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A new estimator for quantile-oriented sensitivity indices Thomas - - PowerPoint PPT Presentation

Oct 03, 2023 298 likes •519 views

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 - Universit Paris

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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

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Sensitivity Analysis - Introduction

Numerical code g. Random inputs (X1, . . . , Xd) ∼ (f1, . . . , fd) iid . Random output Y ∈ R such that Y = g (X1, . . . , Xd) . 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

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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

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Sensitivity analysis

Several potential uses : − → Better understanding of the model, − → Neglect Xi’s distribution if not influential, − → Feedback on the inputs - reducing Xi’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

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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 Xi’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

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GOSA

One by one strategy : condition the code g by Xi and compute θ(Y | Xi) Set xi realization of Xi − → g (X1, . . . , xi, . . . , Xd) − → θ(Y | Xi = xi), Condition g by all the possible values xi, regarding fi : − → θ(Y | Xi)’s distribution, random variable function of Xi.

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

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GOSA - Schema

Conditional simulation model : f(XlXi) Random inputs θ(YlXi) pdf’s X1 X2 X3

Runs of f(X l X1) Runs of f(X l X2) Runs of f(X l X3)

θ(YlX1) θ(YlX2) θ(YlX3)

Respective influences of each input over θ(Y)

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

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Contrast functions

Use contrast functions to quantify θ(Y | Xi)’s variability. Simple contrasts : ∀(y, θ) ∈ R2 ϕ(y, θ) ≥ 0 quantify a "distance" between two real components. Mean contrasts : for Y r.r.v. φY(θ) = EY[ϕ(Y, θ)]. Y’s feature : θ(Y) := arg minθ∈R φY(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

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Contrast functions : mean and quantiles

If ϕ(y, θ) = m(y, θ) =| y − θ |2 : therefore φY(θ) = EY[| Y − θ |2], − → θ(Y) = E[Y]. If, for α ∈]0; 1[, ϕ(y, θ) = cα(y, θ) = (y − θ)(α − 1y≤θ) : therefore φY(θ) = EY[(Y − θ)(α − 1Y≤θ)], − → θ(Y) = qα(Y), α-quantile de Y. We focus on ϕ = cα : θ(Y) = qα(Y). N.B. : min

θ

φ (θ | Xi = xi) = E [cα (Y, qα(Y | Xi)) | Xi = xi] .

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

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Sensitivity analysis with respect to a contrast

Need to quantify the variability of θ(Y | Xi). Sensitivity indices based on contrasts [Fort et al., 2013] SXi

ϕ (Y) = min θ∈R φY(θ) − E

  • min

θ∈R φY (θ | Xi)

  • .

− → quantifies the influence of the input Xi on θ(Y). SXi

ϕ (Y) ≥ 0.

We divide SXi

ϕ (Y) by min θ∈R φY(θ) so that 0 ≤ SXi ϕ (Y) ≤ 1.

Si

cα(Y) = 0 ⇔ θ(Y | Xi) = θ(Y) a.s.

Si

cα(Y) = 1 ⇔ (Y | Xi = xi) = constant(xi) 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

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Sensitivity analysis with respect to a contrast

0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 alpha s1

Y = X1 + X2 with X1 ∼ Exp(1) and X2 ∼ −Exp(1) independent. SX1

m = SX2 m = 0.5 (Sobol indices).

Both inputs are influential on the mean E[Y] ! SX1

cα : X1’s influence on Y’s

α-quantile. SX2

cα : X2’s influence on Y’s

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

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Estimation of the quantile-oriented index

Goal : from a n-sample (X 1

i , Y 1), . . . , (X n i , Y n), estimation of

SXi

cα(Y)

= min

θ

φY(θ) − E

  • min

θ∈R φY (θ | Xi)

  • .

= E [cα(Y, qα(Y))] − EXi

  • min

θ∈R E [cα(Y, θ) | Xi]

  • .

1st term estimation : min

θ

1 n

n

  • j=1

cα(Y j, θ) = 1 n

n

  • j=1

cα

  • Y j, ˆ

qα(Y)

  • ,

where ˆ qα(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

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Estimation for the second term

Second term : EXi

  • min

θ∈R E [cα(Y, θ) | Xi]

  • .

Several issues :

  • Double expectation
  • Conditional expectation
  • Minimization problem.

We use the following asymptotic result [Fan et al., 1994], for xi any possible realization of Xi : arg min

θ

1 fi(xi)

n

  • j=1

cα

  • Y j, θ
  • Kh(n)
  • X j

i − xi

  • P

− →

n→∞ arg min θ

E [cα(Y, θ) | Xi = xi] , where fi is the pdf of Xi 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

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Estimation for the second term

We define the estimator as : ˆ Vn = 1 n

n

  • k=1

1 k min

θ

1 fi(X k

i ) k

  • j=1

cα

  • Y j, θ
  • Kh(k)
  • X j

i − X k i

  • Useful points :
  • As
  • θ → k

j=1 cα

  • Y j, θ
  • Kh(k)
  • X j

i − X k i

  • is a piecewise linear

function whose “angles” are the Y 1, ..., Y k − → its minimizer is among Y 1, ..., Y k.

  • ˆ

Vn is built recursively, ie if we know it, we also know V1, . . . , Vn−1. We prove : ˆ Vn

P

− →

n→∞ EXi

  • min

θ

E [cα(Y, θ) | Xi]

  • .

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

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Numerical experiments

100 200 300 400 500 0.0 0.2 0.4 0.6 0.8 1.0

Influences of the inputs over the output quantiles

size of sample SX Inputs X1 X2 X3

Defect detection : wave control through a structure to study. Sensitivity analysis over the random defect a90, function of the inputs X, which we detect with a probability of 90%. Influence of the inputs over q0.25(a90). 3 random inputs :

  • X1 : the thickness of the structure
  • X2 : the angle of the control
  • X3 : 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

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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 Vn ? 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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Sketch of proof for the consistency

We define a parallel “estimator”, Vn, by substituting the minimum, for each k ∈ {1, ..., n}, by : 1 fi(X k

i ) k

  • j=1

cα

  • Y j, qα

Y | X k Kh(k)

  • X j

i − X k i

  • .

As we express the increment of (Vn)n∈N, we get : ∀n ∈ N∗ Vn − Vn−1 1/n = (V ∗ − Vn−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 (Vn)n∈N such that : ∀n ∈ N l(n

k=1 1/k) = Vn. Then :

l(n

k=1 1 k ) − l(n−1 k=1 1 k )

1/n ≃

+∞

  • V ∗ − l

n−1

  • k=1

1 k

  • .

17 / 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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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

t→+∞l(t) = V ∗ a.s., since V ∗ is the limit of every solution of ODE*.

This leads to : Vn

P

− →

n→∞ V ∗.

By using : ∀n ∈ N ˆ Vn ≤ Vn and proving E

  • |Vn − ˜

Vn|

  • →

n→∞ 0

= ⇒ ˆ Vn

P

− →

n→∞ V ∗. 18 / 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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  • N. Rachdi

Statistical Learning and Computer Experiments PhD thesis, Université Paul Sabatier, France, 2011.

  • J. C. Fort, T. Klein, N. Rachdi

New sensitivity analysis subordinated to a contrast Communication in Statistics : Theory and Methods, In press, 2013.

  • J. Fan, T. Hu and Y. K. Truong

Robust Non-Parametric Function Estimation Scandinavian Journal of Statistics,Vol. 21, No. 4, pp. 433-446, 1994.

19 / 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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  • H. J. Kushner, D. S. Clark

Stochastic Approximations for Constrained and Unconstrained Systems Springer, Berlin, 1978.

20 / 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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Les Houches, c’est bien.

21 / 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