Generalized Sobol indices for dependent variables

Transcription

1 Generalized Sobol indices for dependent variables Gaelle Chastaing Fabrice Gamboa Clémentine Prieur July 1st, /28 Gaelle Chastaing Sensitivity analysis and dependent variables

2 Contents 1 Context and motivation 2 Generalized functional decomposition 3 Generalized sensitivity indices 4 Procedure of estimation 5 Numerical example 2/28 Gaelle Chastaing Sensitivity analysis and dependent variables

3 1 Context and motivation Contents 2 Generalized functional decomposition Construction Illustration 3 Generalized sensitivity indices 4 Procedure of estimation Construction of the approximation spaces Least-squared estimation Estimation of sensitivity indices 5 Numerical example 3/28 Gaelle Chastaing Sensitivity analysis and dependent variables

4 Mathematical framework Disturbance Input variables X = (X 1,,X p ) Model η(x) Output Y Sensitivity analysis X is a random vector of distribution P X, not necessarily independent, η L 2 R (Rp ) i.e., for all h 1,h 2 L 2 R (Rp ), h 1,h 2 = h 1 (x)h 2 (x)dp X (x) = E[h 1 (X)h 2 (X)] R p Y is in value in R, and V (Y ) 0. 4/28 Gaelle Chastaing Sensitivity analysis and dependent variables

5 Use of functional decomposition When X independent, the Hoeffding decomposition gives p Y = η(x) = η + η i (X i ) + + η 1,,p (X) (1) i=1 (1) exists and is unique iff E[η u (X u )η v (X v )] = 0 u, v {1,, p}, u v with X u = (X u1,, X ut ) if u = (u 1,, u t ) {1,, p} 5/28 Gaelle Chastaing Sensitivity analysis and dependent variables

6 Moreover η = E(Y ) When X independent, Then, The Sobol index V (Y ) = i V [η i (X i )]+ i<j V [η ij (X i, X j )]+ +V [η 1,,p (X)] The Sobol index of a group of variables Xu is S u = V (η u) V (Y ) η i = E(Y X i ) η η ij = E(Y X i, X j ) η i η j η S u = V [E(Y X u)] v u ( 1) u v V [E(Y X v )] V (Y ) 6/28 Gaelle Chastaing Sensitivity analysis and dependent variables

7 Functional decomposition When X no more independent, The usual approach [XG08, LRY + 10, Can12] is Y u {1,,p} η u (X u ) + ε, max u = d 2, 3 Use of surrogate models to approximate each components (splines, polynomial chaos,...); ANOVA decomposition V (Y ) u d Sensitivity index [ V (η u (X u )) + v ucov(η ] u (X u ), η v (X v )) S u = V (η u(x u )) + v u Cov(η u(x u ), η v (X v )) V (Y ) Degree of implication of meta models? 7/28 Gaelle Chastaing Sensitivity analysis and dependent variables

8 1 Context and motivation Contents 2 Generalized functional decomposition Construction Illustration 3 Generalized sensitivity indices 4 Procedure of estimation Construction of the approximation spaces Least-squared estimation Estimation of sensitivity indices 5 Numerical example 8/28 Gaelle Chastaing Sensitivity analysis and dependent variables

9 Generalized functional decomposition Our objective : Propose a decomposition for the initial model η ; Ensure the existence and uniqueness of such decomposition Context Y = η(x), X P X p X the density function of X p Xu et p X u marginal densities of X u and X u = X \ X u 9/28 Gaelle Chastaing Sensitivity analysis and dependent variables

10 Generalized functional decomposition Condition 0 < M 1, u {1,..,p}, p X M p Xu p X u (C.1) Theorem Under (C.1), we have η(x) = η + i η i (X i ) + i<j η ij (X i,x j ) + + η 1,,p (X) Moreover, this decomposition is unique iff η u Hu, 0 with Hu 0 = { h u (X u ), E[h u (X u )h v (X v )] = 0, v u,h v Hv 0 } 0/28 Gaelle Chastaing Sensitivity analysis and dependent variables

11 Illustration with p = 3 inputs Functional decomposition η(x) = η + η 1 (X 1 ) + η 2 (X 2 ) + η 3 (X 3 ) + η 12 (X 1,X 2 ) +η 13 (X 1,X 3 ) + η 23 (X 2,X 3 ) + η 123 (X) Orthogonality relation representation for P X = P X1 P Xp (Hoeffding decomposition) For general P X (Generalized decomposition) 11/28 Gaelle Chastaing Sensitivity analysis and dependent variables

12 Illustration Hoeffding decomposition η 123 Generalized decomposition η 123 η 12 η 23 η 13 η 12 η 23 η 13 η 1 η 2 η 3 η 1 η 2 η 3 η η 12/28 Gaelle Chastaing Sensitivity analysis and dependent variables

13 1 Context and motivation Contents 2 Generalized functional decomposition Construction Illustration 3 Generalized sensitivity indices 4 Procedure of estimation Construction of the approximation spaces Least-squared estimation Estimation of sensitivity indices 5 Numerical example 13/28 Gaelle Chastaing Sensitivity analysis and dependent variables

14 Generalized Sobol sensitivity indices We have Y = η(x) = u η u(x u ), E[η u (X u )η v (X v )] = 0 whether u v or v u As a consequence, V (Y ) = u [V (η u ) + u v u,v Sensitivity index for the group X u is Cov(η u,η v )] S u = V (η u(x u )) + u v u,v Cov(η u(x u ), η v (X v )) V (Y ) and S u = 1 u 14/28 Gaelle Chastaing Sensitivity analysis and dependent variables

15 1 Context and motivation Contents 2 Generalized functional decomposition Construction Illustration 3 Generalized sensitivity indices 4 Procedure of estimation Construction of the approximation spaces Least-squared estimation Estimation of sensitivity indices 5 Numerical example 15/28 Gaelle Chastaing Sensitivity analysis and dependent variables

16 Procedure Remind that Y = u η u(x u ) with Idea : η u Hu 0 := {h u (X u ), E[h u (X u )h v (X v )] = 0, v u} Hu 0 Hv, 0 v u 1 Construct H 0,L u = Span { φ u 1,,φu L u } H 0 u, u; 2 Estimate components (η u ) u by the LSE (η u ) u Arg min η u H 0 u E[(Y u {1,,p} η u (X u )) 2 ]; 3 Estimate empirically the generalized sensitivity indices S u = V (η u(x u )) + u v u,v Cov(η u(x u ),η v (X v )). V (Y ) 6/28 Gaelle Chastaing Sensitivity analysis and dependent variables

17 Step 1 : Construction of the approximation spaces H 0 u = { h u (X u ), E[h u (X u )h v (X v )] = 0, v u,h v H 0 v}, u Definition of the finite approximation spaces H 0,L u use of truncated orthonormal basis (φ i l i ) L l i =0 of L 2 (R, B(R),P Xi ), i = 1,,p; inductive procedure on u ; exploitation of extended basis to reconstitute the hierarchical orthogonality. H 0 u, u by 17/28 Gaelle Chastaing Sensitivity analysis and dependent variables

18 Procedure 1 Initialization : for any i {1,,p}, construct Proceed as H 0,L i H 0 i := {h i (X i ), E[h i (X i )] = 0} choose a truncated orthonormal basis (φ i li ) L l of i=0 L 2 (R, B(R), P Xi ) with φ i 0 = 1. Then, Set Finally, H 0,L i E(φ i l i ) = 0, l i = 1,, L = Span { φ i 1,,φ i } L L i h i (X i ) = βl i i φ i l i (X i ), and E[h i (X i )] = 0 l i =1 8/28 Gaelle Chastaing Sensitivity analysis and dependent variables

19 Procedure(2) 2 For any i j {1,,p}, construct H 0,L ij Hij 0 := {h ij (X i,x j ), E[h ij h i ] = E[h ij h j ] = E[h ij ] = 0} Proceed recursively as set L L φ ij l ij (X i, X j ) = φ i l i φ j l j (X i, X j )+ λ i kφ i k(x i )+ λ j k φj k (X j)+c k=1 k=1 compute the coefficients (C, λ i 1,, λ i L, λj 1,, λj L ) by solving Set H 0,L ij = Span E[φ ij l ij φ i k ] = 0, k {1,, L} E[φ ij l ij φ j k ] = 0, k {1,, L} E[φ ij l ij ] = 0 { φ ij 1, φij L ij }, L ij = L L. 3 The same idea for any u = k 9/28 Gaelle Chastaing Sensitivity analysis and dependent variables (2)

20 Suppose that Construction in practice we get a i.i.d. n-sample (y l,x l ) l=1,,n define the empirical expected value E n ( ) as n L E n (g) = 1 n n g(x l ) The same procedure can be applied by replacing E( ) by E n ( ). Finally, H 0,L u,n = Span {ˆφu 1,n,, ˆφ } u L u,n, u l=1 0/28 Gaelle Chastaing Sensitivity analysis and dependent variables

21 We get Step 2 : Least-squared estimation η u (X u ) L u l u=1 β u l u ˆφu lu,n (X u), u Estimate components (η u ) from the minimization min β u lu 1 n n s=1[y l u L u l u=1 The total number of components (β u l u ) is pl + ( ) p L Unfeasible scheme for large p! β u l u ˆφu lu,n (xl u )]2 ( ) p L /28 Gaelle Chastaing Sensitivity analysis and dependent variables

22 Solution Order of ANOVA d p, i.e. Y η u (X u ) u d Use of a penalization to select a small number of (ˆφ u l u,n ), min β u lu 1 n n [y l s=1 L u u d l u=1 β u l u ˆφu lu,n (xl u)] 2 + λj(β 1 1,,β u l u, ) J(β) = β 0 := j 1(β j 0) by greedy algorithm ; Discrete process ; sparse selection from a dictionary ; J(β) = β 1 := j β j by LARS algorithm. Convex relaxation of the l 0 penalty; build up the regression function by successive steps; 2/28 Gaelle Chastaing Sensitivity analysis and dependent variables

23 Step 3 : Estimation of sensitivity indices We get ˆη u = L u l u=1 ˆβ u l u ˆφu lu,n, u, u d Empirical estimation of sensitivity indices S u with Ŝ u = V (ˆη u (X u )) + u v u,v Ĉov(ˆη u(x u ), ˆη v (X v )). V (Y ) Ĉov(ˆη u (X u ), ˆη v (X v )) = 1 n V (Y ) = 1 n n ˆη u (x l u)ˆη v (x l v), u,v l=1 n (y l ȳ) 2, ȳ = 1 n l=1 23/28 Gaelle Chastaing Sensitivity analysis and dependent variables n l=1 y l

24 1 Context and motivation Contents 2 Generalized functional decomposition Construction Illustration 3 Generalized sensitivity indices 4 Procedure of estimation Construction of the approximation spaces Least-squared estimation Estimation of sensitivity indices 5 Numerical example 24/28 Gaelle Chastaing Sensitivity analysis and dependent variables

25 River flood inundation Aim : prevent from inundation The maximal overflow is modelized by eight parameters Why randomness and uncertainty? spatio-temporal variability ; errors of measurements. 25/28 Gaelle Chastaing Sensitivity analysis and dependent variables

26 River flood inundation Test with n = 500 observations, d = 2; pairwise correlations ; Hermite basis of degree 10, i.e. P = 2880 parameters Greedy algorithm LARS algorithm Quantiles Quantiles Q K s Z v Z m C b H d L B Sensitivity indices Q K s Z v Z m C b H d L B 26/28 Gaelle Chastaing Sensitivity analysis and dependent variables Sensitivity indices

27 Thank you for your attention! 27/28 Gaelle Chastaing Sensitivity analysis and dependent variables

28 Y. Caniou. Analyse de sensibilité globale pour les modèles imbriqués et multiéchelles. PhD thesis, Université Blaise Pascal - Clermont II, G. Chastaing, F. Gamboa, and C. Prieur. Generalized hoeffding-sobol decomposition for dependent variables -Application to sensitivity analysis. Electronic Journal of Statistics, 6 : , G. Chastaing, F. Gamboa, and C. Prieur. Generalized sobol sensitivity indices for dependent variables : Numerical methods. Available at http ://arxiv.org/abs/ , G. Li, H. Rabitz, P.E. Yelvington, O. Oluwole, F. Bacon, Kolb C.E., and J. Schoendorf. Global sensitivity analysis with independent and/or correlated inputs. Journal of Physical Chemistry A, 114 : , I.M. Sobol. Sensitivity estimates for nonlinear mathematical models. Mathematical Modeling and Computational Experiment, 1(4) : , C. Xu and G. Gertner. Uncertainty and sensitivity analysis for models with correlated parameters. Reliability Engineering & System Safety, 93 : , /28 Gaelle Chastaing Sensitivity analysis and dependent variables