Kernel-based sensitivity analysis for (excursion) sets

TL;DR

This article proposes to use the Hilbert-Schmidt Independence Criterion with a kernel designed for set-valued outputs to perform sensitivity analysis with set-valued outputs and proposes to use HSIC-ANOVA indices to measure the contribution of each input.

Abstract

In this paper, we aim to perform sensitivity analysis of set-valued models and, in particular, to quantify the impact of uncertain inputs on feasible sets, which are key elements in solving a robust optimization problem under constraints. While most sensitivity analysis methods deal with scalar outputs, this paper introduces a novel approach for performing sensitivity analysis with set-valued outputs. Our innovative methodology is designed for excursion sets, but is versatile enough to be applied to set-valued simulators, including those found in viability fields, or when working with maps like pollutant concentration maps or flood zone maps. We propose to use the Hilbert-Schmidt Independence Criterion (HSIC) with a kernel designed for set-valued outputs. After proposing a probabilistic framework for random sets, a first contribution is the proof that this kernel is characteristic, an essential property in a kernel-based sensitivity analysis context. To measure the contribution of each input, we then propose to use HSIC-ANOVA indices. With these indices, we can identify which inputs should be neglected (screening) and we can rank the others according to their influence (ranking). The estimation of these indices is also adapted to the set-valued outputs. Finally, we test the proposed method on three test cases of excursion sets.

Figures (20)

• Figure 1: Kernel mean embedding
• Figure 2: Excursion set of the constraint $g \leq 0$ for $U_1 \in \{-5,-2.5,0,2.5,5\}$ and $U_2=0$ (first row) and for $U_1 =0$ and $U_2 \in \{-5,-2.5,0,2.5,5\}$. (second row)
• Figure 3: Acceptance rates ($\%$) over $20$ independence tests with a risk of $5\%$
• Figure 4: Estimations of $\hat{\hat{S}}^{\operatorname{H}_{set}}_{i}$ and $\hat{\hat{S}}^{\operatorname{H}_{set}}_{T_i}$
• Figure 6: Acceptance rates ($\%$) over $20$ independence tests with a risk of $5\%$

Theorems & Definitions (22)

• Definition 2.1
• Definition 2.2
• Definition 2.3: Orthogonal and ANOVA kernel
• Theorem 2.1: ANOVA decomposition of $\operatorname{HSIC}$
• Definition 2.4: HSIC-ANOVA indices
• Lemma 3.1
• Definition 3.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3