A new estimator for quantile-oriented sensitivity indices

Transcription

1 A new estimator for quantile-oriented sensitivity indices Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) EDF R&D-MRI Chatou - Université Paris 5 April 19th, 2016, Les Houches 1 / 21

2 Sensitivity Analysis - Introduction Numerical code g. Random inputs (X 1,..., X d ) (f 1,..., f d ) iid. Random output Y R such that Y = g (X 1,..., X d ). Main goal : for i {1,..., d}, how does Xi s uncertainty propagate through g? 2 / 21

3 Sensitivity analysis - Schema 3 / 21

4 Sensitivity analysis Several potential uses : Better understanding of the model, Neglect X i s distribution if not influential, Feedback on the inputs - reducing X i s distribution if too much influential. Global analysis : most relevant way? 4 / 21

5 Goal-oriented sensitivity analysis In practice, Y s distribution does not need to be fully known. Choice of a probability feature θ(y ) (mean, quantiles etc... ) which may be relevant. Goal-oriented sensitivity analysis (GOSA) [N. Rachdi, 2011] : For i {1,..., d}, quantification of X i s influence over θ(y ). 5 / 21

6 GOSA One by one strategy : condition the code g by X i and compute θ(y X i ) Set x i realization of X i g (X 1,..., x i,..., X d ) θ(y X i = x i ), Condition g by all the possible values x i, regarding f i : θ(y X i ) s distribution, random variable function of X i. 6 / 21

7 GOSA - Schema Respective influences of each input over θ(y) Random inputs Conditional simulation model : f(xlxi) θ(ylxi) pdf s θ(ylx1) X1 Runs of f(x l X1) X2 Runs of f(x l X2) θ(ylx2) X3 Runs of f(x l X3) θ(ylx3) 7 / 21

8 Contrast functions Use contrast functions to quantify θ(y X i ) s variability. Simple contrasts : (y, θ) R 2 ϕ(y, θ) 0 quantify a "distance" between two real components. Mean contrasts : for Y r.r.v. φ Y (θ) = E Y [ϕ(y, θ)]. Y s feature : θ(y ) := arg min θ R φ Y (θ). 8 / 21

9 Contrast functions : mean and quantiles If ϕ(y, θ) = m(y, θ) = y θ 2 : therefore φ Y (θ) = E Y [ Y θ 2 ], θ(y ) = E[Y ]. If, for α ]0; 1[, ϕ(y, θ) = c α(y, θ) = (y θ)(α 1 y θ ) : therefore φ Y (θ) = E Y [(Y θ)(α 1 Y θ )], θ(y ) = q α (Y ), α-quantile de Y. We focus on ϕ = c α : θ(y ) = q α (Y ). N.B. : min θ φ (θ X i = x i ) = E [c α (Y, q α (Y X i )) X i = x i ]. 9 / 21

10 Sensitivity analysis with respect to a contrast Need to quantify the variability of θ(y X i ). Sensitivity indices based on contrasts [Fort et al., 2013] [ ] Sϕ Xi (Y ) = min φ Y (θ) E min φ Y (θ X i ). θ R θ R quantifies the influence of the input X i on θ(y ). S Xi ϕ (Y ) 0. We divide S Xi ϕ (Y ) by min θ R φ Y (θ) so that 0 S Xi ϕ (Y ) 1. S i c α (Y ) = 0 θ(y X i ) = θ(y ) a.s. S i c α (Y ) = 1 (Y X i = x i ) = constant(x i ) a.s. 10 / 21

11 Sensitivity analysis with respect to a contrast s alpha Y = X 1 + X 2 with X 1 Exp(1) and X 2 Exp(1) independent. S X 1 m = S X 2 m = 0.5 (Sobol indices). Both inputs are influential on the mean E[Y ]! S X 1 c α : X 1 s influence on Y s α-quantile. S X 2 c α : X 2 s influence on Y s α-quantile. Sensitivity changes regarding the level of quantile α. 11 / 21

12 Estimation of the quantile-oriented index Goal : from a n-sample (X 1 i, Y 1 ),..., (X n S Xi c α (Y ) = min φ Y (θ) E θ i, Y n ), estimation of ]. [ min θ R φ Y (θ X i ) ] = E [c α(y, q α (Y ))] E Xi [min E [cα(y, θ) X i]. θ R 1st term estimation : 1 min θ n n c α(y j, θ) = 1 n j=1 n j=1 ) c α (Y j, ˆq α (Y ), where ˆq α (Y ) is the classical empirical quantile estimator this estimator converges a.s. 12 / 21

13 Estimation for the second term ] Second term : E Xi [min E [cα(y, θ) X i]. θ R Several issues : -Double expectation -Conditional expectation -Minimization problem. We use the following asymptotic result [Fan et al., 1994], for x i any possible realization of X i : arg min θ 1 f i (x i ) n j=1 ) ( ) c α (Y j, θ K h(n) X j i x i P arg min E [c n α(y, θ) X i = x i ], θ where f i is the pdf of X i with a compact support, K a 2-order positive kernel and (h(n)) n N a bandwidth sequence such that h(n) 0 while n h(n). 13 / 21

14 Estimation for the second term We define the estimator as : ˆV n = 1 n n k=1 1 k min 1 θ f i (Xi k ) k j=1 ) ( ) c α (Y j, θ K h(k) X j i Xi k Useful points : ( -As θ k ( j=1 cα Y j, θ ) ( )) K h(k) X j i Xi k is a piecewise linear function whose angles are the Y 1,..., Y k its minimizer is among Y 1,..., Y k. - ˆV n is built recursively, ie if we know it, we also know V 1,..., V n 1. We prove : ˆV n [ ] P E X n i min E [c α(y, θ) X i ]. θ 14 / 21

15 Numerical experiments S X Influences of the inputs over the output quantiles size of sample Inputs X1 X2 X3 Defect detection : wave control through a structure to study. Sensitivity analysis over the random defect a 90, function of the inputs X, which we detect with a probability of 90%. Influence of the inputs over q 0.25 (a 90 ). 3 random inputs : -X 1 : the thickness of the structure -X 2 : the angle of the control -X 3 : the depth of the defect. 15 / 21

16 Conclusions Relevant information for the sensitivity analysis - useful alternative to Sobol indices! Estimator not so expensive to compute regarding classical estimators in SA. Convergence criterion for ˆV n? Perspective : extension to SA over random cumulative distribution functions (ouch!) 16 / 21

17 Sketch of proof for the consistency We define a parallel estimator, V n, by substituting the minimum, for each k {1,..., n}, by : 1 f i (X k i ) k j=1 ( ( c α Y j, q α Y X k)) ( ) K h(k) X j i Xi k. As we express the increment of (V n) n N, we get : n N V n V n 1 1/n = (V V n 1 )+ε(n), where 1/n is the time-step and ε(n) is a small enough" residual. Let us define a real function l that interpolates (V n) n N such that : n N l( n k=1 1/k) = Vn. Then : l( n 1 k=1 ) l( n 1 1 k k=1 ) ( ( n 1 )) k V 1 l. 1/n + k k=1 17 / 21

18 Sketch of proof for the consistency Under the right conditions, the Kushner-Clark theorem [3] states that, with a probability of 1, g behaves asymptotically like a solution of the associated ODE* : l = V l lim l(t) = V a.s., since V is the limit of every solution of ODE*. t + P This leads to : V n V. n By using : n N ˆV ] n V n and proving E [ V n Ṽn 0 n = ˆV P n V. n 18 / 21

19 N. Rachdi Statistical Learning and Computer Experiments PhD thesis, Université Paul Sabatier, France, J. C. Fort, T. Klein, N. Rachdi New sensitivity analysis subordinated to a contrast Communication in Statistics : Theory and Methods, In press, J. Fan, T. Hu and Y. K. Truong Robust Non-Parametric Function Estimation Scandinavian Journal of Statistics,Vol. 21, No. 4, pp , / 21

20 H. J. Kushner, D. S. Clark Stochastic Approximations for Constrained and Unconstrained Systems Springer, Berlin, / 21

21 Les Houches, c est bien. 21 / 21