A Generalized ANOVA Dimensional Decomposition for Dependent Probability Measures

TL;DR

This paper introduces a generalized ANOVA dimensional decomposition for dependent random variables, providing new formulas, properties, and a constructive method to analyze multivariate functions with dependent measures.

Contribution

It develops a generalized ADD framework for dependent variables, including new properties, equations, and a constructive polynomial-based method for component function determination.

Findings

Correlation significantly affects component functions and sensitivity indices.

The generalized ADD reproduces classical ADD for independent variables.

Application to eigenvalue analysis demonstrates practical usefulness.

Abstract

This article explores the generalized analysis-of-variance or ANOVA dimensional decomposition (ADD) for multivariate functions of dependent random variables. Two notable properties, stemming from weakened annihilating conditions, reveal that the component functions of the generalized ADD have \emph{zero} means and are hierarchically orthogonal. By exploiting these properties, a simple, alternative approach is presented to derive a coupled system of equations that the generalized ADD component functions satisfy. The coupled equations, which subsume as a special case the classical ADD, reproduce the component functions for independent probability measures. To determine the component functions of the generalized ADD, a new constructive method is proposed by employing measure-consistent, multivariate orthogonal polynomials as bases and calculating the expansion coefficients involved from the solution of linear algebraic equations. New generalized formulae are presented for the second-moment characteristics, including triplets of global sensitivity indices, for dependent probability distributions. Furthermore, the generalized ADD leads to extended definitions of effective dimensions, reported in the current literature for the classical ADD. Numerical results demonstrate that the correlation structure of random variables can significantly alter the composition of component functions, producing widely varying global sensitivity indices and, therefore, distinct rankings of random variables. An application to random eigenvalue analysis demonstrates the usefulness of the proposed approximation.