Narrow Results

We present an approach to automatically extract a pertinent subset of soft output classifiers, and to aggregate them into a global decision rule using the Choquet integral. This approach relies on two key points. The first is a learning algorithm that uses a measure of the confusion between the categories to be recognized. The second is a selection scheme that discards weak or redundant decision rules, keeping only the most relevant subset. An experimental study, based on real world data, is then described. It analyzes the improvements achieved by these points first when used independently, then when combined together.

In this paper we propose an operational interpretation of general fuzzy measures. On the basis of this interpretation, we define the concept of coherence with respect to a partial information, and propose a rule of inference similar to the natural extension ⁷. We also suggest a definition of independence for fuzzy measures.

This paper discusses multiattribute preference relations compatible with a value/utility function represented by the Choquet integral with respect to a fuzzy measure, and shows that the additivity of the fuzzy measure is equivalent to each of mutual preferential independence, mutual weak difference independence, mutual difference independence, mutual utility independence, and additive independence.

The concept of entropy of a discrete fuzzy measure has been recently introduced in two different ways. A first definition was proposed by Marichal in the aggregation framework, and a second one by Yager in the framework of uncertain variables. We present a comparative study between these two proposals and point out their properties. We also propose a definition for the entropy of an ordinal fuzzy measure, that is, a fuzzy measure taking its values in an ordinal scale in the sense of measurement theory.

First a balancing property on a fuzzy measure is introduced. After that, the conditions of when an additive fuzzy measure is balanced are given and similar results are presented for 0-1 and S–decomposable fuzzy measures where S is a continuous t–conorm. Moreover, the concept of distance between two additive fuzzy measures is presented and some results related to the distance and the balancing property are developed.

In this paper, we discuss the defuzzification problem. We first propose a set defuzzification method, (from a fuzzy set to a crisp set) by using the Aumann integral. From the obtained set to a point, we have two methods of defuzzification. One of these uses the mean value method and the other uses a fuzzy measure. In the first case, we compare our mean value method with the method of the center of gravity. In the second case, we compare fuzzy measure method with the Choquet integral method. We also give there a sufficient condition so that the results in the last two methods are equivalent.

Previous work has shown that the Choquet integral w.r.t. a symmetric k-order additive fuzzy measure is characterized as a weighted sum, with possibly negative weights, of k specific OWA operators, and that the Sugeno integral w.r.t. a symmetric k-order maxitive fuzzy measure is characterized as a weighted maximum of k order statistics. In this paper, these characterizations are extended to the general, non-symmetric case, thereby providing additional insight in the symmetric case. More specifically, a new general decomposition of the Choquet integral w.r.t. a k-order additive fuzzy measure and of the Sugeno integral w.r.t. a k-order maxitive fuzzy measure are established.

The intepretation of aggregation functions in multicriteria decision making is often based on indices such as importance indices that measure the importance of one criterion. For instance, for the Choquet integral, the importance index is the so-called Shapley value. The use of an index must always be limited to the specific context it has been designed for. Here, we are interested in determining on which criteria acts should be improved if we want their global evaluation to increase as much as possible. The Shapley value is not suited for describing this. So, we introduce a new index of importance which represents the mean worth for acts to reach higher scores in a set of criteria. This index is defined for general aggregation functions with the help of several axioms. This importance index is then applied to the Choquet integral. In particular, we computer the worth to reach higher levels in one attribute, and in a couple of attributes. Interestingly, this leads to quantities that are closely related to the Shapley and interaction indices.

Three new versions of the most typical value (MTV)^1,2 of the population (generalized weighted averages) are introduced. The first version, WFEV_g, is a generalization of the weighted fuzzy expected value (WFEV)³ for any fuzzy measure g on a finite set and it coincides with the WFEV when a sampling probability distribution is used. The second and the third version are respectively the weighted fuzzy expected intervals WFEI and WFEI_g which are generalizations of the WFEV, namely, MTVs of the population for a sampling distribution and for any fuzzy measure g on a finite set, respectively, when the fuzzy expected interval (FEI)⁴ exists but the fuzzy expected value (FEV)⁴ does not. The construction process is based on the Friedman-Schneider-Kandel (FSK)³ principle and results in the new MTVs called the WFEI and the WFEI_g when the combinatorial interval extension of a function⁵ is used.

The weighted fuzzy expected value (WFEV) of the population for a sampling distribution was introduced in ¹. In ² the notion of WFEV is generalized for any fuzzy measure on a finite set (WFEV_g). The latter paper also describes the notions of weighted fuzzy expected intervals WFEI and WFEI_g which are an interval extension of WFEV and WFEV_g, respectively, when due to ''scarce'' data the fuzzy expected value (FEV) ³ does not exist, but the fuzzy expected interval (FEI) ³ does. In this paper, The generalizations GWFEV_g and GWFEI_g of WFEV_g and WFEI_g, respectively, are introduced for any fuzzy measure space. Furthermore, the generalized weighted fuzzy expected value is expressed in terms of two monotone expectation (ME)⁴ values with respect to the Lebesgue measure on [0,1]. The convergence of iteration processes is provided by an appropriate choice of a ''weight'' function. In the interval extension (GWFEI_g) the so-called combinatorial interval extension of a function ⁵ is successfully used, which is clearly illustrated by examples. Several examples of the use of the new weighted averages are discussed. In many cases these averages give better estimations than classical estimators of central tendencies such as mean, median or the fuzzy ''classical'' estimators FEV, FEI and ME.

Fuzzy measures and possibilistic measures taking their values in partially ordered sets, in particular, in semilattices and lattices, are introduced. Suggested and investigated in more detail are Cartesian products and marginals of poset-valued fuzzy and possibilistic measures. The conditions under which the resulting set functions possess the properties of poset-valued fuzzy and possibilistic measures are analyzed and some relevant assertions are stated and proved.

In this paper, the well-known Egoroff's theorem in classical measure theory is established on monotone non-additive measure spaces. Taylor's theorem, which concerns almost everywhere convergence of measurable function sequence in classical measure theory, is also generalized. The converse problem of the theorems are discussed, and a necessary and sufficient condition for the Egoroff's theorem is obtained on semicontinuous fuzzy measure space with S-compactness.

We provide a survey of recent developments about capacities (or fuzzy measures) and cooperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.

Some theorems concerning representation of fuzzy measure and the Choquet integral are shown independently by Murofushi and Sugeno (J. Math. Anal. Appl. 159, No 2, (1991), 532-549) and by Denneberg (Fuzzy sets and Systems, 92, (1997), 139-156). Using the mediator for representations, it is shown that the theorems are partially equivalent. The difference between them are also stated with some examples.

The techniques of fuzzy measure and fuzzy integral have been successfully applied in various real-world applications. The determination of fuzzy measures is the most difficult part in problem solving. Signed efficiency measure, which is a special kind of fuzzy measure with the best representation ability but the highest complexity, is even harder to determine. Some methodologies have been developed for solving this problem such as artificial neural networks (ANNs) and genetic algorithms (GAs). However, none of the existing methods can outperform the others with unique advantages. Thus, there is a strong need to develop a new technique for learning distinct signed efficiency measures from data. Extreme learning machine (ELM) is a new learning paradigm for training single hidden layer feed-forward networks (SLFNs) with randomly chosen input weights and analytically determined output weights. In this paper, we propose an ELM based algorithm for signed efficiency measure determination. Experimental comparisons demonstrate the effectiveness of the proposed method in both time and accuracy.

In this paper, a two-dimensional Hardy type inequality for fuzzy integrals is proved. Also, some illustrative examples are presented.

In this paper, two uncertain linguistic Choquet aggregation operators called the induced uncertain linguistic hybrid generalized Shapley Choquet weighted averaging (IULHGSCWA) operator and the induced uncertain linguistic hybrid generalized Shapley Choquet geometric mean (IULHGSCGM) operator are defined. To reduce the complexity of solving a fuzzy measure, the induced uncertain linguistic hybrid 2-additive generalized Shapley Choquet weighted averaging (IULHAGSCWA) operator and the induced uncertain linguistic hybrid 2-additive generalized Shapley Choquet geometric mean (IULHAGSCGM) operator are presented. When the weight information is incompletely known, models for the optimal fuzzy measures on an expert set, on an attribute set and on their ordered sets are built, respectively. Furthermore, an approach to multi-attribute group decision making under uncertain linguistic environment is developed, which can cope with the interdependent or correlative characteristics between elements. Finally, a practical application is selected to illustrate the validity and practicability of the developed procedure.

Intuitionistic fuzzy sets can describe the uncertainty and complexity of the world flexibly, so it has been widely used in multi-attribute decision making. Traditional intuitionistic fuzzy aggregation operators are usually based on the probability measure, namely, they consider that the attributes of objects are independent. But in actual situations, it is difficult to ensure the independence of attributes, so these operators are unsuitable in such situations. Fuzzy measure is able to depict the relationships among the attributes more comprehensively, so it can complement the traditional probability measure in dealing with the multi-attribute decision making problems. In this paper, we first analyze the existing intuitionistic fuzzy operators based on fuzzy measure, then introduce two novel additive intuitionistic fuzzy aggregation operators based on the Shapley value and the Choquet integral, respectively, and show their advantages over other ones.

This paper propose a Berwald type inequality and a Favard type inequality for Sugeno integrals. That is, we first show that

In this paper, we propose a concept of comonotonicity of random sets, which is a set inclusion relation and generalized notion of comonotonicity of real-valued random variables. Then we study some elementary properties of comonotonicity of random sets and comonotonic additivity of real-valued Choquet integral for random set mappings. After this, some other properties of this kind of real-valued Choquet integral for random set mappings are characterized by the comonotonic additivity, for instance, translation invariance, sup-norm continuous, positive homogeneity.