Computation of Choquet integrals: Analytical approach for continuous functions

Abstract

In the continuous case, analytical computations of the Choquet integral are limited, despite being commonly used in various applications. One can either use the definition, which is computationally demanding and impractical, or apply already existing formulas restricted only to monotone nonnegative functions on a real interval starting at zero. This article aims to present more convenient computational formulas for continuous functions without imposing restrictions on their monotonicity given any real interval. First, a more general approach to monotone functions is provided for both positive and negative functions. Then, reordering techniques are introduced to compute the Choquet integral of an arbitrary continuous function, and with these, a monotone equivalent to every function can be constructed. This equivalent function preserves the final Choquet integral value, implying that only formulas for monotone functions are required. In addition to general fuzzy measures, the article assumes particular cases of distorted Lebesgue measures and distorted probabilities as the most commonly used fuzzy measures.