The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of [0, 1]. We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism...

We consider a collection F of subsets of a finite set N together with a capacity v : F → R+ and call a function f : N → R measurable if its level sets belong to F . Only in this case, the classical definition of the Choquet integral works in our wider context. In this article, we provide a general framework for a Choquet integral that works also for non-measurable functions f and includes the i...

The concepts of the Choquet and Sugeno integrals, based on a fuzzy measure, can be adopted as useful tools in estimation of the total effectiveness of a drug when appreciating its positive influence on a collection of symptoms typical of a considered diagnosis. The expected effectiveness of the medicine is evaluated by a physician as a verbal expression for each distinct symptom. By converting ...

In this paper, a new fuzzy density function, called N-density, is proposed. A real data set about Students Valuing Science with 5-fold cross-validation RMSE is conducted, for comparing the performances of the Choquet integral regression model with respect to six measures, Pmeasure, λ-measure, L-measure, extensional L-measure, completed L-measure and extensional completed L-measure based on this...

Given a finite set of generalized lotteries, that is random quantities equipped with a belief function, and a partial preference relation on them, a necessary and sufficient condition (Choquet rationality) has been provided for its representation as a Choquet expected utility of a strictly increasing utility function. Here we prove that this condition assures the extension of the preference rel...

This paper presents an ordered sorting procedure based on the Choquet integral as a discriminant function. It uses information provided by the Decision Maker (dm) in terms of a set of prototypes (alternatives well-known to the dm). The capacities of the Choquet integral are assessed through the solving of a linear program or a quadratic program. An interpretation of the results is provided by m...

In the context of multicriteria decision making whose aggregation process is based on the Choquet integral, bi-capacities can be regarded as a natural extension of capacities when the underlying evaluation scale is bipolar. The notion of entropy, recently generalized to capacities to measure their uniformity, is now extended to bi-capacities. We show that the resulting entropy measure has a ver...

In this paper, we introduce a kernel for monotone data derived from the Choquet integral with its underlying fuzzy measure. While a naïve computation of this kernel has a complexity that is exponential in the number of data attributes, we propose a more efficient approach with quadratic time complexity. Kernel PCA and SVM classification are employed to illustrate characteristics and benefits of...