[Paper Review] Nonparametric adaptive estimation of order 1 Sobol indices in stochastic models, with an application to Epidemiology

Viet Chi Tran, Gwénaëlle Castellan+2 more|arXiv (Cornell University)|Nov 22, 2016

Probabilistic and Robust Engineering Design32 references4 citations

TL;DR

This paper proposes a nonparametric adaptive estimator for first-order Sobol indices in stochastic models using warped wavelets, eliminating the need for metamodels. The method achieves optimal convergence rates dependent on the model's regularity, demonstrating an elbow effect in mean squared error as the sample size increases, with application to Hepatitis C transmission modeling in epidemiology.

ABSTRACT

Global sensitivity analysis is a set of methods aiming at quantifying the contribution of an uncertain input parameter of the model (or combination of parameters) on the variability of the response. We consider here the estimation of the Sobol indices of order 1 which are commonly-used indicators based on a decomposition of the output's variance. In a deterministic framework, when the same inputs always give the same outputs, these indices are usually estimated by replicated simulations of the model. In a stochastic framework, when the response given a set of input parameters is not unique due to randomness in the model, metamodels are often used to approximate the mean and dispersion of the response by deterministic functions. We propose a new non-parametric estimator without the need of defining a metamodel to estimate the Sobol indices of order 1. The estimator is based on warped wavelets and is adaptive in the regularity of the model. The convergence of the mean square error to zero, when the number of simulations of the model tend to infinity, is computed and an elbow effect is shown, depending on the regularity of the model. Applications in Epidemiology are carried to illustrate the use of non-parametric estimators.

Motivation & Objective

• To develop a nonparametric, adaptive estimator for first-order Sobol indices in stochastic models where output variability arises from internal randomness.
• To eliminate reliance on metamodels for estimating mean and variance of stochastic responses.
• To establish convergence rates for the estimator that adapt to the unknown regularity of the underlying conditional expectation function.
• To demonstrate the method's efficiency and robustness through application to an SIR model of Hepatitis C transmission among people who inject drugs (PWID).

Proposed method

• Employs warped wavelet expansions to nonparametrically estimate the conditional expectation E[Y|Xℓ], which is central to computing first-order Sobol indices.
• Uses a model selection procedure based on penalized empirical risk minimization to adapt to the unknown smoothness of the function hℓ(xℓ) = E[Y|Xℓ].
• Applies a data-driven penalty term pen(J) to balance bias and variance in the wavelet coefficient estimation.
• Derives convergence rates for the mean squared error of the estimator by decomposing the error into approximation, estimation, and model selection components.
• Uses Bernstein-type inequalities to control the deviation of empirical wavelet coefficients from their true values.
• Establishes that the estimator adapts to the regularity of the function hℓ, achieving optimal rates up to logarithmic factors.

Experimental results

Research questions

• RQ1 Can a nonparametric, adaptive estimator be developed for first-order Sobol indices in stochastic models without requiring a metamodel?
• RQ2 How does the convergence rate of the proposed estimator depend on the regularity of the underlying function hℓ(xℓ) = E[Y|Xℓ]?
• RQ3 Does the estimator achieve optimal convergence rates in terms of sample size n, adapting to unknown smoothness?
• RQ4 What is the impact of model selection on the estimator's performance, and how is the penalty term calibrated?
• RQ5 How does the method compare to existing approaches in terms of computational cost and accuracy in stochastic settings?

Key findings

• The proposed estimator achieves a mean squared error of order O(log²(n)/n³/²) under regularity conditions, with convergence rates that adapt to the unknown smoothness of the conditional expectation function.
• An elbow effect is observed in the convergence rate: when the function hℓ belongs to a Hölder ball B(α, 2, ∞), the optimal rate is n⁻⁸ᵃ⁄⁽⁴ᵃ⁺¹⁾, which transitions from n⁻¹ to faster rates as α increases.
• The estimator's performance adapts to the regularity of the model, achieving optimal rates up to logarithmic factors, without requiring prior knowledge of smoothness.
• The method significantly reduces the number of model simulations required compared to classical Monte Carlo estimators, which require n(p+1) calls.
• Theoretical bounds show that the model selection penalty term pen(J) is sufficient to control overfitting and ensure consistency.
• The method is validated on an SIR model of Hepatitis C transmission among PWID, demonstrating practical applicability in real-world epidemiological modeling.