10 - Behavioral analysis of aggregation functions

Summary

Introduction

Given a function F : ⁿ → ℝ, possibly an aggregation function, it is useful to define values or indices that offer a better understanding of the general behavior of F with respect to its variables. These indices may constitute a kind of identity card of F and enable one to classify the aggregation functions according to their behavioral properties.

For example, given an internal aggregation function A (recall that “internal” means Min ≤ A ≤ Max), it might be convenient to appraise the degree to which A is conjunctive, that is, close to Min. Similarly, it might be very instructive to know which variables, among x₁, …, x_n, have the greatest influence on the output value A(x).

In this chapter we present various indices, such as: andness and orness degrees of internal functions, idempotency degrees of conjunctive and disjunctive functions, importance and interaction indices, tolerance indices, and dispersion indices.

Sometimes different indices can be considered to measure the same behavior. In that case it is often needed to choose an appropriate index according to the nature of the underlying aggregation problem.

To keep the exposition concise the proofs of many results from this chapter have been omitted.

Expected values and distribution functions

A very informative treatment of a given function F : ⁿ → ℝ consists in applying it to a random input vector and examining the behavior of the output signal by computing its distribution function.