SLAW, a package for Scaling LAWs from statistical data Version
Contents
1. Save this experiment Screen Capture Cancel Input any error level here and press Enter to show rounded Model E 1 Process User Input _ Screen Capture Cancel Figure 5 Summary plot window 13 Message The following files have been saved to ExampleCase vn Please move this folder to another location HHHH Lox Figure 6 Saving confirmation window A Appendix Below is the list of the programs that form the suite and the brief description of each provided in the file 1 readdata m Function readdata Purpose Loads files containing 1 the statistical data of the process 2 the description of the relation of variables to the units and sets up the appropriate data structures Input names of files containing problem archdats file that contains the actual sample data it has p rows one for each data measurement n cols the dependent variable and then n 1 independent vars archR matrix of units n cols units for the dependent var and then n 1 independent vars 14 k rows one for each unit in the problem m kg sec Data structures to perform the desired constrained linear regression log dependent variable p times 1 vector 1 log indep var p times n matrix In the first column is the data associated to the intercept matrix with description of units for 1 constant coeff and n 1 indep vars k times n matrix rhs vector of units describing the units f2. vnorm L2 norm of vdiff rss values of residual sum of squares for each regression function call vout vdiff vnorm rss dr4 y X R b cte Function automod m Purpose To obtain a concise model that captures the significant portion of the dimensional analysis linear regression We want to automaticaly detect what model will the algorithm output We currently sequentially remove variables from the model until we can remove no more due to the units constraint We want to stop removing variables when removing an additional variable greatly reduces the quality of the model We have 3 criteria 1 Let RSS_k be the residual sum of squares of a model with k variables We will stop when removing an additional variable causes RSS_ k 1 gt RSS_k AMOUNT 16 when the RSS increases by more than AMOUNT Typically we would set AMOUNT to 5 or 10 meaning that if the RSS increases by more than 500 or 1000 removing this extra variable is not worth it When removing an extra variable causes an absolute change in the RSS that exceed a threshold amount Here we would need some estimation of the values of the RSS for each particular problem Values such as AMOUNT 0 00001 would keep the whole model AMOUNT 1000 would keep only the model required to satisfy the units constraint Recommend of AMOUNT 1 When removing an extra variable causes an absolute change in RSS p 1 2 This quantity can be interpreted as an aver
3. be the length of the pendulum the mass the density of the medium etc Note that we are not interested in variables used to describe the evolution in a specific realization of the process such as position and time We further assume that the characteristic value is given by the following power law of the problem parameters Y a Jj x gt 1 j 1 SLAW identifies this power law by separating it into the most represen tative scaling law and m dimensionless groups ranked by their significance to the characteristic value X5 I where I x 2 1 i 1 j 1 Y a s J where a is a numerical constant The model postulated also considers the uncertainties which arise from working with experimental data and consider ing only n independent variables and disregarding the possibly tiny effect of other variables By taking the logarithm of Equation 1 and considering the existing uncertainties in the model we arrive at the following multivariate linear regression model logY bo X Blog X e 3 j l where is an error term and the coefficients are given by m bo loga bj E Xaj i 0 Experimental data is used to estimate the coefficients in this multivari ate linear regression model by solving different linearly constrained linear regressions i e minimizing the squared sums of errors subject to linear con straints For example the standard linear regression suggested by Equation 3 produces a model tha
4. data can be summarized generalized extrapolated and simplified with great accuracy and physical meaning One advantage of this approach is that it enables engineers to simplify a prob lem by eliminating physical parameters which are sometimes very difficult to measure or obtain Other statistical simplifications such as the principal directions of the matrix of correlation reduce the mathematical complexity of the problem but still require the use of all parameters regardless of their importance For example the thermal expansion of a solid might be very difficult to determine yet not relevant in a problem Strictly mathematical approaches might still require this parameter while the experimental data could indicate that thermal expansion is not relevant and thus SLAW would generate scaling laws in which that parameter does not appear This user s manual first describes the model assumptions the inputs and what are the outputs of SLAW We then describe the operation of SLAW either through the Matlab files or through the Java interface In the appendix we present a list of all program files of the SLAW algorithm with their input and output 2 Program Objectives The objective of SLAW is to derive the characteristic value Y X of some process that only depends on n problem parameters X X For example in the problem of determining the period of a pendulum the characteristic quantity would be the period while problem parameters can
5. vA 1 if the linearly const regression found the solution function call h coeff stat lclregr y X M g Function roundmod m Purpose To give an engineering meaning to a consise model by producing rounded coefficients such that still satisfy the unit constraints Input The linear regression model obtained by the method and 18 the data that originates the model model coefficients of the model to be rounded vA y log dependent variable p times 1 vector X 1 log indep var p times n matrix In the first column vA is the data associated to the intercept R matrix with description of units for 1 constant coeff and n 1 indep vars k times n matrix vA b rhs vector of units describing the units for the dependent variable k times 1 vector hh Output rmod a rounded version of model only with exponents ending in 00 25 1 3 5 2 3 75 function call rmod roundmod model y X R b roundvct m Returns a rounded vector each entry rounded to the closest of sOy 25 173 25 273 2755 Input a vector v Output a rounded vector r of the same dimension Function call h r roundvct v 19
6. 087277713 112028962 031239833 112028962 06241881 031239833 oo ooo Mo Roo MoM eRe Moo Roo Ro Mo Ro Moo Roo Mo Ro Roo MoM eRe Moo Moo Ro Mo Mo Mono Mono momo Momo mome diameter 0399034 0399034 0399034 0399034 0345948 0345948 0345948 0345948 0345948 0345948 0345948 0184658 0184658 0184658 0184658 0184658 0184658 0184658 0184658 0184658 0184658 0184658 01397 01397 01397 01397 01397 01397 01397 0085344 0085344 0085344 0085344 0085344 0085344 0085344 0050038 0050038 0050038 0050038 0184658 0184658 0184658 01397 01397 01397 PRPRPRB RRR RRP RR BRP BPR BRR BRP RRP BRR PP RPP BEEP fluid density kg m3 m s2 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 195056201 1 195056201 998 9 81 998 998 998 998 998 81 81 81 81 81 O owog OCOOWDDHDDDDDHDDHDDDODHDDHDDODDDDDDDODODDHDDDDDDHDDOOO gravity 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 8
7. 1 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 e Units Data A second text file must indicate the information that is contained in Table 1 This text file contains a matrix with one row for each unit used and one column for each parameter where the entry in row 2 and column j contains the exponent of unit i for the units of variable j For the pendulum example we use the following file period m 1 theta diamt density gravity s kg m rad m kg m3 m s2 0 ie 1 0 1 3 1 A m o 1 0 0 o 1 0 kg 1 0 0 0 0 0 2 s 3 3 Outputs The principal outputs of SLAW are the rounded coefficients of the scaling law selected in the text file rmod txt and a ranked list of the dimensionless groups that capture the residuals of the model in rmod txt in the text file eout txt For the pendulum example these files turn out to be e rmod txt 1 87228 0 00000 0 50000 0 00000 0 00000 0 00000 0 50000 e eout txt 0 00096 0 17759 0 19462 0 12507 0 00000 0 00005 0 02083 0 02122 0 01616 0 00000 0 00235 0 00577 0 00000 0 00000 0 00000 0 02308 0 02501 0 02155 0 00000 0 00000 0 00449 0 05672 0 06367 0 04847 0 00000 0 02099 0 02083 0 02122 0 01616 0 00000 0 00220 0 00000 0 00000 0 00000 0 00000 Each column in eout txt contains the coefficients of a different dimen sionless group where the first column on the left is the dimensionless group which least affects the residual error when
8. 828 558 434 446 528 605 924 667 671 053 044 059 668 678 4265 4335 044 049 066 4795 818 868 884 918 944 896 NNNNNNONNNOCCCONNNOCORKRKBRBRODOORBRBBRHBBREPEBRERBRROORKRHBRENNAD m kg SOC OOOOC OC OOOO OOO OOOO OOOO OOOO OC OOOO OC OOOO OOOO OCC Oo 295 295 295 7295 1475 1475 1475 1475 1475 1475 1475 05 05 05 05 05 05 05 05 05 05 05 02 02 02 02 02 02 02 005 005 005 005 005 005 005 001 001 001 001 05 05 05 02 02 02 PRPPRPRRORRROODOOCORHBRRODOO OCOD OOOO COCO OOO COCO OC OC OORKE KH DAAAAAD 265 265 435 435 84 84 192 192 287 287 287 816 816 816 816 354 354 354 354 158 158 085 502 502 502 502 202 ead wit 025 025 025 106 106 042 042 037 037 037 052 theta rad SOCOCCORD COCO DOO COCO OOOO OOOO COO OCC OOO OOOO OOO OCC OC OC COO 085953704 171423406 245519496 466402423 129040859 255037 516337133 278987417 362964317 189342043 651811167 067300169 132792341 262203488 491673135 154134896 298697423 554015696 655149327 334982296 603897219 307567108 109126499 411364964 716285835 785398163 494826586 244978663 244978663 105943307 210493613 40300633 253837789 253837789 785398163 785398163 104726344 208127938 568847858 785398163
9. SLAW a package for Scaling LAWs from statistical data Version 0 93 User s Guide Fernando Ord ez Timothy Li and Patricio Mendez Abstract Program SLAW derives representative scaling laws of a process from sensitivity analysis experimental data The approach integrates dimensional analysis with a multivariate linear regression backward elimination procedure In addition to the scaling laws the program provides a set of dimensionless groups ranked by relevance The algorithm is coded as a group of Matlab m files This user s manual describes the different programs in the SLAW suite and its Java interface discussing its usage through an illustrative example 1 Introduction The objective of SLAW is to derive representative scaling laws of a process from sensitivity analysis data from experiments of the process The ap proach integrates dimensional analysis with a multivariate linear regression backward elimination procedure In addition to the scaling laws the pro gram provides a set of dimensionless groups ranked by relevance With the Industrial and Systems Engineering University of Southern California GER 247 Los Angeles CA 90089 0193 USA email fordon usc edu Industrial and Systems Engineering University of Southern California Los Angeles CA 90089 0193 USA email timothy usc edu Exponent 21 Strathmore Road Natick MA 01760 USA email pmendez exponent com scaling laws obtained the whole set of
10. age residual error That is an average over all observations 1 p of the residuals difference of data and model prediction A cutoff of amount means that removing a variable from the model causes the increase in this average residual error to be larger th an amount Input All output from the dimregress function vout matrix with column vector of iterates of the method vdiff differences of vout vnorm L2 norm of vdiff rss values of residual sum of swuares for each regression criteria 1 check a change in rss to stop 2 check an absolute increase in rss to stop 3 check an absolute increase in rss n 1 2 to stop amount the or tolerance to check depending if criteria is 1 2 or 3 p number of observations in the data Output model a column vector of coefficients for each parameter variable in the final model function call 17 vA J model automod vout vdiff vnorm rss criteria amount p Iclregr m Purpose Obtain the coefficients that minimize y X beta _2 such that M beta g vA This program solves the problem min_v y X v t yX v VA M v g Input y X data for linear regression M g data for additional linear constraints that new regression coefficients must satisfy output VA coeff vector of differences from beta0 the solution to this vA problen stat O if the linearly const regression does not have a solution
11. ar In Linux systems the command prompt gt java jar SlawEdit jar starts the program To finish the installation the correct Matlab path must be provided for SLAW JI The Matlab path can be keyed in or browsed from SLAW Process gt Setup Matlab Path menu see Figure 2 This updates the file matlabPath slw and must be done prior to executing SLAW Alter natively the path can be provided by modifying this text file by hand This process completes the installation To verify that you are running properly please execute one of the examples in the examples directory 4 2 2 Running SLAW Java Interface Through the use of an installed Java 2 Runtime Environment execute pro gram SlawEdit jar that starts the SLAW Java Interface This opens the main SLAW program window Figure 2 which contains three menu options 9 E SLAW Java Interface 0 93 a a File Help Process Data Setup Matlab Path Figure 2 Initial SLAW JI window e File Allows to create and manipulate data files and also exit SLAW JI e SLAW Process Enables the execution of SLAW and analysis of its results e Help Provides contact information and credits SLAW Process Clicking the option SLAW Process leads to two alternatives e Setup Matlab Path which enables the user to key in or browse for the path to the executable Matlab program This modifies the text file matlabPath slw which is used to run Matlab e Process Data whic
12. h enables the user to key in or browse for the input data files This alternative first asks for the location of the experiment data file after confirming the file given it asks for the location of the units data file Once both files have been loaded it automatically opens Matlab and runs SLAW Processing SLAW Data After imputing the experimental and units data file SLAW JI opens up a plot 10 i ESI amp amp Equation Error Graph Jeg Input any error level here and press Enter to show rounded Model Figure 3 Average residual errors of possible models that displays the errors of the different possible models to select Figure 3 The program then waits for user input to identify the model that represents the scaling law for the data inputed On this window the user can 1 key in an acceptable error level and press the process user input button this selects the simplest model the one involving fewest variables that has an error smaller than the one entered 2 click on the error column of a model this selects the corresponding model 3 press the screen capture button to generate a copy of this picture 4 cancel and return to the main SLAW window If either option 1 or 2 are selected SLAW is executed to round the model and output rmod txt and eout txt and the model is written in an interactive window Figure 4 This window allows the user to 1 select the model and generate a
13. model in sorted adimensional groups eout ediff enorm ess dr5 y X rmod X R zeros length b 1 1 Output fid fopen eout txt w for i 1 length eout 1 formatted fprintf fid 10 5f eout i formatted fprintf fid n end fclose fid 4 2 Using the Java interface The main difference between using the Matlab code directly and the SLAW Java Interface is that SLAW JI executes the algorithm in an interactive fash ion It allows the user to look at a plot of the increasing estimation errors for further simplified models and allows the user to select a model which achieves an acceptable balance between simplicity and accuracy The inter face then is able to display the rounded model and a plot that illustrates the quality of the fit of the simple rounded model 4 2 1 Installing SLAW Java Interface The tar ball SLAW javadistrib tar gz downloaded from the SLAW web page 2 contains e SlawEdit jar The java executable program e matlabPath slw An utility text file which contains the path executable command that must be executed from this directory to run Matlab e examples Directory containing the examples in 1 e matlabFiles Directory containing the SLAW Matlab m files e sourceFiles Directory containing the source files for the java inter face To install the software simply unzip the tar ball in the desired direc tory SLAW JI is started by executing SlawEdit j
14. or the dependent variable k times 1 vector This data will be used to solve linear regressions to determine the model y beta_O beta_1x_1 beta_ n 1 x_ n 1 where in addition we require the coefficients to satisfy R beta b and successive elimitations of variables function call y X R b readdata archdats archR Function dimregress version 5 Purpose To find the decomposition in terms of adimensional groups of the linear regression of an independent variable y on n independent variables Includes a flag to determine if we allow the constant coefficient to be eliminated or not Program uses subroutine lclregr to compute the coefficients of a linearly constrained linear regression Input y data structures describing the problem log dependent variable p times 1 vector 15 X 1 log indep var p times n matrix In the first column is the data associated to the intercept R matrix with description of units for 1 constant coeff and n 1 indep vars k times n matrix b rhs vector of units describing the units for the dependent variable k times 1 vector cte 0 1 flag to determine if we can eliminate the contant term default is 0 to allow contant to be eliminated or we cannot eliminate the constant term cte 1 Output vout matrix with column vector of iterates of the method first column contains the original regression vdiff differences of vout
15. plot illustrating the fit of the rounded model to observed data in an interactive window 2 press the screen capture button to generate a copy of this picture 3 cancel and return to the error plot window 11 Rounded Model Selection The Rounded Model You Have Selected is 5 e 1 87228 x 00 x 0 5 input any error level here and press Enter to show rounded Model Process User Input Figure 4 Model selected window The interactive plot window Figure 5 allows the user to 1 save the model that generates this plot All relevant output files are placed in the folder ExampleCase and the following confirmation window appears Figure 6 In order to preserve the answer saved these files should be stored in a different location or under a different name before saving a different model otherwise the files will be overwritten 2 press the screen capture button to generate a copy of this picture 3 cancel and return to the rounded model interactive window References 1 P F Mendez and F Ordonez Scaling laws from statistical data and dimensional analysis Working Paper 2003 04 Industrial and Systems Engineering USC 2003 2 P F Mendez and F Ord ez SLAW a package for Scaling LAWs from statistical data World Wide Web http illposed usc edu pat SLAW 2004 12 2 The Rounded Model You Have Selected is 5 e 1 87228 4 00 x05 observed 1 15 predicted
16. removed Thus remov ing the dimensionless groups toward the right most affect the residual error In addition by running the Java interface of SLAW you can generate plots of the relative significance of different models that can be selected as well as plots of how significant the model selected is with respect to the observed data This will be outlined in more detail in a later section 4 Running SLAW 0 93 Program SLAW has been coded through Matlab m files and can be executed in two different modes The first is to call the program files through Matlab the second is a java interface which runs appropriate Matlab code We have tested SLAW 0 93 both in Microsoft Windows and in Linux platforms The program requires that Matlab is properly installed and executable in your computer The Java interface has been executed on Java 2 Runtime Environment Standard Edition Version 1 4 2 4 1 Using Matlab The SLAW package is based on the Matlab m files automod m dr5 m lclregr m readdata m roundmod m and roundvct m which are pre sented individually in the Appendix 4 1 1 Installing SLAW for use through Matlab To install SLAW simply put the program files in a directory that is accessible form the Matlab executable path This can be achieved with the MATLAB path command Assuming that the files are placed in the directory SLAWDIR under the home directory the command would be for Linux gt path SLAWDIR 4 1 2 Running SLAW
17. s the mass the initial angle and the density of the surrounding fluid In Table 1 we list a few parameters that could be assumed to influence the period of a pendulum and their units Table 1 Parameters to explain the period of a pendulum symbol units description Y IT S period dependent variable Xilm kg mass of the bob X 1 m length of the pendulum X31 0 rad initial angle X4 d m characteristic dimension of the bob Xs p kg m density of fluid surrounding the bob Xe g m s acceleration of gravity 3 2 Inputs The inputs to SLAW are two text files containing the experimental data and the units information data respectively e Experimental data A text file with the experiment data one experi ment per row with the value of the dependent variable Y in the first column and the n independent variables in the next n columns This file should be in a format that is readable by the Matlab function load Below we present the file used for the pendulum example Note that the experiment are conducted in order to show the effect on the period of varying one parameter at a time SLAW requires that the experimental data explores how the dependent variable is affected by each parameter through such a sensitivity analysis We refer to such data as sensitivity analysis data period 248 26 328 337 844 84 912 891 093 084 z421 821 836 84 866 197 198 234 255 823
18. t gives estimates of the dependent variable that do not have the same units as Y We explicitly impose a linear units constraint which constrains our search of coefficients only to those that give models with the correct units Additional linear constraints are also used to identify the dimensionless groups in order of relevance to the characteristic value We refer the reader to 1 for a detailed description of the model and the procedure used to identify the scaling law and dimensionless groups 3 Inputs and Outputs Before describing the input data for SLAW and the output obtained from it we discuss in a bit more detail the problem of determining the period of a pendulum This is a well known physics example that will be used to illustrate the use of the program 3 1 Period of a Pendulum For a pendulum such as the one described in Figure 1 when the only relevant force acting on the pendulum is the force of gravity and for small oscillations Figure 1 Representation of a simple pendulum and its elements it is well known that the period of a pendulum is given by the formula T ml g 4 where T is the period the dependent variable Y in this problem l is the length of the string and g is the acceleration of gravity If Expression 4 were unknown and should be discovered from experimental data it is reasonable to assume that additional independent parameters X might be considered to describe the period of a pendulum such a
19. through Matlab To usage of SLAW is then conducted through calling the individual files In the distribution we include a file SLAWrun m which exemplifies this usage and which we list below This file contains a sample run of SLAW It assumes that the six Matlab m files which code SLAW Cautomod m dr5 m lclregr m readdata m roundmod m roundvct m are on the current directory or accessible through path Substitute the real path and file names of the data file and units file for datafile txt and unitsfile txt below Loading statistical data from files Dful txt and Rful txt Ly X R b readdata datafile txt unitfile txt Generating sequences of models vout vdiff vnorm rss dr5 y X R b O Automatically selecting model this in addition uses criteria 1 check a change in rss to stop 2 check an absolute increase in rss to stop 3 check an absolute increase in rss n 1 2 to stop amount the or tolerance to check for each criteria 1 2 or 3 model automod vout vdiff vnorm rss 3 0 5 length y hh Determining the closest rounded version of model We sequentially fix the coordinates that are closest to being rounded rmod roundmod model y X R b Output of rounded model fid fopen rmod txt w formatted fprintf fid 10 5f rmod fclose fid Check how we can sort the deviation of the rounded
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