In this paper, we introduce the notion of ordered directionally monotone function as a type of function which allows monotonicity along different directions in different points. In particular, these ...functions take into account the ordinal size of the coordinates of the inputs in order to fuse them. We show several examples of these functions and we study their properties. Finally, we present an illustrative example of an application which justifies the introduction and the study of the concept of ordered directional monotonicity.
Overlap functions are aggregation operators specially introduced to be used in applications involving the overlap problem and/or when the associativity property is not strongly required for the ...aggregation operator, as in classification problems and decision making based on fuzzy preference relations. This paper considers the existent results on residual implication induced by fuzzy conjunctions to introduce the concept of residual implication derived from overlap functions O, denoted by RO-implication, preserving the residuation property. RO-implications are weaker than R-implications constructed from positive and continuous t-norms, in the sense that RO-implications do not necessarily satisfy certain properties satisfied by such R-implications, as the exchange principle, but only weaker versions of these properties. However, in general, such properties are not demanded for many applications. The objectives of this paper are: (a) to analyse the main properties satisfied by RO-implications, establishing under which conditions of an overlap function O the derived RO-implication satisfies the properties of fuzzy implications and (b) to provide two particular characterization of RO-implications derived from (i) the sub-class of overlap functions O that have 1 as neutral element and (ii) the more general sub-class of overlap functions O satisfying the condition O(x,1)⩽x.
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3.
Preaggregation Functions: Construction and an Application Lucca, Giancarlo; Sanz, Jose Antonio; Dimuro, Gracaliz Pereira ...
IEEE transactions on fuzzy systems,
2016-April, 2016-4-00, 20160401, Volume:
24, Issue:
2
Journal Article
Peer reviewed
Open access
In this paper, 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 preaggregation 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 such as the maximum and the Choquet integral.
Overlap functions are a particular type of aggregation functions, given by increasing continuous commutative bivariate functions defined over the unit square, satisfying appropriate boundary ...conditions. Overlap functions are applied mainly in classification problems, image processing and in some problems of decision making based on some kind of fuzzy preference relations, in which the associativity property is not strongly required. Moreover, the class of overlap functions is reacher than the class of t-norms, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap functions. This flexibility of overlap functions increases their applicability. The aim of this papers is to introduce the concept of Archimedean overlap functions, presenting a study about the cancellation, idempotency and limiting properties, and providing a characterization of such class of functions. The concept of ordinal sum of overlap functions is also introduced, providing constructing/representing methods of certain classes of overlap functions related to idempotency, cancellation, limiting and Archimedean properties.
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This work aims to evaluate explainable classification methods for the detection of fish species from hydroacoustic data acquired by echo sounders at a region near the coastline of south and ...southeastern Brazil. Decision trees and fuzzy rule-based methods were adopted. The fitted models were evaluated by quality measures based on the performance of the classifiers and also by an expert which analyzed the usefulness of the rules on describing the schools. The models learned by the algorithms performed well for the available data and were able to represent the documented behavior of the species considered in the studied region, according to the literature.
Overlap functions and grouping functions are special kinds of aggregation operators that have been recently proposed for applications in classification problems, like, e.g., imaging processing. ...Overlap and grouping functions can also be applied in decision making based on fuzzy preference relations, where the associativity property is not strongly required and the use of t-norms or t-conorms as the combination/separation operators is not necessary. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow the application in several different contexts. This paper introduces new interesting results related to overlap and grouping functions, investigating important properties, such as migrativity, homogeneity, idempotency and the existence of generators. De Morgan triples are introduced in order to study the relationship between those dual concepts. In particular, we introduce important results related to the action of automorphisms on overlap and grouping functions, analyzing the preservation of those properties and also the Lipschitzianity condition.
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•We make a revision of recent generalizations of the Choquet integral that appear in the literature.•We show some of the most relevant theoretical features of these extensions.•We also discuss some ...applications where these extensions have provided good results.
In 2013, Barrenechea et al. used the Choquet integral as an aggregation function in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems. After that, starting from 2016, new aggregation-like functions generalizing the Choquet integral have appeared in the literature, in particular in the works by Lucca et al. Those generalizations of the Choquet integral, namely CT-integrals (by t-norm T), CF-integrals (by a fusion function F satisfying some specific properties), CC-integrals (by a copula C), CF1F2-integrals (by a pair of fusion functions (F1, F2) under some specific constraints) and their generalization gCF1F2-integrals, achieved excellent results in classification problems. The works by Lucca et al. showed that the aggregation task in a FRM may be performed by either aggregation, pre-aggregation or just ordered directional monotonic functions satisfying some boundary conditions, that is, it is not necessary to have an aggregation function to obtain competitive results in classification. The aim of this paper is to present and discuss such generalizations of the Choquet integral, offering a general panorama of the state of the art, showing the relations and intersections among such five classes of generalizations. First, we present them from a theoretical point of view. Then, we also summarize some applications found in the literature.
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General Interval-valued Grouping Functions da Cruz Asmus, Tiago; Pereira Dimuro, Gracaliz; Bustince, Humberto ...
2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE),
2020-July
Conference Proceeding
Grouping functions are aggregation functions used in decision making based on fuzzy preference relations in order to express the measure of the amount of evidence in favor of either of the two ...alternatives when performing pairwise comparisons. They have been also used as a disjunction operator in some important problems, such as image thresholding and the construction of a class of implication functions for the generation of fuzzy subsethood and entropy measures. Some generalizations of this concept were recently proposed, such as n-dimensional and general grouping functions, which allowed their application in ndimensional problems, such as fuzzy community detection. Also the concept of interval-valued overlap functions was presented, in order to deal with the uncertainty when defining membership functions. The aim of this paper is to introduce the concepts of n-dimensional interval-valued grouping functions and general interval-valued grouping functions, studying representability, characterization and construction methods.
In this paper, we point out several problems in the different definitions (and related results) of penalty functions found in the literature. Then, we propose a new standard definition of penalty ...functions that overcomes such problems. Some results related to averaging aggregation functions, in terms of penalty functions, are presented, as the characterization of averaging aggregation functions based on penalty functions. Some examples are shown, as the penalty functions based on spread measures, which happen to be continuous. We also discuss the definition of quasi-penalty functions, in order to deal with non-monotonic (or weakly/directionally monotonic) averaging functions.
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Overlap functions are aggregation functions that express the overlapping degree between two values. They have been used both as a conjunction in several practical problems (e.g., image processing and ...decision making), and to generate overlap indices between two fuzzy sets, which can be used to construct fuzzy confidence values to be applied in fuzzy rule based classification systems. Some generalizations of overlap functions were recently proposed, such as n-dimensional and general overlap functions, which allowed their application in n-dimensional problems. More recently, the concept of interval-valued overlap functions was presented, mainly to deal with uncertainty in providing membership functions. In this paper, we introduce: (i) the concept of n-dimensional interval-valued overlap functions, studying their representability, (ii) the definition of general interval-valued overlap functions, providing their characterization and some construction methods. Moreover, we also define the concept of interval-valued overlap index, as well as some constructing methods. In addition, we show an illustrative example where those new concepts are applied.
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