Combinatorial Structure of the Polytope of 2-Additive Measures

Combinatorial structure of polytopes associated to fuzzy measures, 2020

This PhD thesis is devoted to the study of geometric and combinatorial aspects of polytopes associated to fuzzy measures. Fuzzy measures are an essential tool, since they generalize the concept of probability. This greater generality allows applications to be developed in various fields, from the Decision Theory to the Game Theory. The set formed by all fuzzy measures on a referential set is a polytope. In the same way, many of the most relevant subfamilies of fuzzy measures are also polytopes. Studying the combinatorial structure of these polytopes arises as a natural problem that allows us to better understand the properties of the associated fuzzy measures. Knowing the combinatorial structure of these polytopes helps us to develop algorithms to generate points uniformly at random inside these polytopes. Generating points uniformly inside a polytope is a complex problem from both a theoretical and a computational point of view. Having algorithms that allow us to sample uniformly in polytopes associated to fuzzy measures allows us to solve many problems, among them the identification problem, i.e. estimate the fuzzy measure that underlies an observed data set. Many of these polytopes associated with subfamilies of fuzzy measures are order polytopes and their combinatorial structure depends on a partially ordered set (poset). In this way we can transform problems of a geometric nature into problems of Combinatorics and Order Theory. In this thesis, we start by introducing the most important results about posets and order polytopes. Next, we focus on the problem of generating uniformly at random linear extensions in a poset. To this end, we have developed a method, which we have called Bottom-Up, which allows generating linear extensions for several posets quickly and easily. The main disadvantage of this method is that it is not applicable to every poset. The posets for which we can apply Bottom-Up are called BU-feasibles. For this reason we study another method that we call Ideal-based method. This method is more general than Bottom-Up and is applicable to any poset, however its computational cost is much higher. en los momentos más duros. Ahora que nuestra familia está creciendo, nos embarcamos juntos en otra etapa de descubrimiento y aprendizaje, seguiremos luchando juntos codo con codo como hemos hecho hasta ahora.

Fuzzy Sets and Systems, 2013

In this paper we deal with the problem of obtaining a random procedure for generating fuzzy measures. We use the fact that the polytope of fuzzy measures is an order polytope, so that it has special properties that allow to build a uniform algorithm. First, we derive an exact procedure based on an existing procedure to generate random linear extensions; then, we study the applicability of this algorithm to the polytope of fuzzy measures, showing that the complexity grows dramatically with the cardinality of the referential set. Next, we study other heuristics appearing in the literature for the polytope of fuzzy measures; our results seem to mean that these procedures cannot be applied to this case either. Finally, we propose another heuristic that reduces the complexity and could be used instead of the other procedures. We finish comparing the performance of this heuristic with the other possibilities, showing that our alternative seems to work better for the polytope of fuzzy measures.

Fuzzy Sets and Systems, 2008

In this paper we deal with the problem of studying the structure of the polytope of non-additive measures for finite referential sets. We give a necessary and sufficient condition for two extreme points of this polytope to be adjacent. We also show that it is possible to find out in polynomial time whether two vertices are adjacent. These results can be extended to the polytope given by the convex hull of monotone Boolean functions. We also give some results about the facets and edges of the polytope of non-additive measures; we prove that the diameter of the polytope is 3 for referentials of three elements or more. Finally, we show that the polytope is combinatorial and study the corresponding properties; more concretely, we show that the graph of non-additive measures is Hamilton connected if the cardinality of the referential set is not 2.

… of the 10th International Conference on …, 2004

In this paper we extend our geometric approach to the theory of evidence in order to include other important classes of nite fuzzy measures. In particular we describe the geometric counterparts of possibility measures or fuzzy sets, represented as consonant belief functions. The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of simplicial complex.

IEEE Transactions on Fuzzy Systems, 2000

The generation of fuzzy measures is an important question arising in the practical use of these operators. In this paper, we deal with the problem of developing a random generator of fuzzy measures. More concretely, we study some of the properties that any random generator should satisfy. These properties lead to some theoretical problems concerning the group of isometries that we tackle in this paper for some subfamilies of fuzzy measures.

Journal of Mathematical Analysis and Applications, 1992

2006

In this paper we extend the geometric approach to the theory of evidence in order to include other important finite fuzzy measures. In particular we describe the geometric counterparts of the class of possibility measures represented by consonant belief functions. The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of simplicial complex. Exploiting the duality of the associated norms we consider the consonant approximation problem in a simple case study, compare the geometry of the solutions with that of inner and outer approximations, and formulate conjectures on their general behavior. We also propose an optimization criterion based on Dempster's rule of combination. Abstract

Information Sciences, 1998

We address the (generalized) extension problem for possibility measures: given a map de ned on a family of (fuzzy) sets, is it possible to extend it to a (generalized) possibility measure? The extension problem for possibility measures is known to be equivalent to a system of sup-T equations, with T a t-norm. A key role is played by the greatest solution (of type inf-I, with I a border implicator). When the family of sets considered is a semi-partition, another important solution (of type sup-T , with T a t-norm) can be identi ed. In the treatment of the generalized possibilistic extension problem, we show that a fuzzi cation of the greatest solution also plays a central role. On the other hand, an immediate fuzzi cation of the sup-T type solution is investigated. General necessary and su cient conditions for this fuzzi cation to be a solution are established. This fuzzication is then further discussed in the case of a T -semi-partition or a T -partition. Finally, we investigate possible criteria for extendability, inspired by Wang's classical criterion of P-consistency.

International Journal of Intelligent Systems, 2011

In assigning weights and scores in a decision problem usually we assume that they are finitely additive normalized measures, i.e., from the formal point of view, finitely additive probabilities. The normalization requirement sometimes appears as an actual restriction. For instance, it happens when the weights and scores can not be assigned in precise numerical terms or some logical and numerical issues arising from the given problem imply conditions that are different from the property of additivity. We must then consider extensions of the concept of probability. One, introduced by Zadeh, is to express the probabilities with fuzzy numbers; another extension, considered by Sugeno, Weber, and others, is first to replace the additivity with the condition of monotonicity, much weaker, and then to identify conditions "intermediate" between monotonicity and additivity. In any case, by assigning weights and scores, they must be consistent with the point of view considered. The conditions of consistency of finitely additive probabilities and their generalizations were discussed in several papers. This paper proposes an extension of the concept of finitely additive probability from a purely geometric point of view. Specifically, the environment of Euclidean Geometry, as de Finetti used to define the consistency of an assignment of probabilities, is replaced by the more general environment of Join Geometry by Prenowitz and Jantosciak. In this context, we introduce the concept of coherent join measure, i.e., normalized measure that is consistent with a join system, in particular, a join space or a join geometry. We show that decomposable measures with respect to a t-conorm are special cases of join coherent measures. Finally, we present some applications, significant special cases, and possible lines of research.

This paper presents a state of art on the latest concepts of measure, from the additive measures, to monotone fuzzy measures and the latest mono- tone measures in relation to a preorder that gives an ordering for a measurable characteristic.