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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
Sensitivity analysis - Schema 3 / 21 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
GOSA - Schema Respective influences of each input over θ (Y) Random Conditional simulation θ (YlXi) pdf’s inputs model : f(XlXi) θ (YlX1) X1 Runs of f(X l X1) θ (YlX2) X2 Runs of f(X l X2) Runs of f(X l X3) X3 θ (YlX3) 7 / 21 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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. : φ ( θ | X i = x i ) = E [ c α ( Y , q α ( Y | X i )) | X i = x i ] . min θ 9 / 21 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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] � � S Xi ϕ ( Y ) = min θ ∈ R φ Y ( θ ) − E min θ ∈ R φ Y ( θ | X i ) . − → quantifies the influence of the input X i on θ ( Y ) . S Xi ϕ ( Y ) ≥ 0. We divide S Xi θ ∈ R φ Y ( θ ) so that 0 ≤ S Xi ϕ ( Y ) by min 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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
Sensitivity analysis with respect to a contrast Y = X 1 + X 2 with X 1 ∼ Exp ( 1 ) and X 2 ∼ − Exp ( 1 ) 0.8 independent. S X 1 m = S X 2 0.6 m = 0 . 5 (Sobol indices). Both inputs are influential on the s1 0.4 mean E [ Y ] ! S X 1 c α : X 1 ’s influence on Y ’s 0.2 α -quantile. S X 2 0.0 0.2 0.4 0.6 0.8 1.0 c α : X 2 ’s influence on Y ’s alpha α -quantile. Sensitivity changes regarding the level of quantile α . 11 / 21 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
Estimation of the quantile-oriented index Goal : from a n -sample ( X 1 i , Y 1 ) , . . . , ( X n i , Y n ) , estimation of � � S Xi c α ( Y ) = min φ Y ( θ ) − E θ ∈ R φ Y ( θ | X i ) . min θ � � = E [ c α ( Y , q α ( Y ))] − E X i min θ ∈ R E [ c α ( Y , θ ) | X i ] . 1st term estimation : n n 1 c α ( Y j , θ ) = 1 � � � � Y j , ˆ q α ( Y ) min c α , n n θ j = 1 j = 1 where ˆ q α ( Y ) is the classical empirical quantile estimator − → this estimator converges a . s . 12 / 21 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
Estimation for the second term � � Second term : E X i min θ ∈ R E [ c α ( Y , θ ) | X i ] . 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 : n 1 � � � � � Y j , θ X j P i − x i − → E [ c α ( Y , θ ) | X i = x i ] , arg min c α K h ( n ) n →∞ arg min f i ( x i ) θ θ j = 1 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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
Estimation for the second term We define the estimator as : n k V n = 1 1 1 � � � � ˆ � � Y j , θ X j i − X k k min c α K h ( k ) i n f i ( X k i ) θ k = 1 j = 1 Useful points : � � �� θ �→ � k Y j , θ X j i − X k � � -As j = 1 c α K h ( k ) is a piecewise linear i function whose “angles” are the Y 1 , ..., Y k → its minimizer is among Y 1 , ..., Y k . − - ˆ V n is built recursively, ie if we know it, we also know V 1 , . . . , V n − 1 . We prove : � � ˆ P V n n →∞ E X i − → min E [ c α ( Y , θ ) | X i ] . θ 14 / 21 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
Numerical experiments Influences of the inputs over the output quantiles Defect detection : wave control 1.0 Inputs through a structure to study. X1 X2 0.8 X3 Sensitivity analysis over the random 0.6 defect a 90 , function of the inputs X , S X which we detect with a probability of 0.4 90 % . 0.2 Influence of the inputs over 0.0 q 0 . 25 ( a 90 ) . 0 100 200 300 400 500 size of sample 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 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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 ˆ V n ? Perspective : extension to SA over random cumulative distribution functions (ouch !) 16 / 21 Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) A new estimator for quantile-oriented sensitivity indices
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