NISP Toolbox Manual - Forge
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1. a 3 n ot o a Figure 6 4 Monte Carlo Sampling Histogram and exact distribution functions for the first variable The following script performs the same comparison for the second variable scf histplot 50 sampling 2 xtitle Empirical histogram of X2 x linspace 2 3 1000 y ones 1000 1 plot x y r The previous script produces the figure 6 5 6 2 2 A Monte Carlo design with 2 variables In this section we create a Monte Carlo design with 2 variables We are going to use the exponential distribution function which is not defined in Scilab The following exppdf function computes the probability distribution function of the exponential distribution function 39 Empirical histogram of X2 Figure 6 5 Monte Carlo Sampling Histogram and exact distribution functions for the second variable function p exppdf x lambda p lambda exp lambda x endfunction The following script creates a Monte Carlo sampling where the first variable is Normal and the second variable is Exponential Then we compare the empirical histogram and the exact distribution function We use the dno2. Figure 6 9 Latin Hypercube Sampling Normal random variable We can use the mean and variance on each random variable and check that the expected result is computed We insist on the fact that the mean and variance functions are not provided by the NISP library these are pre defined functions which are available in the Scilab library That means that any Scilab function can be now used to process the data generated by the toolbox for ivar 1 2 m mean sampling ivar mprintf Variable 4d LMeany 04f n ivar m v variance sampling ivar mprintf Variable 4d Variance Af n ivar v end The previous script produces the following output Variable 1 Mean 1 000000 Variable 1 Variance 0 249925 Variable 2 Mean 2 500000 Variable 2 Variance 0 083417 Our numerical simulation is now finished but we must destroy the objects so that the memory managed by the toolbox is deleted 40 Variable 2 Uniforme 2 0 3 0 Figure 6 10 Latin Hypercube Sampling Uniform random variable the LHS sampling is not a Monte Carlo sampling randvar_destroy vul randvar_destroy vu2 setrandvar_destroy srv 6 2 4 Other types of DOEs The following Scilab session allows to generate a Monte Carlo sampling with two unifor
3. 0 6 4 0 5 1 0 0 8 0 6 0 4 0 2 0 0 0 2 0 4 0 6 0 8 1 0 Figure 6 13 Latin Hypercube Sampling Second uniform variable in 1 1 Petras Sampling VITI TTT UE VES TTT 0 7 7 IN E do U o 9 pe 2047 mes o o 9 e o o e 8 O O soso sa 2 5 gt ist ic D g 2 8 8 2 4 2934 AG did id 4122424 L 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 Variable 1 Uniforme 0 0 1 0 Figure 6 14 Petras sampling Two uniform variables in the interval 1 1 43 Ouadrature Sampling 1 0 TT 3 T T T T T T UL ERT FT Fi ss o N N o gt o o ee ese o o e e o o e ee 0 9 ho 0 o o o o ee ho e 6 6 6 6 6 6 LU ee PRES ee dal es e o e ee Fe e o k Ta Se ea e et boe o o o e o o ooo Q S 064 g po o o L e oe Seiad e tas te Me e teh cide een Teng M pee o o eee 047 2 B ooo o e o o e ee S gt 0 30 0 0 e o o eo WR EE Wa 0 oo 027 boss o o o o o ee ke s 0 o o o o e ee 0 1 o e e e 6 6 6 6 6 6 e 6 eee
4. 1 exp 2u 0 5 2 It is possible to invert these formulas in the situation where the given parameters are the expected value and the variance of the Log Normal random variable We can invert completely the previous equations and get u In E X In 1 Ka 5 3 ees Go l 1 yn 5 4 In particular the expected value y of with the Normal random variable satisfies the equation p In E X o 5 5 5 1 3 Uniform random number generation In this section we present the generation of uniform random numbers The goal of this section is to warn users about a current limitation of the library Indeed the random number generator is based on the compiler so that its guality cannot be guaranted The Uniforme law is associated with the parameters a b R with a lt b It produces real values uniform in the interval a b To compute the uniform random number X in the interval a b a uniform random number in the interval 0 1 is generated and then scaled with X a b a X 5 6 Let us now analyse how the uniform random number X 0 1 is computed The uniform random generator is based on the C function rand which returns an integer n in the interval 24 0 RAND_MAX The value of the RAN D M AX variable is defined in the file stdlib h and is compiler dependent For example with the Visual Studio C 2008 compiler the value is RAND_MAX 2 1 32767 5 7 A uniform valu
5. end mymean mean values mysigma st_deviation values myvariance variance values mprintf Meanvisu u fu expected 0 f n mymean mu mprintf Standard deviationyisuy u 4fu expected f n mysigma sigma mprintf Variance isy 1 4fu expected 0 4f n myvariance sigma 2 randvar_destroy rv histplot 50 values xtitle Histogram of X X P x The previous script produces the following output Mean is 0 988194 expected 1 000000 Standard deviation is 0 505186 expected 0 500000 Variance is 0 255213 expected 0 250000 5 3 2 Variable transformations In this section we present the transformation of uniform random variables into other types of variables The transformations which are available in the randvar class are presented in figure 5 5 We begin the analysis by a presentation of the theory required to perform transformations Then we present some of the many the transformations which are provided by the library We now present some additionnal details for the function randvar_getvalue rv rv2 value2 This method allows to transform a random variable sample from one law to another The statement value randvar_getvalue rv rv2 value2 28 Empirical normal probability distribution function P x Figure 5 4 The histogram of a Normal random variable with 1000 samples Source Target Source Target Normale LogNormale Norm
6. CI Usecase 1 Usecase 2 Randvar 1 Randvar 2 Setrandvar 1 Setrandvar Lhs design Setrandvar Monte Carlo samplin Figure 1 4 Demonstrations provided with the NISP toolbox Chapter 2 Theory In this chapter we extremely briefly present the theory which is used in the library 2 1 Sensitivity analysis Consider the model Y f X 2 1 where X Dx C R is the input and Y Dy CR is the output of the model The mapping f is presented in figure 2 1 Dy Dy Figure 2 1 Global analysis The assume that the input X is a random variable so that the output variable Y is also a random variable We are interested in measuring the sensitivity of the output depending on the uncertainty of the input More precisely we are interested in knowing e the input variables X which generate the most variability in the output Y e the input variables X which are not significant e a sub space of the input variables where the variability is maximum e if input variables interacts Consider the mapping presented in figure 2 1 The f mapping transforms the domain Dy into the domain Dy If f is sufficiently smooth small perturbations of X generate small perturbations of Y The local sensitivity analysis focuses on the behaviour of the mapping in the neighbourhood of on particular point X Ox toward a particular point Y Dy The global sensisitivity analysis models the propagation of u
7. 2 o 0 0 eee ee ee OS RETTE da Soe BS Crr Ae Bal Apample SESSION i saca eS DAE A SOE Re ds bow Variable o er ys soros sos Oe Oi O dop dE do dE ei 6 The setrandvar class o Lis aie Lei YI YH EU YSW B O U TOCE HE YU GYS HEO Wy 0 2 Examples 22 6b sua RO E RR Bw DG AEDES RD 6 2 1 A Monte Carlo design with 2 variables H Co Co 6 10 11 11 12 12 14 15 15 16 17 22 23 23 23 24 24 25 25 25 27 28 28 6 2 2 A Monte Carlo design with 2 variables Dar r 6 Soe SR oS EDS ce ED a Da I CI 624 Other types of DOES bedded eee dee w D t de a 7 The polychaos class Tl TO e e a eee AW eee e Re ONS SE HN SEE EUR OB HD SEE Ta RAN Aia FR EDD RD ed de eed ND DER 721 Product of two random variables 2 2 24 ili 72 2 Thelehigam test case o coc ocaeca RR ACH mu ACH RAE AR R8 8 Thanks Bibliography Chapter 1 Introduction 1 1 The OPUS project The goal of this toolbox is to provide a tool to manage uncertainties in simulated models This toolbox is based on the NISP library where NISP stands for Non Intrusive Spectral Projection This work has been realized in the context of the OPUS project http opus project fr Open Source Platform for Uncertainty treatments in Simulation funded by ANR the french Agence Nationale pour la Recherche http www agence nationale recherche fr The toolbox is released under the Lesser General Public Licence LGPL as all componen
8. 93 Fonction Ishigami Indice de sensibilit CCI totale CU premier ordre Figure 7 4 Ishigami function Sensitivity indices 54 Chapter 8 Thanks Many thanks to Allan Cornet who helped us many times in the creation of this toolbox 99 Bibliography 1 T Homma and A Saltelli Importance measures in global sensitivity analysis of non linear models Reliability Engineering and System Safety 52 1 17 1996 2 Julien Jacques Contributions l analyse de sensibilit et l7analyse discriminante g n ral is e 2005 3 Didier Pelat Bases et m thodes pour le traitement des donn es Bruits et Signaux Master M2 Recherche Astronomie astrophysique 2006 4 I M Sobol Sensitivity estimates for nonlinear mathematical models Mathematical Modelling and Computational Experiments 1 407 414 2003 96
9. ans 2 5030984 The prevous script also produces the figures 6 2 and 6 3 We may now want to add the exact distribution to these histograms and compare The Normal distribution function is not provided by Scilab but is provided by the Stixbox module Indeed the dnorm function of the Stixbox module computes the Normal probability distribution function 33 Empirical histogram of X1 3 a o o 3 a Figure 6 2 Monte Carlo Sampling Normal random variable Empirical histogram of X2 0 0 2 0 24 22 23 24 2 5 2 6 2 7 2 8 2 9 3 0 Figure 6 3 Monte Carlo Sampling Uniform random variable 34 In order to install this module we can run the atomsInstall function as in the following script atomsInstall stixbox The following script compares the empirical and theoretical distributions scf histplot 50 sampling 1 xtitle Empirical histogramuofuX1 x linspace 15 15 1000 y dnorm x 1 3 plot x y r legend Empirical Exact The previous script produces the figure 6 4 E Empirical Exact nd FIN 0 05 4
10. 1000 experiments The function nisp_initseed is used to set the value of the seed to zero so that the re sults can be reproduced The setrandvar_new function is used to create a new set of ran dom variables Then we create two new random variables with the randvar_new function These two variables are added to the set with the setrandvar_addrandvar function The setrandvar_buildsample allows to build the design of experiments which can be retrieved as matrix with the setrandvar_getsample function The sampling matrix has np rows and 2 columns one for each input variable nisp_initseed 0 rvui randvar_new Normale 1 3 rvu2 randvar_new Uniforme 2 3 srvu setrandvar_new setrandvar_addrandvar srvu rvul setrandvar_addrandvar srvu rvu2 77 np 5000 setrandvar_buildsample srvu MonteCarlo np sampling setrandvar_getsample srvu Check sampling of random variable 1 mean sampling 1 Expectation 1 Check sampling of random variable 2 mean sampling 2 Expectation 2 5 scf histplot 50 sampling 1 xtitle Empirical histogram jof X1 scf histplot 50 sampling 2 xtitle Empirical histogram of X2 Clean up setrandvar_destroy srvu randvar_destroy rvul randvar_destroy rvu2 The previous script produces the following output gt mean sampling 1 Expectation 1 ans 1 0064346 gt mean sampling 2 Expectation 2 5
11. 3 1 Introduction There are two possible ways of installing the NISP toolbox in Scilab e use the ATOMS system and get a binary version of the toolbox e build the toolbox from the sources The next two sections present these two ways of using the toolbox Before getting into the installation process let us present some details of the the internal components of the toolbox The following list is an overview of the content of the directories tbanisp demos demonstration scripts tbanisp doc the documentation tbanisp doc usermanual the TEXsources of this manual tbznisp etc startup and shutdow scripts for the toolbox tbrnisp help inline help pages tbanisp macros Scilab macros files sci tbznisp sci_gateway the sources of the gateway tbrnisp src the sources of the NISP library tbinisp tests tests tbanisp tests nonreg_tests tests after some bug has been identified tbanisp tests unit_tests unit tests The current version is based on the NISP Library v2 1 15 3 2 Installing the toolbox from ATOMS The ATOMS component is the Scilab tool which allows to search download install and load toolboxes ATOMS comes with Scilab v5 2 The Scilab NISP toolbox has been packaged and is provided mainly by the ATOMS component The toolbox is provided in binary form depending on the user s operating system The Scilab NISP toolbox is available for the following platforms e Windows 32 bits e Linux 32 bits 64 bits e
12. A LHS design In this section we present the creation of a Latin Hypercube Sampling In our example the DOE is based on two random variables the first being Normal with mean 1 0 and standard deviation 0 5 and the second being Uniform in the interval 2 3 We begin by defining two random variables with the randvar_new function vui randvar_new Normale 1 0 0 5 vu2 randvar_new Uniforme 2 0 3 0 Then we create a collection of random variables with the setrandvar_new function which creates here an empty collection of random variables Then we add the two random variables to the collection 37 Empirical histogram of X2 Empirical Exact Figure 6 7 Monte Carlo Sampling Histogram and exact distribution functions for the second variable srv setrandvar_new setrandvar_addrandvar srv vul setrandvar_addrandvar srv vu2 We can now build the DOE so that it is a LHS sampling with 1000 experiments setrandvar_buildsample srv Lhs 1000 At this point the DOE is stored in the memory space of the NISP library but we do not have a direct access to it We now call the setrandvar_getsample function and store that DOE into the sampling matrix sampling setrandvar_getsample srv The sampling matrix has 1000 rows corresponding to each experiment
13. are normalized we use the default parameters of the randvar_new function The normalized collection is stored in the variable srvx vx1 randvar_new Normale vx2 randvar_new Uniforme srvx setrandvar_new setrandvar_addrandvar srvx vxl setrandvar_addrandvar srvx vx2 We create a collection of two uncertain parameters We explicitely set the parameters of each random variable that is the first Normal variable is associated with a mean equal to 1 0 and a standard deviation equal to 0 5 while the second Uniform variable is in the interval 1 0 2 5 This collection is stored in the variable srvu vul randvar_new Normale 1 0 0 5 vu2 randvar_new Uniforme 1 0 2 5 srvu setrandvar_new setrandvar_addrandvar srvu vul setrandvar_addrandvar srvu vu2 46 Methods output polychaos_gettarget pc np polychaos_getsizetarget pc polychaos_getsample pc k ovar polychaos_getquantile pc k polychaos_getsample pc polychaos_getquantile pc alpha polychaos_getoutput pc polychaos_getmultind pc polychaos_getlog pc polychaos_getinvquantile pc threshold polychaos_getindextotal pc polychaos_getindexfirst pc ny polychaos_getdimoutput pc nx polychaos_getdiminput pc p polychaos_getdimexp pc no polychaos_getdegree pc polychaos_getcovariance pc polychaos_getcorrelation pc polychaos_getanova
14. file and configure the path to the toolboxes stored in the SCILABTBX variable exec C tbxnisp loader sce The following script presents the messages which are generated when the unit tests script of the toolbox is launched gt exec C tbxnisp runtests sce Tests beginning the 2009 11 18 at 12 47 45 TMPDIR C Users baudin AppData Local Temp SCI_TMP_6372_ 001 004 tbxnisp nisp passed ref created 002 004 tbxnisp polychaosl passed ref created 003 004 tbxnisp randvarl passed ref created 004 004 tbxnisp setrandvarl passed ref created Summary tests 4 100 passed 0 0 failed 0 0 skipped 0 o length 3 84 sec Tests ending the 2009 11 18 at 12 47 48 end verbatim 21 Chapter 4 Configuration functions In this section we present functions which allow to configure the NISP toolbox The nisp_ functions allows to configure the global behaviour of the toolbox These func tions allows to startup and shutdown the toolbox and initialize the seed of the random number generator They are presented in the figure 4 1 nisp_startup Starts up the NISP toolbox nisp_shutdown Shuts down the NISP toolbox level nisp_verboselevelget Returns the current verbose level nisp_verboselevelset level Sets the value of the verbose level nisp_initseed seed Sets the seed of the uniform random number generator nisp_ destroyall Des
15. o o o o o o o o Mes th Ap ATI I 0 0 0 1 0 2 0 3 04 0 5 0 6 0 7 0 8 0 9 Variable 1 Uniforme 0 0 1 0 Figure 6 15 Quadrature sampling Two uniform variables in the interval 1 1 Sobol Sampling Variable 2 Uniforme 1 0 1 0 Variable 1 Uniforme 1 0 1 0 Figure 6 16 Sobol sampling Two uniform variables in the interval 1 1 44 Chapter 7 The polychaos class 7 1 Introduction The polychaos class allows to manage a polynomial chaos expansion The coefficients of the expansion are computed based on given numerical experiments which creates the association between the inputs and the outputs Once computed the expansion can be used as a regular function The mean standard deviation or quantile can also be directly retrieved The tool allows to get the following results e mean e variance e quantile e correlation etc Moreover we can generate the C source code which computes the output of the polynomial chaos expansion This C source code is stand alone that is it is independent of both the NISP library and Scilab It can be used as a meta model The figure 7 1 presents the most commonly used methods available in the polychaos class More methods are presented in figure 7 2 The inline help contains the detailed calling sequence for each function and will not be repeated here More than 50 methods are available and most of them will not be presented here More inform
16. randvar_destroy rv2 The transformation depends on the mother random variable rv1 and on the daughter ran dom variable rv Specific transformations are provided for all many combinations of the two distribution functions These transformations will be analysed in the next sections 30 Chapter 6 The setrandvar class In this chapter we presen the setrandvar class The first section gives a brief outline of the features of this class and the second section present several examples 6 1 Introduction The setrandvar class allows to manage a collection of random variables and to build a Design Of Experiments DOE Several types of DOE are provided e Monte Carlo e Latin Hypercube Sampling e Smolyak Once a DOE is created we can retrieve the information experiment by experiment or the whole matrix of experiments This last feature allows to benefit from the fact that Scilab can natively manage matrices so that we do not have to perform loops to manage the complete DOE Hence good performances can be observed even if the language still is interpreted The figure 6 1 presents the methods available in the setrandvar class A complete description of the input and output arguments of each function is available in the inline help and will not be repeated here More informations about the Oriented Object system used in this toolbox can be found in the section 5 2 2 6 2 Examples In this section we present examples of use of the s
17. 03 013 1403 pipa pupo 2 38 13 The last two terms of the previous eguality can be simplified so that we get V Y 0703 0413 103 2 39 The sensitivity indices can be computed from the definitions V E Y X1 S 2 40 V E Y X2 Ba n 2 41 2 VY 2 41 We have E Y X E X2 X 2X1 Similarily F Y X2 11 X2 Hence V u2X1 A 2 42 V X2 Sa 2 43 TR 2 43 We get 2 HV Xi RER o 2 44 2 LV X2 SS oe 2 45 2 45 Finally the first order sensitivity indices are 2 2 H201 gie 2 46 2 2 H102 S2 2 47 FEL 247 Since 0703 gt 0 we have H301 107 lt V Y oj 03 072 1303 2 48 We divide the previous inequality by V Y and get 2 2 2 2 H201 H192 l lt 1 2 49 vo vY S gi Therefore the sum of the first order sensitivity indices satisfies the ineguality Che cree 2 50 Hence in this example one part of the variance V Y cannot be explained neither by X alone or by X alone because it is caused by the interactions between X and X2 We define by S12 the sensitivity index associated with the group of variables X1 X2 as 010 Si2 1 S1 62 CT 2 51 2 5 2 Ishigami function 14 Chapter 3 Installation In this section we present the installation process for the toolbox We present the steps which are reguired to have a running version of the toolbox and presents the several checks which can be performed before using the toolbox
18. Mac OS X The ATOMS component allows to use a toolbox based on compiled source code without having a compiler installed in the system Installing the Scilab NISP toolbox from ATOMS reguires the following steps e atomsList prints the list of current toolboxes e atomsShow prints informations about a toolbox e atomsInstall installs a toolbox on the system e atomsLoad loads a toolbox Once installed and loaded the toolbox will be available on the system from session to session so that there is no need to load the toolbox again it will be available right from the start of the session In the following Scilab session we use the atomsList function to print the list of all ATOMS toolboxes gt atomsList ANN_Toolbox ANN Toolbox dde_toolbox Dynamic Data Exchange client for Scilab module_lycee Scilab pour les lyc l es NISP Non Intrusive Spectral Projection plotlib Matlab like Plotting library for Scilab scipad Scipad 7 20 sndfile_toolbox Read amp write sound files stixbox Statistics toolbox for Scilab 5 2 In the following Scilab session we use the atomsShow function to print the details about the NISP toolbox gt atomsShow NISP Package NISP Title NISP Summary Non Intrusive Spectral Projection Version 2 1 Depend Category ies Optimization Maintainer s Pierre Marechal lt pierre marechal scilab org gt Michael Baudin lt michael baudin scilab org gt 16 Ent
19. NISP Toolbox Manual Michael Baudin INRIA Jean Marc Martinez CEA Version 0 2 January 2011 Abstract This document is a brief introduction to the NISP module We present the installation process of the module in binary from ATOMS or from the sources We present the configuration functions and the randvar setrandvar and polychaos classes Contents 1 Introduction LI Nar UG Moet ceso SLR MN ee a O AI ita ta The NIGP DDE ili RAE RELA RD HI A Re la The NIGP mod le e st ik A CR TETTO Tr 2 Theory 21 pensiya AUS o e da d m a me AR AR aa 2 2 Standardized regression coefficients of linear models 2 3 Link with the linear correlation o ER ES ER DR GR 4 2 4 Sensitivity analysis for nonlinear models 20 AIN DEE E cu ET 291 PIONEER iunctio si c go accro pidea pi RE REE RR HS Dee I OR it o 20 e e a Lu as le a Lu BAR Eu UA LG 3 Installation Li C5 O se iene EY ewes GU ems oY ems EU ome UUN 3 2 Installing the toolbox from ATOMS Lili A ERW A a A A a 3 3 Installing the toolbox from the sources 4 Configuration functions 5 The randvar class 8 1 The distribution functions lt a o eres RW WR Thi Rd co e ee EE KS Se ei 5 1 2 Parameters of the Log normal distribution 5 1 3 Uniform random number generation dl RISE au acilia Wyd wnh Bre Hb dy RR ea rs ae A ATA ee Se ORE ERE DE 5 2 2 The Oriented Object system
20. The NISP toolbox is available under the following operating systems e Linux 32 bits e Linux 64 bits e Windows 32 bits e Mac OS X The following list presents the features provided by the NISP toolbox 4 e Manage various types of random variables uniform normal exponential log normal Generate random numbers from a given random variable Transform an outcome from a given random variable into another e Manage various Design of Experiments for sets of random variables Monte Carlo Sobol Latin Hypercube Sampling various samplings based on Smolyak designs Manage polynomial chaos expansion and get specific outputs including mean variance guantile correlation etc e Generate the C source code which computes the output of the polynomial chaos expansion This User s Manual completes the online help provided with the toolbox but does not replace it The goal of this document is to provide both a global overview of the toolbox and to give some details about its implementation The detailed calling seguence of each function is provided by the online help and will not be reproduced in this document The inline help is presented in the figure 1 2 For example in order to access to the help associated with the randvar class we type the following statements in the Scilab console help randvar The previous statements opens the Help Browser and displays th
21. Variable X2 4f n polychaos_getindexfirst pc 2 mprintf uuuuVariablewX3u u4fAn polychaos_getindexfirst pc 3 mprintf Indice de sensibilite Totale n mprintf Guu uVariable X1 4f n polychaos_getindextotal pc 1 mprintf QuuuVariable X2 f n polychaos_getindextotal pc 2 mprintf uuuuVariablewX3u u4fAn polychaos_getindextotal pc 3 The previous script produces the following output Mean 3 500000 Variance 13 842473 Indice de sensibilitAl du ter ordre Variable X1 0 313953 Variable X2 0 442325 Variable X3 0 000000 Indice de sensibilite Totale Variable X1 0 557675 Variable X2 0 442326 Variable X3 0 243721 We now focus on the variance generated by the variables 1 and 3 We set the group to the empty group with the polychaos_setgroupempty function and add variables with the polychaos_setgroupaddvar function groupe 1 3 51 polychaos_setgroupempty pc polychaos_setgroupaddvar pc groupe 1 polychaos_setgroupaddvar pc groupe 2 mprintf Part de la variance d un groupe de variables n mprintf uuuuGroupe X1 etyX2 4f n polychaos_getgroupind pc The previous script produces the following output Part de la variance d un groupe de variables Groupe X1 et X2 0 557674 The function polychaos_getanova prints the functionnal decomposition o
22. ale Normale Uniforme Uniforme Exponentielle Exponentielle LogNormale LogNormale LogUniforme LogUniforme Source Target Source Target Uniforme LogUniforme Uniforme Uniforme Normale Normale Exponentielle Exponentielle LogNormale LogNormale LogUniforme LogUniforme Source Target Exponentielle Exponentielle Figure 5 5 Variable transformations available in the randvar class 29 returns a random value from the distribution function of the random variable rv by transformation of value2 from the distribution function of random variable rv2 In the following session we transform a uniform random variable sample into a LogUniform variable sample We begin to create a random variable rv from a LogUniform law and parameters a 10 b 20 Then we create a second random variable rv2 from a Uniforme law and parameters a 2 b 3 The main loop is based on the transformation of a sample computed from rv2 into a sample from rv The mean allows to check that the transformed samples have an mean value which corresponds to the random variable rv nisp_initseed O a 10 0 b 20 0 rv randvar_new LogUniforme a rv2 randvar_new Uniforme 2 nbshots 1000 values zeros nbshots for i 1 nbshots value2 randvar_getvalue rv2 values i randvar_getvalue rv rv2 value2 end computed mean values mu b a log b log a expected mu computed should be close to expected randvar_destroy rv
23. and 2 columns cor responding to each input random variable The following script allows to plot the sampling which is is presented in figure 6 8 my_handle scf clf my_handle reset plot sampling 1 sampling 2 my_handle children children children line_mode off my_handle children children children mark_mode on my_handle children children children mark_size 2 my_handle children title text Latin Hypercube Sampling my _handle children x_label text Variable 1 Normale 1 0 0 5 my _handle children y_label text Variable 2 Uniforme 2 0 3 0 The following script allows to plot the histogram of the two variables which are presented in figures 6 9 and 6 10 Plot Var 1 38 Latin Hypercube Sampling Variable 2 Uniforme 2 0 3 0 1 0 0 5 0 0 0 5 1 0 1 5 2 0 2 5 3 0 Variable 1 Normale 1 0 0 5 Figure 6 8 Latin Hypercube Sampling The first variable is Normal the second variable is Uniform 39 my_handle scf clf my_handle reset histplot 50 sampling 1 my_handle children title text Variable 1 Normale 1 0 0 5 Plot Var 2 my_handle scf clf my_handle reset histplot 50 sampling 2 my_handle children title text Variable 2 Uniforme 2 0 3 0 Variable 1 Normale 1 0 0 5
24. ate a cleaner file Generating loader_gateway sce Building help Building the master document C tbxnisp help en_US Building the manual file javaHelp C tbxnisp help en_US Please wait building Generating loader sce in this can take a while The following script presents the messages which are generated when the loader of the toolbox 20 is launched The loader script performs the following steps e load the gateway and the NISP library e load the help e load the demo gt exec C tbxnisp loader sce Start NISP Toolbox Load gateways Load help Load demos It is now necessary to setup your Scilab system so that the toolbox is loaded automatically at startup The way to do this is to configure the Scilab startup configuration file The directory where this file is located is stored in the Scilab variable SCIHOME In the following Scilab session we use Scilab v5 2 0 beta 1 in order to know the value of the SCIHOME global variable gt SCIHOME SCIHOME C Users baudin AppData Roaming Scilab scilab 5 2 0 beta 1 On my Linux system the Scilab 5 1 startup file is located in home myname Scilab scilab 5 1 scilab On my Windows system the Scilab 5 1 startup file is located in C Users myname AppData Roaming Scilab scilab 5 1 scilab This file is a regular Scilab script which is automatically loaded at Scilab s startup If that file does not already exist create it Copy the following lines into the scilab
25. ations about the Oriented Object system used in this toolbox can be found in the section 5 2 2 7 2 Examples In this section we present to examples of use of the polychaos class 7 2 1 Product of two random variables In this section we present the polynomial expansion of the product of two random variables We analyse the Scilab script and present the methods which are available to perform the sensi 45 Constructors pc polychaos_new file pc polychaos_new srv ny pc polychaos_new pc nopt varopt Methods polychaos_setsizetarget pc np polychaos_settarget pc output polychaos_setinput pc invalue polychaos_setdimoutput pc ny polychaos_setdegree pc no polychaos_getvariance pc polychaos_getmean pc Destructor polychaos_destroy pc Static methods tokenmatrix polychaos_tokens nb polychaos_ size Figure 7 1 Outline of the methods of the polychaos class tivity analysis This script is based on the NISP methodology which has been presented in the Introduction chapter We will use the figure 1 1 as a framework and will follow the steps in order In the following Scilab script we define the function Example which takes a vector of size 2 as input and returns a scalar as output function y Exemple x y 1 x 1 x 2 endfunction We now create a collection of two stochastic normalized random variables Since the ran dom variables
26. chaos_getindextotal pc 2 The previous script produces the following output Mean 1 750000 Variance 1 000000 Indice de sensibilit l du ter ordre Variable X1 0 765625 49 Variable X2 0 187500 Indice de sensibilite Totale Variable X1 0 812500 Variable X2 0 234375 In order to free the memory required for the computation it is necessary to delete all the objects created so far polychaos_destroy pc randvar_destroy vul randvar_destroy vu2 randvar_destroy vx1 randvar_destroy vx2 setrandvar_destroy srvu setrandvar_destroy srvx 7 2 2 The Ishigami test case In this section we present the Ishigami test case The function Exemple is the model that we consider in this numerical experiment This function takes a vector of size 3 in input and returns a scalar output function y Exemple x a 7 b 0 1 si sin x 1 s2 sin x 2 y 1 s1 a s2 s2 b x 3 x 3 x 3 x 3 s1 endfunction We create 3 uncertain parameters which are uniform in the interval r m and put these random variables into the collection srvu rvui randvar_new Uniforme pi pi rvu2 randvar_new Uniforme pi pi rvu3 randvar_new Uniforme pi pi srvu setrandvar_new setrandvar_addrandvar srvu rvul setrandvar_addrandvar srvu rvu2 setrandvar_addrandvar srvu rvu3 The collection of stochastic variabl
27. destroy 4 ans A gt randvar_tokens ans L 5 3 Examples In this section we present to examples of use of the randvar class The first example presents the simulation of a Normal random variable and the generation of 1000 random variables The second example presents the transformation of a Uniform outcome into a LogUniform outcome 27 5 3 1 A sample session We present a sample Scilab session where the randvar class is used to generate samples from the Normale law In the following Scilab session we create a Normale random variable and compute samples from this law The nisp_initseed function is used to initialize the seed for the uniform random variable generator Then we use the randvar_new function to create a new random variable from the Normale law with mean 1 and standard deviation 0 5 The main loop allows to compute 1000 samples from this law based on calls to the randvar_getvalue function Once the samples are computed we use the Scilab function mean to check that the mean is close to 1 which is the expected value of the Normale law when the number of samples is infinite Finally we use the randvar_destroy function to destroy our random variable Once done we plot the empirical distribution function of this sample with 50 classes nisp_initseed O mu 1 0 sigma 0 5 rv randvar_new Normale mu sigma nbshots 1000 values zeros nbshots for i 1 nbshots values i randvar_getvalue rv
28. e Compilation of utils cpp Compilation of blasi_d cpp Compilation of dcdflib cpp Compilation of faure cpp Compilation of halton cpp Compilation of linpack_d cpp Compilation of niederreiter cpp Compilation of reversehalton cpp Compilation of sobol cpp Building shared library be patient Generate a cleaner file Generate a loader file Generate a Makefile Running the Makefile Compilation of nisp_gc cpp Compilation of nisp_gva cpp Compilation of nisp_ind cpp Compilation of nisp_index cpp Compilation of nisp_inv cpp Compilation of nisp_math cpp Compilation of nisp_msg cpp Compilation of nisp_conf cpp Compilation of nisp_ort cpp Compilation of nisp_pc cpp Compilation of nisp_polyrule cpp 18 Compilation Compilation Compilation Compilation Compilation Compilation of of of of of of nisp_qua cpp nisp_random cpp nisp_smo cpp nisp_util cpp nisp_va cpp nisp_smolyak cpp Building shared library be patient Generate a cleaner file Building gateway Generate a gateway file Generate a loader file Generate a Makefile Running the Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilation Compilatio
29. e variables gt randvar_destroy vul gt randvar_destroy vu2 gt randvar_destroy vu3 We can finally check that there are no random variables left in the memory space 26 gt nb randvar_size nb 0 gt tokenmatrix randvar_tokens tokenmatrix Scilab is a wonderful tool to experiment algorithms and make simulations It happens some times that we are managing many variables at the same time and it may happen that at some point we are lost The static methods provides tools to be able to recover from such a situation without closing our Scilab session In the following session we create two random variables gt vui randvar_new Uniforme vul 3 gt vu2 randvar_new Uniforme vu2 4 Assume now that we have lost the token associated with the variable vu2 We can easily simulate this situation by using the clear which destroys a variable from Scilab s memory space gt clear vu2 gt randvar_getvalue vu2 error 4 Undefined variable vu2 It is now impossible to generate values from the variable vu2 Moreover it may be difficult to know exactly what went wrong and what exact variable is lost At any time we can use the randvar_tokens function in order to get the list of current variables Deleting these variables allows to clean the memory space properly without memory loss gt randvar_tokens ans 3 4 gt randvar_destroy 3 ans 3 gt randvar_
30. e X in the range 0 1 is computed from L N where N RAND_MAX and n 0 RAND_MAX 5 8 5 2 Methods In this section we give an overview of the methods which are available in the randvar class 5 2 1 Overview The figure 5 3 presents the methods available in the randvar class The inline help contains the detailed calling sequence for each function and will not be repeated here Constructors rv randvar_new type options Methods value randvar_getvalue rv options randvar_getlog rv Destructor randvar_destroy rv Static methods rvlist randvar_tokens nbrv randvar_size Figure 5 3 Outline of the methods of the randvar class 5 2 2 The Oriented Object system In this section we present the token system which allows to emulate an oriented object program ming with Scilab We also present the naming convention we used to create the names of the functions The randvar class provides the following functions e The constructor function randvar_new allows to create a new random variable and returns a token rv e The method randvar_getvalue takes the token rv as its first argument In fact all methods takes as their first argument the object on which they apply 25 e The destructor randvar_destroy allows to delete the current object from the memory of the library e The static methods randvar_tokens and randvar_size allows to guiery the current object wh
31. e helps page presented in figure Several demonstration scripts are provided with the toolbox and are presented in the figure 1 4 These demonstrations are available under the question mark in the menu of the Scilab console Finally the unit tests provided with the toolbox cover all the features of the toolbox When we want to know how to use a particular feature and do not find the information we can search in the unit tests which often provide the answer Help Browser NISP Toolbox Table of Contents nisp Functions to configure the NISP library overview An overview of the NISP toolbox polychaos A class to manage a Polynomial Chaos expansion randvar A class to manage a random variable setrandvar A class to manage a set of random variables Figure 1 2 The NISP inline help Help Browser Nom randvar A class to manage a random variable SYNOPSIS tokens randvar tokens randvar size randvar new name a b randvar new name a randvar new name randvar destroy rv value randvar getvalue rv value randvar getvalue rv rv2 value2 randvar getlog rv Figure 1 3 The online help of the randvar function GUI Genetic Algorithms Simulated Annealing Graphics Signal Processing CACSD Optimization and Simulation Polynomials Simulation Scicos Metanet Tcl Tk Sound file handling Random Spreadsheet Nisp Sci
32. es is created with the function setrandvar_new The calling sequence srvx setrandvar_new nx allows to create a collection of nx 3 random variables uniform in the interval 0 1 Then we create a Petras DOE for the stochastic collection srvx and transform it into a DOE for the uncertain parameters srvu nx setrandvar_getdimension srvu srvx setrandvar_new nx degre 9 setrandvar_buildsample srvx Petras degre setrandvar_buildsample srvu srvx 90 We use the polychaos_new function and create the new polynomial pc with 3 inputs and 1 output noutput 1 pc polychaos_ new srvx noutput The next step allows to perform the simulations associated with the DOE prescribed by the collection srvu np setrandvar_getsize srvu polychaos_setsizetarget pc np for k 1 np inputdata setrandvar_getsample srvu k outputdata Exemple inputdata polychaos_settarget pc k outputdata end We can now compute the polynomial expansion by integration polychaos_setdegree pc degre polychaos_computeexp pc srvx Integration Everything is now ready so that we can do the sensitivy analysis as in the following script average polychaos_getmean pc var polychaos_getvariance pc mprintf Meanuuuuuuucu 4fAMn average mprintf Varianceuuuucu 4fAn var mprintf Indice de sensibilitAl du ler jordre n mprintf u uuuyVariable X1 fin polychaos_getindexfirst pc 1 mprintf Guu
33. etrandvar class In the first example we present a Scilab session where we create a Latin Hypercube Sampling In the second part we present various types of DOE which can be generated with this class 6 2 1 A Monte Carlo design with 2 variables In the following example we build a Monte Carlo design of experiments with 2 input random variables The first variable is associated with a Normal distribution function and the second 31 Constructors srv setrandvar_new srv setrandvar_new n srv setrandvar_new file Methods setrandvar_setsample setrandvar_setsample setrandvar_setsample srv k value setrandvar_setsample srv value setrandvar_save srv file np setrandvar_getsize srv sample setrandvar_getsample srv k sample setrandvar_getsample srv k sample setrandvar_getsample srv setrandvar_getlog srv nx setrandvar_getdimension srv setrandvar_freememory srv setrandvar_buildsample srv srv2 setrandvar_buildsample srv name np 7 setrandvar_buildsample srv name np ne setrandvar_addrandvar srv rv Destructor setrandvar_destroy srv Static methods tokenmatrix setrandvar_tokens nb setrandvar_size srv name np srv k i value L L L i 7 Figure 6 1 Outline of the methods of the setrandvar class 32 variable is associated with a Uniform distribution function The simulation is based on
34. f the normalized variance polychaos_getanova pc The previous script produces the following output 0 O 0 313953 0 442325 1 55229e 009 8 08643e 031 0 243721 7 26213e 031 1 6007e 007 KA o HO HO KA KA O O KK A KA KAK H KA O We can compute the density function associated with the output variable of the function In order to compute it we use the polychaos_buildsample function and create a Latin Hypercube Sampling with 10000 experiments The polychaos_getsample function allows to guiery the polynomial and get the outputs We plot it with the histplot Scilab graphic function which produces the figure 7 3 polychaos_buildsample pc Lhs 10000 0 sample_output polychaos_getsample pc histplot 50 sample_output xtitle Fonction Ishigami LHistogramme normalis l We can plot a bar graph of the sensitivity indices as presented in figure 7 4 for i 1 nx indexfirst i polychaos_getindexfirst pc i indextotal i polychaos_getindextotal pc i end my_handle scf 10002 bar indextotal 0 2 blue bar indexfirst 0 15 yellow legend totale premier ordre pos 1 xtitle Fonction Ishigami uIndice de sensibilit l 92 Fonction Ishigami Histogramme normalis 0 12 0 10 0 08 0 06 0 04 0 02 dll 10 5 0 0 00 r 10 15 20 Figure 7 3 Ishigami function Histogram of the output variable on a LHS design with 10000 experiments
35. hen the srvu has been deduced from srvx based on random variable transformations We now use the polychaos_new function and create a new polynomial pc The number of input variables corresponds to the number of variables in the stochastic collection srvx that is 2 and the number of output variables is given as the input argument ny In this particular case the number of experiments to perform is equal to np 9 as returned by the setrandvar_getsize function This parameter is passed to the polynomial pc with the polychaos_setsizetarget function ny 1 pc polychaos_new srvx ny np setrandvar_getsize srvx polychaos_setsizetarget pc np 48 In the next step we perform the simulations prescribed by the DOE We perform this loop in the Scilab language and make a loop over the index k which represents the index of the current experiment while np is the total number of experiments to perform For each loop we get the input from the uncertain collection srvu with the setrandvar_getsample function pass it to the Exemple function get back the output which is then transferred to the polynomial pc by the polychaos_settarget function for k 1 np inputdata setrandvar _getsample srvu k outputdata Exemple inputdata mprintf Experiment d uinputu sl tuoutputu f n k strcat string inputdata outputdata polychaos_settarget pc k outputdata end The previous script produces the following output Experime
36. ich are in use More specifically the randvar_size function returns the number of current randvar objects and the randvar_tokens returns the list of current randvar objects In the following Scilab sessions we present these ideas with practical uses of the toolbox Assume that we start Scilab and that the toolbox is automatically loaded At startup there are no objects so that the randvar_size function returns 0 and the randvar_tokens function returns an empty matrix gt nb randvar_size nb 0 gt tokenmatrix randvar_tokens tokenmatrix We now create 3 new random variables based on the Uniform distribution function We store the tokens in the variables vu1 vu2 and vu3 These variables are regular Scilab double precision floating point numbers Each value is a token which represents a random variable stored in the toolbox memory space gt vul randvar_new Uniforme vul 0 gt vu2 randvar_new Uniforme vu2 1 gt vu3 randvar_new Uniforme vu3 2 There are now 3 objects in current use as indicated by the following statements The tokenmatrix is a row matrix containing regular double precision floating point numbers gt nb randvar_size nb 3 gt tokenmatrix randvar_tokens tokenmatrix 0 1 2 We assume that we have now made our job with the random variables so that it is time to destroy the random variables We call the randvar_destroy functions which destroys th
37. ity CEA DIGITEO WebSite License LGPL Scilab Version gt 5 2 0 Status Not installed Description This toolbox allows to approximate a given model which is associated with input random variables This toolbox has been created in the context of the OPUS project http opus project fr within the workpackage 2 1 1 Construction de m l ta mod les This project has received funding by Agence Nationale de la recherche http www agence nationale recherche fr See in the help provided in the help en_US directory of the toolbox for more information about its use Use cases are presented in the demos directory In the following Scilab session we use the atomsInstall function to download and install the binary version of the toolbox corresponding to the current operating system gt atomsInstall NISP ans INISP 2 1 allusers D Programs SC3623 1 contrib NISP 2 1 I The allusers option of the atomsInstall function can be used to install the toolbox for all the users of this computer We finally load the toolbox with the atomsLoad function gt atomsLoad NISP Start NISP Toolbox Load gateways Load help Load demos ans INISP 2 1 D Programs SC3623 1 contrib NISP 2 1 Now that the toolbox is loaded it will be automatically loaded at the next Scilab session 3 3 Installing the toolbox from the sources In this section we present the steps which are required in order to install the toolbox from the so
38. m variables in the interval 1 1 The figure 6 11 presents this sampling and the figures 6 12 and 6 13 present the histograms of the two uniform random variables vui randvar_new Uniforme 1 0 1 0 vu2 randvar_new Uniforme 1 0 1 0 srv setrandvar_new setrandvar_addrandvar srv vul setrandvar_addrandvar srv vu2 setrandvar_buildsample srv MonteCarlo 1000 sampling setrandvar_getsample srv randvar_destroy vul randvar_destroy vu2 setrandvar_destroy srv It is easy to change the type of sampling by modifying the second argument of the setrandvar_buildsa function This way we can create the Petras Ouadrature and Sobol sampling presented in figures 6 14 6 15 and 6 16 41 Monte Carlo Sampling Variable 2 Uniforme 1 0 1 0 1 0 0 8 0 6 0 4 0 2 0 0 0 2 0 4 0 6 0 8 1 0 Variable 1 Uniforme 1 0 1 0 Figure 6 11 Monte Carlo Sampling Two uniform variables in the interval 1 1 Variable 1 Uniforme 1 0 1 0 0 8 0 7 7 0 67 0 5 0 4 0 3 0 2 0 1 0 0 Figure 6 12 Latin Hypercube Sampling First uniform variable in 1 1 42 Variable 2 Uniforme 1 0 1 0 0 8 0 7 7
39. n Compilation Compilation Compilation Compilation Compilation Makelib makefile of of of of of of of of of of of of of of of of of of of of of of of of of of of of of of of of of of of of nisp_gettoken cpp nisp_gwsupport cpp nisp_PolynomialChaos_map cpp nisp_RandomVariable_map cpp nisp_SetRandomVariable_map cpp sci_nisp_startup cpp sci_nisp_shutdown cpp sci_nisp_verboselevelset cpp sci_nisp_verboselevelget cpp sci_nisp_initseed cpp sci_randvar_new cpp sci_randvar_destroy cpp sci_randvar_size cpp sci_randvar_tokens cpp sci_randvar_getlog cpp sci_randvar_getvalue cpp sci_setrandvar_new cpp sci_setrandvar_tokens cpp sci_setrandvar_size cpp sci_setrandvar_destroy cpp sci_setrandvar_freememory cpp sci_setrandvar_addrandvar cpp sci_setrandvar_getlog cpp sci_setrandvar_getdimension cpp sci_setrandvar_getsize cpp sci_setrandvar_getsample cpp sci_setrandvar_setsample cpp sci_setrandvar_save cpp sci_setrandvar_buildsample cpp sci_polychaos_new cpp sci_polychaos_destroy cpp sci_polychaos_tokens cpp sci_polychaos_size cpp sci_polychaos_setdegree cpp sci_polychaos_getdegree cpp sci_polychaos_freememory cpp 19 Compilation of sci_polychaos_getdimoutput cpp Compilation of sci_polychaos_setdimoutput cpp Compilation of sci_polychaos_getsizetarget cpp Compilation of sci_polychaos_setsizetarget cpp Compilation of sci_polychaos_freememtarget cpp Compilation of sci_polychaos_settarget cpp Compilation
40. ncertainties transforming the whole set Dx into the set Dy In the following we assume that there is only one output variable so that Y R There are two types of analysis that we are going to perform that is uncertainty analysis and sensitivity analysis In uncertainty analysis we assume that fx is the probability density function of the variable X and we are searching for the probability density function fy of the variable Y and by its cumulated density function Fy y P Y lt y This problem is difficult in the general case and this is why we often are looking for an estimate of the expectation of Y as defined by EV ida 2 2 and an estimate of its variance Var Y y E Y fy y dy 2 3 We might also be interested in the computation of the probability over a threshold In sensitivity analysis we focus on the relative importance of the input variable X on the uncertainty of Y This way we can order the input variables so that we can reduce the variability of the most important input variables in order to finally reduce the variability of Y More details on this topic can be found in the papers of Homma and Saltelli 1 or in the work of Sobol 4 The Thesis by Jacques 2 presents an overview of sensitivity analysis 2 2 Standardized regression coefficients of linear models Assume that the random variables X are independent with mean E X and finite variances V X for i 1 2 p Let us conside
41. nt 1 input 0 1339746 1 1690525 output 0 156623 Experiment 2 input 0 1339746 1 75 output 0 234456 Experiment 3 input 0 1339746 2 3309475 output 0 312288 Experiment 4 input 1 1 1690525 output 1 169052 Experiment 5 input 1 1 75 output 1 750000 Experiment 6 input 1 2 3309475 output 2 330948 Experiment 7 input 1 8660254 1 1690525 output 2 181482 Experiment 8 input 1 8660254 1 75 output 3 265544 Experiment 9 input 1 8660254 2 3309475 output 4 349607 We can compute the polynomial expansion based on numerical integration so that the coeffi cients of the polynomial are determined This is done with the polychaos_computeexp function which stands for compute the expansion polychaos_setdegree pc degre polychaos_computeexp pc srvx Integration Everything is now ready for the sensitivity analysis Indeed the polychaos_getmean returns the mean while the polychaos_getvariance returns the variance average polychaos_getmean pc var polychaos_getvariance pc mprintf Meanguuucu AfAn average mprintf Varianceyyuu sf n var mprintf Indice de sensibilitAl du ier ordre n mprintf u uuuyVariable X1 fin polychaos_getindexfirst pc 1 mprintf Quu Variable X2 4f n polychaos_getindexfirst pc 2 mprintf Indice de sensibilite Totale n mprintf uuuuVariable X10 014f n polychaos_getindextotal pc 1 mprintf uuuuVariable X20 14f n poly
42. of sci_polychaos_gettarget cpp Compilation of sci_polychaos_getdiminput cpp Compilation of sci_polychaos_getdimexp cpp Compilation of sci_polychaos_getlog cpp Compilation of sci_polychaos_computeexp cpp Compilation of sci_polychaos_getmean cpp Compilation of sci_polychaos_getvariance cpp Compilation of sci_polychaos_getcovariance cpp Compilation of sci_polychaos_getcorrelation cpp Compilation of sci_polychaos_getindexfirst cpp Compilation of sci_polychaos_getindextotal cpp Compilation of sci_polychaos_getmultind cpp Compilation of sci_polychaos_getgroupind cpp Compilation of sci_polychaos_setgroupempty cpp Compilation of sci_polychaos_getgroupinter cpp Compilation of sci_polychaos_getinvguantile cpp Compilation of sci_polychaos_buildsample cpp Compilation of sci_polychaos_getoutput cpp Compilation of sci_polychaos_getguantile cpp Compilation of sci_polychaos_getguantwilks cpp Compilation of sci_polychaos_getsample cpp Compilation of sci_polychaos_setgroupaddvar cpp Compilation of sci_polychaos_computeoutput cpp Compilation of sci_polychaos_setinput cpp Compilation of sci_polychaos_propagateinput cpp Compilation of sci_polychaos_getanova cpp Compilation of sci_polychaos_setanova cpp Compilation of sci_polychaos_getanovaord cpp Compilation of sci_polychaos_getanovaordco cpp Compilation of sci_polychaos_realisation cpp Compilation of sci_polychaos_save cpp Compilation of sci_polychaos_generatecode cpp Building shared library be patient Gener
43. pc polychaos_generatecode pc filename funname polychaos_computeoutput pc polychaos_computeexp pc srv method polychaos_computeexp pc pc2 invalue varopt polychaos_buildsample pc type np order Figure 7 2 More methods from the polychaos class 47 The first design of experiment is build on the stochastic set srvx and based on a Quadrature type of DOE Then this DOE is transformed into a DOE for the uncertain collection of parameters srvu degre 2 setrandvar_buildsample srvx Guadrature degre setrandvar_buildsample srvu srvx The next steps will be to create the polynomial and actually perform the DOE But before doing this we can take a look at the DOE associated with the stochastic and uncertain collection of random variables We can use the setrandvar_getsample twice and get the following output gt setrandvar_getsample srvx ans 1 7320508 0 1127017 1 7320508 0 5 1 7320508 0 8872983 0 0 1127017 0 0 5 0 0 8872983 1 7320508 0 1127017 1 7320508 0 5 1 7320508 0 8872983 gt setrandvar_getsample srvu ans 0 1339746 1 1690525 0 1339746 1 75 0 1339746 2 3309475 1 1 1690525 1 1 75 Li 2 3309475 1 8660254 1 1690525 1 8660254 1 75 1 8660254 2 3309475 These two matrices are a 9x2 matrices where each line represents an experiment and each column represents an input random variable The stochastic normalized srvx DOE has been created first t
44. r the random variable Y as an affine function of the variables X Y bo y BiXi 2 4 i 1 2 p where 6 are real parameters for 1 2 p The expectation of the random variable Y is E Y f0 y BE X 2 5 i 1 2 p Since the variables X are independent the variance of the sum of variables is the sum of the variances Hence V Y V Bo gt V BX 2 6 i 1 2 p which leads to the equality VW BWV X 2 7 i 1 2 p 10 Hence each term 82V X is the part of the total variance V Y which is caused by the variable Xi The standardized regression coefficient is defined as BEV Xi SRC gt 2 8 VY 2 8 for i LZ ius Hence the sum of the standardized regression coefficients is one SRC SRG LE SRA i 2 9 2 3 Link with the linear correlation Assume that the random variables X are independent with mean E X and finite variances V X for i 1 2 p Let us consider the random variable Y which depends linearily on the variables X by the relationship 2 4 The linear correlation coefficient between Y and X is Cov Y Xi e i 2 10 VV VV x for 1 2 p In the particular case of the affine model 2 4 we have Cov Y X Cou bo Xi BCou X 1 Xi B2C00 X2 Xi BiCou Xj Xi B Cov X XI 2 4 Sensitivity analysis for nonlinear models Let us focus on one particular input X of the model f with 1 2 p If we set X to a partic
45. rm function defined in the Stixbox module nisp_initseed O rvi randvar_new Normale 1 0 0 5 rv2 randvar_new Exponentielle 5 D l finition d un groupe de variables al l atoires srv setrandvar_new setrandvar_addrandvar srv rvi setrandvar_addrandvar srv rv2 np 1000 setrandvar_buildsample srv MonteCarlo np sampling setrandvar_getsample srv Check sampling of random variable 1 mean sampling 1 variance sampling 1 Check sampling of random variable 2 min sampling 2 max sampling 2 Plot scf histplot 40 sampling 1 x linspace 1 3 1000 p dnorm x 1 0 5 plot x p r 36 xtitle Empirical histogram of X1 X P X legend Empirical Exact scf histplot 40 sampling 2 x linspace 0 2 1000 p exppdf x 5 plot x p r xtitle Empirical histogram of X2 X P X legend Empirical Exact Clean up setrandvar_destroy srv randvar_destroy rv1 randvar_destroy rv2 The previous script produces the figures 6 6 and 6 7 Empirical histogram of X1 Empirical Exact Figure 6 6 Monte Carlo Sampling Histogram and exact distribution functions for the first variable 6 2 3
46. sent examples of sensitivity analysis 2 5 1 Product function The goal of this example is to show that in some cases we have to consider the interations between the variables Consider the function Y X X gt 2 19 where X and X are two independent normal random variables with mean 1 and uz and variances o and 03 Let us compute the expectation of the random variable Y The expectation of Y is E Y J f X XoF X1 Xa dx1dx2 2 20 where F x1 2 is the joint probability distribution function of the variables X and X Since X and X are independent variables we have F x1 2 F X 1 Fo X2 2 21 12 where Fj is the probability distribution function of X and F3 is the probability distribution function of X Then we have E Y J i i a X1XXF Xi E ldadr 2 22 XF Aa dm T YK 2 23 E X E X2 2 24 Therefore EV myz 2 25 The variance of Y is V Y E Y E Y 2 26 The expectation E Y is E Y i i Ni ADF 01 2 de de 2 27 T n si XXX FAX deyde 2 28 E GII _ Re 2 29 E X E X2 2 30 Now we have V X1 E XT E X1 2 31 V X2 E X2 E X 2 32 which leads to E X V Xi E X 2 33 E X3 V X2 E X 2 34 Therefore E Y V X1 E X1 V X2 E X2 2 35 07 3 03 1 2 36 Finally we get V Y oi ui 02 13 pupo 2 37 We can expand the previous equality and get V Y 01
47. solver which has the form Y f X where X are input uncertain parameters and Y are output random variables The method is based on the following steps e We begin by defining normalized random variables For example we may use a random variables in the interval 0 1 or a Normal random variable with mean 0 and variance 1 This choice allows to define the basis for the polynomial chaos denoted by YVx x gt o Depending on the type of random variable the polynomials Yx x gt o are based on Hermite Legendre or Laguerre polynomials e We can now define a Design Of Experiments DOE and with random variable transforma tions rules we get the physical uncertain parameters X Several types of DOE are available Monte Carlo Latin Hypercube Sampling etc If N experiments are reguired the DOE define the collection of normalized random variables 6 1 nv Transformation rules allows to compute the uncertain parameters X 1 v which are the input of the numerical solver f e We can now perform the simulations that is compute the collection of outputs Y i N where Y f X e The variables Y are then projected on the polynomial basis and the coefficients y are computed by integration or regression Random Uncertain Numerical Spectral Variable gt Parameter Solver gt Projection 6 X Y 1 X Y 2 y w 5 Figure 1 1 The NISP methodology 13 The NISP module
48. tribution function are presented in the figure 5 2 which also presents the conditions which must be satisfied by the parameters Name Fo EX V X Normale 273 exp 1 cn m o Uniforme po i ni bta Ga Exponentielle asz az y 1 an LogNormale ox Zr O P 2 e iu exp u 507 exp o 1 exp 2u 0 0 ifa lt U LogUniforme TOS E i S TORNO 1 E x Figure 5 1 Distributions functions of the randvar class The expected value is denoted by E X and the variance is denoted by V X 23 Name Parameter 1 a Parameter 2 b Conditions Normale p 0 o 1 o gt 0 Uniforme a 0 bed a lt b Exponentielle 1 LogNormale w 0 1 o 1 0 ms dd LogUniforme a 0 1 GaL a b gt 0 a lt b Figure 5 2 Default parameters for distributions functions 5 1 2 Parameters of the Log normal distribution For the LogNormale law the distribution function is usually defined by the expected value y and the standard deviation o of the underlying Normal random variable But when we create a LogNormale randvar the parameters to pass to the constructor are the expected value of the LogNormal random variable E X and the standard deviation of the underlying Normale random variable o The expected value and the variance of the Log Normal law are given by E X exp r 30 5 1 V X exp o
49. troy all current objects nisp_getpath Returns the path to the current module nisp_printall Prints all current objects Figure 4 1 Outline of the configuration methods The user has no need to explicitely call the nisp_startup and nisp_shutdown func tions Indeed these functions are called automatically by the etc NISP start and etc NISP quit scripts located in the toolbox directory structure The nisp_initseed seed is especially useful when we want to have reproductible re sults It allows to set the seed of the generator at a particular value so that the seguence of uniform pseudo random numbers is deterministic When the toolbox is started up the seed is automatically set to 0 which allows to get the same results from session to session 22 Chapter 5 The randvar class In this section we present the randvar class which allows to define a random variable and to generate random numbers from a given distribution function 5 1 The distribution functions In this section we present the distribution functions provided by the randvar class We especially present the Log normal distribution function 5 1 1 Overview The table 5 1 gives the list of distribution functions which are available with the randvar class 3 Each distribution functions have zero one or two parameters One random variable can be specified by giving explicitely its parameters or by using default parameters The parameters for all dis
50. ts of the OPUS project 1 2 The NISP library The NISP library is based on a set of 3 C classes so that it provides an object oriented framework for uncertainty analysis The Scilab toolbox provides a pseudo object oriented interface to this library so that the two approaches are consistent The NISP library is release under the LGPL licence The NISP library provides three tools which are detailed below e The randvar class allows to manage random variables specified by their distribution law and their parameters Once a random variable is created one can generate random numbers from the associated law e The setrandvar class allows to manage a collection of random variables This collection is associated with a sampling method such as MonteCarlo Sobol Quadrature etc It is possible to build the sample and to get it back so that the experiments can be performed e The polychaos class allows to manage a polynomial representation of the simulated model One such object must be associated with a set of experiments which have been performed This set may be read from a data file The object is linked with a collection of random variables Then the coefficients of the polynomial can be computed by integration guadra ture Once done the mean the variance and the Sobol indices can be directly computed from the coefficients The figure 1 1 presents the NISP methodology The process reguires that the user has a numerical
51. ular value say x for example then the variance of the output Y surely decreases because the variable X is not randome anymore We can then measure the conditionnal variance given Xi denoted by V Y X x Since X is a random variable the conditionnal variance V Y X z is a random variable We are interested in the average value of this variance that is we are interested in E V Y X If X has a large weight in the variance V Y then E V Y X is small The theorem of the total variance states that if V Y is finite then V Y V E Y X E V Y X 2 12 If X has a large weight in the variance V Y then V E Y X is large The first order sensivity indice of Y to the variable X is defined by V E Y X S A 2 13 VY 2 13 fori 1 2 p The sensitivity indice measures the part of the variance which is caused by the uncertainty in X 11 We can compute the sensitivity indice when the function f is linear Assume that the output Y depends linearily on the input X Y fo gt Bi Xi 2 14 i 1 2 p where 5 R for i 0 1 2 p Then E Y X Bo yn GE A 2 15 since the expection of a sum is the sum of expectations Then V E Y Xi V BX 2 16 BV X 2 17 since the variance of a constant term is zero Therefore the sensitivity index of Y to the variable Xi is _Biv X S TU 2 18 for i 1 2 p 2 5 Examples In this section we pre
52. urces In order to install the toolbox from the sources a compiler is required to be installed on the machine This toolbox can be used with Scilab v5 1 and Scilab v5 2 We suppose that the archive has been unpacked in the tbxnisp directory The following is a short list of the steps which are reguired to setup the toolbox 1 build the toolbox run the tbrnisp builder sce script to create the binaries of the library create the binaries for the gateway generate the documentation 2 load the toolbox run the tbrnisp load sce script to load all commands and setup the documentation 17 3 setup the startup configuration file of your Scilab system so that the toolbox is known at startup see below for details 4 run the unit tests run the tbrnisp runtests sce script to perform all unit tests and check that the toolbox is OK 5 run the demos run the tbrnisp rundemos sce script to run all demonstration scripts and get a guick interactive overview of its features The following script presents the messages which are generated when the builder of the toolbox is launched The builder script performs the following steps e compile the NISP C library e compile the C gateway library the glue between the library and Scilab e generate the Java help files from the xml files e gencrate the loader script gt exec C tbxnisp builder sce Building sources Generate a loader file Generate a Makefile Running the Makefil
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