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A new kernel-based index for the global sensitivity analysis of models with correlated inputs

Troy Larsen, Alen Alexanderian

Abstract

We present an HSIC-based approach for global sensitivity analysis of broad classes of models with correlated and possibly function-valued inputs and outputs. To this end, we define the total HSIC sensitivity index: a bounded, interpretable, and moment-independent analogue to the total-effect Sobol' index. These desirable qualities hinge upon the key property of monotonicity under marginalization for the HSIC. We rigorously establish this monotonicity property by using a suitable class of augmented kernels. Furthermore, we provide an efficient algorithm for computing an empirical estimator of the HSIC that significantly reduces computational complexity and storage requirements. The effectiveness and interpretability of the total HSIC sensitivity indices are demonstrated through computational experiments on models that feature nonlinear relationships, correlated inputs, and functional outputs.

A new kernel-based index for the global sensitivity analysis of models with correlated inputs

Abstract

We present an HSIC-based approach for global sensitivity analysis of broad classes of models with correlated and possibly function-valued inputs and outputs. To this end, we define the total HSIC sensitivity index: a bounded, interpretable, and moment-independent analogue to the total-effect Sobol' index. These desirable qualities hinge upon the key property of monotonicity under marginalization for the HSIC. We rigorously establish this monotonicity property by using a suitable class of augmented kernels. Furthermore, we provide an efficient algorithm for computing an empirical estimator of the HSIC that significantly reduces computational complexity and storage requirements. The effectiveness and interpretability of the total HSIC sensitivity indices are demonstrated through computational experiments on models that feature nonlinear relationships, correlated inputs, and functional outputs.
Paper Structure (16 sections, 11 theorems, 46 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 11 theorems, 46 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Reproducing kernel Hilbert spaces
  4. RKHS embeddings of probability measures
  5. Problem setup and the HSIC
  6. Monotonicity of the HSIC under marginalization
  7. Augmented kernel construction
  8. Ensuring monotonicity
  9. The total HSIC sensitivity index
  10. Defining the total HSIC sensitivity index
  11. Computing the total HSIC sensitivity index
  12. Numerical Experiments
  13. Ishigami function
  14. Correlated portfolio model
  15. Function-valued cholera model
  16. ...and 1 more sections

Key Result

Theorem 2.2

For every kernel $k: D\times D\to \mathbb{R}$, there is a unique Hilbert space $\mathcal{H}_k$ of real-valued functions on $D$ so that $\mathcal{H}_k$ is an RKHS with $k$ as its reproducing kernel.

Figures (9)

  • Figure 1: A depiction of the problem setup.
  • Figure 1: Estimated total HSIC sensitivity indices as functions of sample size $n$.
  • Figure 2: A comparison of sensitivity indices for the Ishigami function.
  • Figure 3: Sensitivity indices for the correlated portfolio model as functions of $\rho$.
  • Figure 4: Model reduction informed by Figure \ref{['fig:corr']} at different values of $\rho$.
  • ...and 4 more figures

Theorems & Definitions (21)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5: Szabo18, Theorem 3(i)
  • Definition 2.6
  • Theorem 2.7
  • Lemma 3.1: Durrande11, Proposition 1
  • Definition 3.2
  • Lemma 3.3
  • ...and 11 more