Sensitivity indices for output on a Riemannian manifold

TL;DR

This paper extends sensitivity analysis to outputs on Riemannian manifolds using a geometry-aware criterion and proposes estimators with studied asymptotic properties, supported by numerical examples.

Contribution

It introduces a novel sensitivity index for manifold-valued outputs and develops a Pick-Freeze estimator with theoretical properties.

Findings

Proposed sensitivity indices account for output geometry.

Established asymptotic behavior of the estimators.

Demonstrated effectiveness through numerical examples.

Abstract

In the context of computer code experiments, sensitivity analysis of a complicated input-output system is often performed by ranking the so-called Sobol indices. One reason of the popularity of Sobol's approach relies on the simplicity of the statistical estimation of these indices using the so-called Pick and Freeze method. In this work we propose and study sensitivity indices for the case where the output lies on a Riemannian manifold. These indices are based on a Cram\'er von Mises like criterion that takes into account the geometry of the output support. We propose a Pick-Freeze like estimator of these indices based on an $U$--statistic. The asymptotic properties of these estimators are studied. Further, we provide and discuss some interesting numerical examples.