Preaggregation Functions: Construction and an Application

Abstract

In this work we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build pre-aggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators like the maximum and the Choquet integral.

Generalized Choquet Integral for Face Recognition

“…Hence, we discuss here a large collection of t-norm families which can be applied as the function M x; y ð Þ [see formula (2)]. As the reference, we have used the monograph [3, p. 72, Table 2.6] and paper [29] where simple functions as minimum…”

“…We are interested in a generalization of Choquet integral in the context of introduction of some kind of structural flexibility by applying t-norms instead of product under the integral sign. In [9,29], this weaker class of functions (when compared to the aggregation conditions) was introduced and called pre-aggregation functions. They are monotonic along a fixed direction which is called directional monotonicity.…”

Aggregation and Related Functions in Fuzzy Set Theory: Dualities and Monotonicity-Based Generalizations

“…Recently, we have generalized the class , keeping the boundary conditions and directional monotonicity, but requiring the-increasingness for somē∈ only, see [9].…”

“…There we also recall particular extended Boolean functions characterized by some additional properties, including monotonicity constraints. In Section 3, we recall recently introduced concepts of directional monotonicity [1] and pre-aggregation functions [9]. In Section 4, we generalize some of the discussed classes of extended Boolean functions by relaxing the monotonicity constraints and replacing them by some directional monotonicity.…”

A Fuzzy Approach to Measure the Similarity Between Web Streaming Users

“…The theory of fuzzy sets is the ideal tool to deal with this kind of linguist variable. Since the introduction of fuzzy sets by Zadeh [12], measures of similarity between fuzzy sets have gained attention in many fields [10], such as image processing, pattern recognition, fuzzy reasoning [4,5]. Similarity concepts are a common term in classical set theory, as well as in statistics.…”