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A new paradigm for global sensitivity analysis

Gildas Mazo

TL;DR

The paper reframes global sensitivity analysis by introducing sensitivity maps τ that quantify input-set effects via a two-level factorial perspective, removing the strict dependence on functional decompositions and variance-based measures. By allowing arbitrary input distributions and general divergence-based variability measures, it unifies and extends index construction through weighted factorial effects I(B) = ∑_{A⊂D extbackslash B} p_B(A) Δ_B τ(A). It shows how Sobol-like and Shapley-like decompositions arise as special cases through appropriate weight choices, and introduces the dual sensitivity map τ^* to study symmetry and self-duality properties. The framework also clarifies connections to existing indices, offers insights for implementing factorial designs, and broadens the scope of sensitivity analysis to complex, high-dimensional, or nonstandard settings with flexible weighting schemes.

Abstract

Current theory of global sensitivity analysis, based on a nonlinear functional ANOVA decomposition of the random output, is limited in scope-for instance, the analysis is limited to the output's variance and the inputs have to be mutually independent-and leads to sensitivity indices the interpretation of which is not fully clear, especially interaction effects. Alternatively, sensitivity indices built for arbitrary user-defined importance measures have been proposed but a theory to define interactions in a systematic fashion and/or establish a decomposition of the total importance measure is still missing. It is shown that these important problems are solved all at once by adopting a new paradigm. By partitioning the inputs into those causing the change in the output and those which do not, arbitrary user-defined variability measures are identified with the outcomes of a factorial experiment at two levels, leading to all factorial effects without assuming any functional decomposition. To link various well-known sensitivity indices of the literature (Sobol indices and Shapley effects), weighted factorial effects are studied and utilized.

A new paradigm for global sensitivity analysis

TL;DR

The paper reframes global sensitivity analysis by introducing sensitivity maps τ that quantify input-set effects via a two-level factorial perspective, removing the strict dependence on functional decompositions and variance-based measures. By allowing arbitrary input distributions and general divergence-based variability measures, it unifies and extends index construction through weighted factorial effects I(B) = ∑_{A⊂D extbackslash B} p_B(A) Δ_B τ(A). It shows how Sobol-like and Shapley-like decompositions arise as special cases through appropriate weight choices, and introduces the dual sensitivity map τ^* to study symmetry and self-duality properties. The framework also clarifies connections to existing indices, offers insights for implementing factorial designs, and broadens the scope of sensitivity analysis to complex, high-dimensional, or nonstandard settings with flexible weighting schemes.

Abstract

Current theory of global sensitivity analysis, based on a nonlinear functional ANOVA decomposition of the random output, is limited in scope-for instance, the analysis is limited to the output's variance and the inputs have to be mutually independent-and leads to sensitivity indices the interpretation of which is not fully clear, especially interaction effects. Alternatively, sensitivity indices built for arbitrary user-defined importance measures have been proposed but a theory to define interactions in a systematic fashion and/or establish a decomposition of the total importance measure is still missing. It is shown that these important problems are solved all at once by adopting a new paradigm. By partitioning the inputs into those causing the change in the output and those which do not, arbitrary user-defined variability measures are identified with the outcomes of a factorial experiment at two levels, leading to all factorial effects without assuming any functional decomposition. To link various well-known sensitivity indices of the literature (Sobol indices and Shapley effects), weighted factorial effects are studied and utilized.
Paper Structure (17 sections, 7 theorems, 42 equations, 3 tables)

This paper contains 17 sections, 7 theorems, 42 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Sensitivity maps
  3. General sensitivity maps
  4. Sensitivity maps based on divergences between outputs
  5. An implicit factorial experiment
  6. Weighted factorial effects
  7. General formulas
  8. Examples of weights
  9. Decomposing the total output variability
  10. Sobol-like decomposition
  11. Shapley-like decomposition
  12. The dual sensitivity map
  13. Examples
  14. Sobol indices
  15. Shapley effects
  16. ...and 2 more sections

Key Result

Proposition 1

The map $\tau$ defined in (eq:tau_psi) is a sensitivity map.

Theorems & Definitions (14)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Example 1
  • Remark 2
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Example 2
  • ...and 4 more