Minimizing the Number of Function Evaluations to Estimate Sobol Indices Using Quasi-Monte Carlo

Transcription

1 Minimizing the Number of Function Evaluations to Estimate Sobol Indices Using Quasi-Monte Carlo Lluís Antoni Jiménez Rugama Joint work with: Fred J Hickernell (IIT), Clémentine Prieur (Univ Grenoble), Elise Arnaud (Univ Grenoble), Hervé Monod (INRA), and Laurent Gilquin (Univ Grenoble) Room 120, Bldg E1, Department of Applied Mathematics Illinois Institute of Technology, Chicago, IL ljimene1@hawkiitedu Thursday 26 th May, 2016 ljimene1@hawkiitedu SRC / 21

2 Outline Introduction ANOVA Sobol Indices Quasi-Monte Carlo Methods Replicated Method SRC / 21

3 Outline Introduction ANOVA The ANalysis Of VAriance decomposition Sobol Indices Quasi-Monte Carlo Methods Replicated Method SRC / 21

4 ANOVA For f P L 2 `r0, 1s d, and D t1,, du, fpxq ÿ f u pxq, f µ, uďd where, ż f u pxq fpxqdx u ÿ f v pxq r0,1s d u vău u the cardinality of u u : u c Dzu ljimene1@hawkiitedu SRC / 21

5 Variance Decomposition Under the previous definitions, σ 2 0, σu 2 ż f u pxq 2 dx, r0,1s d σ 2 ż pfpxq µq 2 dx r0,1s d The ANOVA identity is, σ 2 ÿ σu 2 uďd ljimene1@hawkiitedu SRC / 21

6 Outline Introduction ANOVA Sobol Indices Measuring the importance of each input Quasi-Monte Carlo Methods Replicated Method SRC / 21

7 Sobol Indices Sobol introduced the global sensitivity indices which measure the variance explained by any dimension subset u P D: τ 2 u ÿ σv 2, and τ 2 u ÿ σv 2 vďu vxu v PD v PD We have the following properties, τ 2 u ď τ 2 u τ 2 u ` τ 2 u σ 2 ljimene1@hawkiitedu SRC / 21

8 Sobol Indices - Probabilistic Framework For X Ur0, 1s d, Sobol indices can also be presented in the following form, τ 2 u Var re pfpxq X u qs Var pfpxqq E rvar pfpxq X u qs, τ 2 u Var pfpxqq Var re pfpxq X u qs E rvar pfpxq X u qs ljimene1@hawkiitedu SRC / 21

9 The Normalized Sobol Indices One may also use the normalized definition of the Sobol indices, S tot u S u τ 2 u σ 2 Var re pfpxq X uqs Var pfpxqq τ 2 u Var re pfpxq X uqs 1 σ2 Var pfpxqq satisfying 0 ď S u ď S tot u ď 1 1 E rvar pfpxq X uqs, Var pfpxqq E rvar pfpxq X uqs Var pfpxqq ljimene1@hawkiitedu SRC / 21

10 The Normalized Sobol Indices One may also use the normalized definition of the Sobol indices, S tot u S u τ 2 u σ 2 Var re pfpxq X uqs Var pfpxqq τ 2 u Var re pfpxq X uqs 1 σ2 Var pfpxqq 1 E rvar pfpxq X uqs, Var pfpxqq E rvar pfpxq X uqs Var pfpxqq satisfying 0 ď S u ď Su tot ď 1 More specifically, S u is composed by, $ I p1q & I p1q is a 2d u dim integral S u 1, I p2q `Ip3q 2 where I % p2q is a d dim integral I p3q is a d dim integral Error bounds for S u require more care than error bounds for I pkq ljimene1@hawkiitedu SRC / 21

11 Outline Introduction ANOVA Sobol Indices Quasi-Monte Carlo Methods How can we compute high dimensional integrals efficiently? Replicated Method SRC / 21

12 Why Quasi-Monte Carlo? To estimate S u we need to approximate I p1q, I p2q, and I p3q However, in high dimensions we need a suitable technique: Method Convergence Trapezoidal rule: Opn 2{d q Simpson s rule: Opn 4{d q IID Monte Carlo: Opn 1{2 q Quasi-Monte Carlo: Opn 1`ε q (n: number of data points) ljimene1@hawkiitedu SRC / 21

13 Estimating I p1q, I p2q, and I p3q automatically Given ε a and x ÞÑ fpxq, we want Î such that ż fpxq dx Î px ÞÑ fpxq, ε aq r0,1q d ď ε a, where Î px ÞÑ fpxq, ε a q 1 2 m 1 ÿ 2 m fpz i q, i 0 " * Lattice for some automatic and adaptive choice of m and tz i u 8 i 0 P Digital sequence ljimene1@hawkiitedu SRC / 21

14 Examples of Sequences Shifted rank-1 lattice sequence with generating vector p1, 47q Digitally shifted scrambled Sobol sequence ljimene1@hawkiitedu SRC / 21

15 Outline Introduction ANOVA Sobol Indices Quasi-Monte Carlo Methods Replicated Method Reducing the number of function evaluations to compute first-order indices SRC / 21

16 Normalized First-Order Sobol Indices In this particular case, we consider u 1 and want to estimate S u σ 2 u{σ 2 For this purpose, given x, x 1 P r0, 1s d, we define the following point, px u : x 1 uq : px 1 1,, x 1 u 1, x u, x 1 u`1,, x 1 d q P r0, 1sd Thus, one can use the following integral form to build an estimator: S u 1 ş r0,1q 2d 1 g upx,x 1 q: gpx,x 1 q: hkikj hkkkkkkikkkkkkj fpxq pfpxq fpx u : x 1 uqqdxdx1 u ş ş 2 Hpg, g u q r0,1s fpxq 2 dx d r0,1s fpxqdx d ljimene1@hawkiitedu SRC / 21

17 Number of Function Evaluations We will focus on reducing the number of function evaluations, and to estimate σ 2 u{σ 2, only g and g u are evaluated Computing all the indices one by one, if one requires n points for each estimation, the total number of function evaluations of g and g u are 2dn, However, if all indices are computed together, g only needs to be evaluated once Therefore, the number of function evaluations becomes p1 ` dqn, Finally, under a special set of quasi-monte Carlo sequences, this number is decreased to 2n ljimene1@hawkiitedu SRC / 21

18 Replicated Designs Functions g and g u only share input dimension u: gpx, x 1 q fpx 1,, x u 1, x u, x u`1,, x d q, g u px, x 1 q fpx 1 1,, x 1 u 1, x u, x 1 u`1,, x 1 d q Hence, we can construct our points x 1 i as follows, x 0,1 x 0,d x 1 0,1 x1 0,d x π1 p0q,1 x πd p0q,d x n,1 x n,d, x 1 n,1 x1 n,d x π1 pnq,1 x πd pnq,d ljimene1@hawkiitedu SRC / 21

19 The Right Function Values Given the right order of points: x 1 π 1 u p0q x u p0q,1 0,u x 1 π 1 u x 1 x π 1 u pnq,1 n,u x 1 π 1 u x 1 π 1 u pnq x 1 π 1 Therefore, we only need to evaluate g u px, x 1 q once: fpx 1 0 q y 0 g u px 0, x 1 0 q fpx 1 nq y ṇ ùñ g u px n, x 1 nq p0q,d pnq,d y π 1 u p0q y π 1 u pnq ljimene1@hawkiitedu SRC / 21

20 Conclusions We can study how each dimension explains the overall variance of a model using Sobol Indices Our quasi-monte Carlo automatic cubatures can be adapted to estimate these indices automatically First-order Sobol Indices can be estimated using only 2n quasi-monte Carlo function evaluations (not depending on d) ljimene1@hawkiitedu SRC / 21

21 References References I Hickernell, F J and Ll A Jiménez Rugama Reliable adaptive cubature using digital sequences, Monte Carlo and quasi-monte Carlo methods 2014 to appear, arxiv: [mathna] IM, Sobol 2001 Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates, Mathematics and Computers in Simulation (MATCOM) 55, no 1, Joe, Stephen and Frances Y Kuo 2008 Constructing sobol sequences with better two-dimensional projections, SIAM J Scientific Computing 30, no 5, M D McKay, W J Conover, R J Beckman 1979 A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics 21, no 2, Niederreiter, H 1992 Random number generation and quasi-monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia Owen, A B 1998 Scrambling Sobol and Niederreiter-Xing points, J Complexity 14, ljimene1@hawkiitedu SRC / 21

22 References References II Owen, Art B 2013 Variance components and generalized Sobol indices, SIAM/ASA Journal on Uncertainty Quantification 1, no 1, Saltelli, A 2002 Making best use of model evaluations to compute sensitivity indices, Computer Physics Communications 145, Sobol, I M 1967 The distribution of points in a cube and the approximate evaluation of integrals, USSR Comput Math and Math Phys 7, Tissot, Jean-Yves and Clémentine Prieur 2014 A randomized Orthogonal Array-based procedure for the estimation of first- and second-order Sobol indices, Journal of Statistical Computation and Simulation, 1 24 ljimene1@hawkiitedu SRC / 21