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Regularizing nested Monte Carlo Sobol' index estimators to balance the trade-off between explorations and repetitions in global sensitivity analysis of stochastic models

Henri Mermoz Kouye, Gildas Mazo

TL;DR

A regularization method is introduced to bound the mean squared error of functions of nested Monte Carlo estimators and, based on a heuristic, an allocation strategy that seeks to minimize a bias-variance trade-off is proposed.

Abstract

Sobol' sensitivity index estimators for stochastic models are functions of nested Monte Carlo estimators, which are estimators built from two nested Monte Carlo loops. The outer loop explores the input space and, for each of the explorations, the inner loop repeats model runs to estimate conditional expectations. Although the optimal allocation between explorations and repetitions of one's computational budget is well-known for nested Monte Carlo estimators, it is less clear how to deal with functions of nested Monte Carlo estimators, especially when those functions have unbounded Hessian matrices, as it is the case for Sobol' index estimators. To address this problem, a regularization method is introduced to bound the mean squared error of functions of nested Monte Carlo estimators. Based on a heuristic, an allocation strategy that seeks to minimize a bias-variance trade-off is proposed. The method is applied to Sobol' index estimators for stochastic models. A practical algorithm that adapts to the level of intrinsic randomness in the models is given and illustrated on numerical experiments.

Regularizing nested Monte Carlo Sobol' index estimators to balance the trade-off between explorations and repetitions in global sensitivity analysis of stochastic models

TL;DR

A regularization method is introduced to bound the mean squared error of functions of nested Monte Carlo estimators and, based on a heuristic, an allocation strategy that seeks to minimize a bias-variance trade-off is proposed.

Abstract

Sobol' sensitivity index estimators for stochastic models are functions of nested Monte Carlo estimators, which are estimators built from two nested Monte Carlo loops. The outer loop explores the input space and, for each of the explorations, the inner loop repeats model runs to estimate conditional expectations. Although the optimal allocation between explorations and repetitions of one's computational budget is well-known for nested Monte Carlo estimators, it is less clear how to deal with functions of nested Monte Carlo estimators, especially when those functions have unbounded Hessian matrices, as it is the case for Sobol' index estimators. To address this problem, a regularization method is introduced to bound the mean squared error of functions of nested Monte Carlo estimators. Based on a heuristic, an allocation strategy that seeks to minimize a bias-variance trade-off is proposed. The method is applied to Sobol' index estimators for stochastic models. A practical algorithm that adapts to the level of intrinsic randomness in the models is given and illustrated on numerical experiments.
Paper Structure (14 sections, 8 theorems, 54 equations, 4 figures, 2 algorithms)

This paper contains 14 sections, 8 theorems, 54 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Sensitivity analysis of stochastic models
  3. Nested Monte Carlo estimation
  4. Functions of nested Monte Carlo estimators
  5. A regularization method to address functions with unbounded Hessian matrix
  6. Application to Sobol' index estimators
  7. An algorithm of estimation of Sobol' indices for stochastic models
  8. Algorithms
  9. Illustrations
  10. Conclusion
  11. Proof of Proposition \ref{['thm:basic-inequality']}
  12. Proof of Theorem \ref{['res:main']}
  13. A lemma
  14. The Marcinkiewicz-Zygmund inequality

Key Result

Proposition 1

Under Assumptions ass:conv_b and assumption_3, for every $h\in (0,1)$: where $\limsup_{n,m\to \infty}p_{n,m}(h)=0$ and $V_{n,m}(h):=\mathbb{E}(\nabla g_h\left(\mu_m\right)^{\top} (\widehat{\theta}-\mu_m))^2$ and $B_m(h):=g_h\left(\mu_m\right)-g(\theta)$.

Figures (4)

  • Figure 1: Boxplots of sensitivity index estimates of $X_1$ and $X_2$ with respect to the linear model and for different values of $T$. The standard deviation is $\sigma=1$ and regularization parameter $h=10^{-2}$. Three strategies of choice of $m$ are compared: $m=5$ (in red), $m=m_{opt}$ given by the trade-off strategy of Algorithm \ref{['algo']} (in green) and $m=T^{1/2}$ (in blue). The red dashed lines showed the true sensitivity index values, in particular, in this setting $S_1=0.2$ and $S_2=0.8$
  • Figure 2: Boxplots of sensitivity index estimates of $X_1$ and $X_2$ with respect to the linear model and for different values of $T$. The standard deviation and regularization parameter are $\sigma=5$ and $h=10^{-2}$. Three strategies of choice of $m$ are compared: $m=5$ (in red), $m=m_{opt}$ given by the trade-off strategy of Algorithm \ref{['algo']} (in green) and $m=T^{1/2}$ (in blue). The red dashed lines showed the true sensitivity index values, in particular, in this setting $S_1=0.2$ and $S_2=0.8$
  • Figure 3: Boxplots of sensitivity index estimates of $X_1$, $X_2$ and $X_3$ with respect to the stochastic version of Ishigami function (with $b=0.05$) for different values of $T$. Three strategies of choice of $m$ are compared: $m=5$ (in red), $m=m_{opt}$ given by the trade-off strategy of Algorithm \ref{['algo']} (in green) and $m=T^{1/2}$ (in blue). The red dashed lines showed the true sensitivity index values.
  • Figure 4: Boxplots of sensitivity index estimates of $X_1$, $X_2$ and $X_3$ with respect to the stochastic version of Ishigami function (with $b=0.1$) for different values of $T$. Three strategies of choice of $m$ are compared: $m=5$ (in red), $m=m_{opt}$ given by the trade-off strategy of Algorithm \ref{['algo']} (in green) and $m=T^{1/2}$ (in blue). The red dashed lines showed the true sensitivity index values.

Theorems & Definitions (9)

  • Remark 1
  • Proposition 1
  • Corollary 1
  • Theorem 1
  • Corollary 2
  • Lemma A.1
  • Lemma B.1
  • Lemma C.1
  • Theorem D.1: zygmund