As an important class of aggregation operators, the notion of overlap functions was first presented in 2009 in order to be considered for applications in image processing context. Later, many other ...researches arised bringing some variations of those functions for different purposes. Here, our main goal is defining overlap functions on lattices and discuss how a weakned version of it, named quasi-overlaps, works when continuity is eliminated from the definition. Some properties of quasi-overlaps on lattices, namely convex sum, migrativity, homogeneity, idempotency and cancellation law are investigated. Also, Finally, properties related to continuity as “Archimedean” and “limiting” are studied.
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.
naBL-algebras are non-associative generalizations of BL-algebras obtained from non-associative t-norms (nat-norms). In the present paper we propose a further generalization of BL-algebras where ...associativity is not required. Such generalization is based on a subclass of bivariate general overlap functions called inflationary. We call this non-associative generalization inflationary BL-algebras, and we discuss the main differences between the latter and the more specific class of inflationary BL-algebras. We show that the class naBL of non-associative BL-algebras obtained from general overlap functions contains the class naT of naBL-algebras obtained by nat-norms, and we provide a pictorial representation that summarizes these facts. We also prove some related properties, as well as a version of the well-known Chinese Remainder Theorem for these algebras but, under certain restrictions. Moreover, the notions of pseudo-automorphisms, automorphisms and their action on general overlap functions are used to obtain conjugated inflationary BL-algebras, as well as to obtain inflationary BL-algebras by distorting nat-norms by pseudo-automorphisms and, in the converse direction, to obtain naBL-algebras from inflationary BL-algebras via automorphisms.
Ensembles of classifiers have been receiving much attention lately, they consist of a collection of classifiers that process the same information and their output is combined in some manner. The ...combination method is probably the most important part in a ensemble of classifiers however, many works found in literature focus mostly on the classification step, using simple approaches on the combination step, such as majority voting. In this paper, we propose a new combination method based on a generalization of discrete Choquet integrals, combining it with quasi overlap functions to aggregate the outputs of classifiers in an ensemble. We also tested the proposed combination approach in a simple ensemble against other methods in literature to verify if there was a significant gain in performance.
We introduce a new contrapositivisation technique for fuzzy implications constructed from a pair of bivariate aggregation functions and a fuzzy negation, which we call bi-aggregated ...contrapositivisation. We show that the bi-aggregated contrapositivisation generalizes the upper, lower, medium and aggregated contrapositivisations. We characterize this new technique according to some of the main properties commonly associated with fuzzy implications. We study briefly the N-compatibility of the operators bi-aggregated contrapositivisators and we investigate how an automorphism acts under such operators. We also studied some properties that are satisfied by the bi-aggregated contrapositivisator of the main classes of fuzzy implications. Finally, we present three new methods of generating bivariate aggregation functions from bi-aggregated contrapositivisators and fuzzy negations.
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.
Recently, Paiva et al. generalized the notion of overlap functions in the context of lattices and introduced a weaker definition, called quasi-overlap, that originates from the removal of the ...continuity condition. In this paper, we introduce the concept of residuated implications related to quasi-overlap functions on lattices and prove some related properties. We also show that the class of quasi-overlap functions that fulfill the residuation principle is the same class of continuous functions according to a Scott topology on lattices. Scott continuity and the notion of densely ordered posets are used to generalize a classification theorem for residuated quasi-overlap functions on lattices. Conjugated quasi-overlaps are also considered.
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.
This theoretical research in fuzzy correlation analysis integrates data uncertainty analysis by measuring the strength of the linear relationship restricted to two fuzzy sets. As the relevant ...contribution to this research area, this paper presents the axiomatic definition of the n-dimensional generalized fuzzy correlation coefficient (nGCC), assigning to n input fuzzy sets an output value in the interval −1,1. Thus, the properties of general overlap functions and fuzzy negations are studied, discussing binary non-normed restricted dissimilarity functions, and the n-dimensional non-normed conjunctive functions. This study provides new methods for the n-dimensional generalized fuzzy correlation coefficient analysis, regarding their applications in solving multi-criteria and decision-making problems founded on fuzzy logic extensions. The n-dimensional non-normed conjunctive aggregation functions are also introduced, as range domain extensions from 0,1 to −1.1, covering the interpretation of negative to positive variable associations. The proposal correlation analysis promotes a better evaluation and data selection even when more than one algorithm is applied to evaluate the reduction methods in the defuzzification process based on Interval-valued Fuzzy Logic. We also investigate the relevance to determine a reliable result from the fuzzy inference system based on the nGCC methodology. This correlation methodology applied to the Interval Fuzzy Load Balancing for Cloud Computing (Int-FLBCC) model contributes as a flexible approach for virtual machines dynamic consolidation enabling improvements in resource usage and power efficiency, improving the computational system’s energy efficiency. So, the nGCC methodology extends the Int-FLBCC model by adding other degrees of reliability to the results obtained with diverse evaluations through n-dimensional generalized fuzzy correlation coefficient expressions, exploring average aggregations as arithmetic and exponential means and the median operator.
PDF
