Symmetric approximations of pseudo-Boolean functions with applications to influence indexes

TL;DR

This paper introduces a new influence index for pseudo-Boolean functions based on weighted least squares approximations, linking it to concepts in reliability and decision making.

Contribution

It presents a novel influence index derived from least squares approximation, with explicit formulas and properties, unifying concepts in reliability and decision theory.

Findings

Index explicitly computed for pseudo-Boolean functions

Index generalizes system signature and cardinality index

Properties of the index discussed and analyzed

Abstract

We introduce an index for measuring the influence of the k-th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic functions. We give explicit expressions for both the index and the approximation and discuss some properties of the index. Finally, we show that this index subsumes the concept of system signature in engineering reliability and that of cardinality index in decision making.