting with the user Not all scripts are meant to run by themselves off in a dark room somewhere Some scripts will have a real person in front of them and interaction with this user may be necessary There are a few special Extrema commands for this purpose DISPLAY causes Extrema to display a message or other text string INQUIRE prints a message such as a question and waits for the user to provide an answer The answer is kept in a variable and so can be used in subsequent operations In the trivial case this could be a prompt to Hit return to continue and the answer is irrelevant In more sophisticated cases it could prompt for multiple variable names or values to operate on Initialization script You can have Extrema automatically run a script when it first starts This is convenient for setting up physical constants or your favourite graphing parameters and defaults To make an initialization script simply create a file named extrema init in the scripts subfolder of C extrema or wherever you installed Extrema Subroutines Scripts can be executed from within other scripts sometimes the top level script invoked by a person is referred to as the program while the remaining scripts invoked by the script are called the subroutines Other than that a subroutine is just like any oth
43. two sets of data on the same graph but at different Y scales 00 ee ceceecceceeeeeceeceeeees 20 CUSTOMIZING GRAPH PRESENTATION 222 COlOUS audit 2 24 Default Drawing Color 25 Plotting Symbols 25 Line type ccccceeeeeeees 27 Line width 27 fi ener een i eur 28 Axis Labels 00 29 Graph Legend 29 Graph Axes 31 SCALING rss a ER 31 THOS nr nimin nier 32 Logarithmic axes 33 Axis placement 34 DATA ANALYSIS EXAMPLES 220002222236 Operators 36 Functions 37 FAQs 37 Smoothing 38 Fitting to a function 38 Fitting different data segments to different functions 40 BINDIN Gerrans 41 Interpolation 41 2 D interpolation 42 Integration 43 Other functions 44 Data Selection 44 OUTPUT mnnnnnnnnnnnns 47 Printing graphs 47 Exporting graphs for inclusion in other documents 47 PostScript amp EPS 47 PNG nd eme ct 47 JPEG nn nn Rte At 48 Saving Data 48 SCRIPTING EEE Le Creating and editing scripts ET 50 Comments 50 Running scripts 50 Interacting with the user 50 Initialization script 51 Subroutines 51 Branching and looping 52
44. umeric variables can be one of the following types scalar a number vector a one dimensional array of numbers matrix a two dimensional array of numbers SPECIAL _ all indexes the last index tensor a three dimensional array of numbers z be implemented The contents of arrays are indexed sequentially with a starting index of one 1 Except for physical memory limitations there is no limit to the number of variables or to the length of strings or to the size of arrays Addressing parts of arrays To refer to an entire array simply use the variable s name To select an individual element from the array provide the index of the element in square brackets x 8 8th element of vector x y 2 6 value from 2nd row 6th column of matrix y In all of the above cases you are referring to a single value i e a scalar You can also specify a range of indices using the colon character 8220 8th through 20th elements of vector x y 1 10 1 first 10 rows from the first column of y It is also possible to replace any part of an index with a mathematical expression For example ele S202 8th through 20th elements of vector x yl lrpert 100 1 first 10 rows from the first column of y Variables can also be used in indices For example suppose you have a vector z which holds the values 1 2 10 The following are then valid sise Se lF 2 8th through 20th elements of vector x yiz 1
45. ve any convenient checkboxes however and will have to be set manually using a typed command Each drawing parameter has a name To get the value of a parameter use the function GET characteristic This returns a value that can be viewed interactively or stored in a variable To set the value of a parameter use SET characteristic value Specific commonly used examples follow A comprehensive list of parameters is given in the Extrema Command Reference Many drawing parameters refer to positions on the drawing which can be expressed in various units including percentages To interactively determine which position you would prefer simply move your mouse over the drawing and the positions will be displayed below in whatever units have been selected Colors Colors are numbered these numbers are indexes into a color map A color map can hold up to 256 colors Extrema pre loads a default color map that should be adequate for most cases but you can also load your own color map see the SET COLORMAP or the SET COLORMAPFILE command In the GUI select a color map color simply by clicking on the grid of colors that are presented In the command language you will need to know the color number The default color map is stored in the file DefaultColorMap dat and the color numbers can be looked up there In addition to the color map Extrema also predefines a set of colors that are always available no matter what color map is currently loade
46. xtrema Extrema is distributed as a self extracting compressed file You simply need to execute the extremainstall program to begin the installation process After agreeing to the licensing agreement you then select an installation directory the default is C Ext rema and everything else is automatic The installation program places the Extrema icon on the desktop Extrema does not modify the registry so to uninstall Extrema simply delete the files Running Extrema Double click on the Extrema icon to launch the program By default the program raises the visualization graphics window and the analysis command input window Commands may be typed directly into the analysis window if you are familiar with the Extrema command language If not you will probably be more comfortable selecting your actions from the menus and toolbars A typical Extrema session involves the following steps load or generate data to work on graph the data or analyze the data repeat previous two steps until the results are satisfactory customize the presentation of the graph s output the graph s save data for archival purposes or for further analysis Ao eh Once you become familiar with Extrema or with your particular data analysis and presentation requirements many of the above steps can be automated In that case you can build scripts to automatically perform the routine steps in the above sequence and possibly the entire sequence itself
47. y axis to the centre of the number labelling that tic mark controls the distance measured from the base of each large tic mark on the y axis to the centre of the number labelling that tic mark Logarithmic axes To get logarithmic scaling on the x axis use SET XLOGBASE n where n gt 1 0 the x axis will have a logarithmic scale The base will be the integer part of XLOGBASE except for the special case 1 05 e gt XLOGBASE gt 0 95 e where e is the base of the natural logarithms e 2 718281828 in which case the base will be e n lt 1 0 the x axis will have a linear scale 33 If XLOGSTYLE 0 and XLOGBASE gt 1 0 then the numbers labelling the large tic marks on the x axis are displayed in decimal format If XLOGSTYLE 0 and XLOGBASE gt 1 0 then the numbers labelling the large tic marks on the x axis are displayed in exponential format To get logarithmic scaling on the y axis use SET YLOGBASE n where n gt 1 0 the y axis will have a logarithmic scale The base will be the integer part of YLOGBASE except for the special case 1 05 e gt YLOGBASE gt 0 95 where e is the base of the natural logarithms e 2 718281828 in which case the base will be e n lt 1 0 the y axis will have a linear scale If YLOGSTYLE 0 and YLOGBASE gt 1 0 then the numbers labelling the large tic marks on the y axis are displayed in decimal format If YLOGSTYLE 0 and YLOGBASE gt
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