Theory of fuzzy integrals and its applications

The Lebesgue integral is a monotone nonnegative linear function on the space of bounded measurable functions; its construction is strongly linked with the additivity of the underlying measure. However, the additivity of a measure is rather restrictive when modeling several real situations. To overcome this problem, several generalizations of classical probability measures have been proposed, such as pseudoadditive measures, possibility measures, belief functions, k -monotone capacities, and k -order additive measures. All these generalizations are covered by fuzzy measures. These set functions are sometimes called also premeasures. The Choquet integral is defined for nonnegative functions, and it is comonotone additive and homogeneous with respect to the multiplication by nonnegative reals. The Sugeno integral deals with functions whose range is contained with fuzzy sets and with normed fuzzy measures. The properties of the Choquet and the Sugeno integrals are recalled, discussed, and compared in this chapter. The chapter introduces a class of general fuzzy integrals with respect to a general fuzzy measure that has properties similar to those of Choquet or Sugeno integrals. To express the comonotone additivity of the Choquet integral and the comonotone maxitivity of the Sugeno integral, it is convenient to introduce a representation by means of a system of comonotone basic functions. The pseudoadditions and pseudomultiplications are also elaborated in the chapter.

CHAPTER 33 – Monotone Set Functions-Based Integrals

Capacities and overlap indexes with an application in fuzzy rule-based classification systems

Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. We investigate the property of commuting for aggregation operators in connection with their relationship to bisymmetry. In case of bisymmetric aggregation operators we show a sufficient condition ensuring that two operators commute, while for bisymmetric aggregation operators with neutral element we even provide a full characterization of commuting n-ary operators by means of unary distributive functions. The case of associative operations, especially uninorms, is considered in detail.

/pdf/aggregation-operators-and-commuting-4dizrlu4hr.pdf

Aggregation Operators and Commuting

On the distributivity between t-norms and t-conorms

The natural properties of the aggregation operators and the most elementary ones are the idempotence, the monotonicity and the continuity from below. We assume only these properties for the aggregation operators with infinitely many inputs, defined by functionals on the family of measurable functions. A family of fuzzy measures is associated with each aggregation operator. The properties of horizontal or vertical pseudo-additivity are recognizable by means of this family of fuzzy measures.

Aggregation operators and associated fuzzy measures

We give the general solution of the Cauchy equation on a general I-semigroup, i.e. in a given interval 
$ [0, M] \subset \overline{\mathbb{R}}^+ $
 with a pseudo-addition 
$ \oplus $
 . Then we present some examples of solutions.

The Cauchy equation on I-semigroups

In this paper we introduce a general definition of an information measure for fuzzy sets and we study its main properties. Then we describe a procedure to extend any information measure from a σ-algebra of ordinary sets to the σ-algebra of fuzzy sets of the same space.

A General Information for Fuzzy Sets

In an axiomatic way we give some forms of aggregation operators for group of n,n 2 [0,+1] conditional informations without probability on fuzzy setting.

http://www.gimac.uma.es/ipmu08/proceedings/papers/035-VivonaDivari.pdf

Aggregation operators for conditional information without probability

The aim of this paper was a generalization of independence property proposed by J. Kampe de Feriet and B. Forte in Information Theory without probability, called general information. Therefore, its application to fuzzy sets has been presented.

/pdf/an-independence-property-for-general-information-wpcpu5k3zy.pdf

Maria Divari

Papers

Aggregation operators and associated fuzzy measures

The Cauchy equation on I-semigroups

A General Information for Fuzzy Sets

Aggregation operators for conditional information without probability

An Independence Property for General Information