On interval-valued pre-(quasi-)overlap functions
Introduction
As mathematical foundation of the process of aggregation, aggregation functions [12] play an important role in compressing a set of numerical data into a single number. As a special class of aggregation functions, the concept of overlap functions was initially proposed by Bustince et al. [15] to solve problems in image processing. Since then, overlap functions have been used in application fields, especially in classification [31], [29], image processing [15], decision making [28] and power quality [35]. In theory, scholars studied key properties [42], [49], [4], [15] and generally applicable methods for constructing overlap functions, including ordinal sums [20], additive generator pairs [22] and multiplicative generator pairs [41]. A review of the development of overlap functions can be found in [16]. Various extensions of overlap functions, such as quasi-overlap functions [36], have also been investigated.
In recent years, more studies have aimed at broadening the concept of aggregation functions, two important means of which are by allowing for some weaker forms of increasingness and extending it to more general structures. For one thing, after the concept of directional monotonicity [14] was proposed, it was naturally introduced into the context of aggregation functions, which are called pre-aggregation functions [32]. From then on, pre-aggregation functions have become a hot research topic:
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In terms of applications, pre-aggregation functions have wide application in classification and a relatively better performance. In [30], pre-aggregation functions are used to classify sugar level of grape berries. As for fuzzy inference approach of fuzzy rule-based classification systems (FRBCSs), -integral [29] was introduced, which has better results when dealing with classification problems. Researchers have solved optimisation problems where the objective function is expressed as a Choquet integral [11]. The Choquet integral has also been applied to problems in which the relation between elements is needed, such as image processing [33], where directional monotonicity allows for the detection of edges according to the tonal direction. As for multi-criteria decision making [47], pre-aggregation functions were applied to generalization of the Group Modular Choquet Random Technical Order by Preference to Ideal Solution.
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In the case of theory, Dimuro et al. [23] investigated new construction methods and properties of pre-aggregation functions. Properties of several classes of directionally increasing functions, including pre-aggregation functions were analyzed in [17]. Specifically, pre-t-norms [25] and pre-t-conorms [26] were considered, where monotonicity of t-norms and t-conorms is replaced with directional monotonicity, respectively. Recently, Qiao and Gong [39] paid attention to -(quasi-)overlap functions, using directional monotonicity, and they investigated several construction methods.
On the other hand, to cope with a degree of imprecision and uncertainty that may arise in fuzzy set theory when assigning membership degrees or modeling fuzzy sets, researchers have considered fuzzy sets on intervals [48], which have proved to be an adequate tool to model both vagueness (soft class boundaries) and uncertainty (with respect to the membership function). Moore [34] and Sunaga [46] combined interval-valued function evaluations with operators on intervals as a mathematical foundation for interval calculations. In fact, we can figure in a higher dimensional space with interval vectors. To deal with the situations in the interval-valued context, researchers have studied interval-valued fuzzy negations [8], [9], interval-valued implications [8], [6] and interval-valued aggregation functions such as interval-valued t-norms [10], [24], [18] and interval-valued overlap functions [5], [40]. Asmus et al. considered application of general interval-valued overlap functions in classification problems using IV-FRBCSs [2]. Lately, Asmus et al. defined n-dimensional interval-valued overlap functions w.r.t. product order and admissible order, respectively in [2], [4] and applied such functions to compute the interval matching degree in IV-FRBCSs [4]. Researchers also introduced width-limited interval-valued overlap functions [1] to control the information quality by limiting the width of the output interval.
Apart from weak monotonicity on the unit interval, scholars considered more general structures. For example, notion of directional monotonicity on Riesz spaces [43] and particularly, on the interval-valued setting was put forward. In [43], researchers defined directional monotonicity for functions taking value in Riesz spaces and studied a special case when the directions of increasingness are formed by intervals. Sesma-Sara et al. [44] discussed different generalized forms of increasingness in the interval-valued context. Meanwhile, Drygaś et al. [27] proposed a concrete interval-valued pre-aggregation function and its application in decision model of medical diagnosis support with comparably low error. This operator was then used in interval-valued fuzzy relations in decision problems based on preference relations [37].
As we have stated above, both interval-valued pre-aggregation functions and interval-valued overlap functions have relatively good performance in application. To achieve a more satisfactory result that may exist in practical problems, such as computing the interval matching degree in IV-FRBCSs, we take a weaker form of interval-valued overlap functions into account. Apart from classification problems, we also hope such functions can be used to deal with the situations in approximate reasoning [27] and other methods of decision making, for instance, when an expert consider the belief degree of an imprecision by the interval degree of membership functions in practical problems.
It should be pointed out that in this work, we take interval directional monotonicity in [43], where the directions of monotonicity are in the form of interval vectors. The interval directional monotonicity is actually a special case of that in [45], where an n-ary interval-valued function is monotone along a nonzero -dimensional real vector. As Sesma-Sara et al. mentioned [43], all properties of directionally monotone functions on a Riesz space in [43] hold for interval-valued functions for which the vectors of increasingness belong in are interval vectors, with the exception when dealing with the inverse of addition. In this work, however, inverse of addition has no influence on our results. Therefore, for the sake of discussion, we have chosen interval directional monotonicity instead of general directional monotonicity.
Different construction methods for interval-valued functions were put forward. For example, researchers characterized general interval-valued overlap functions through a pair of binary interval-valued functions [3]. Being inspired, we characterize interval-valued pre-(quasi-)overlap functions based on certain binary interval operators, which can also be seen as a construction method. On the other hand, the concept of interval additive generators of interval-valued t-norms was introduced in [24], as interval representations of the standard additive generators of t-norms. Then, interval additive generator pairs of interval-valued overlap functions were proposed in [40]. As we have stated, both additive generator pairs [22] and multiplicative generator pairs [41] of overlap functions were studied. On this basis, we consider interval additive and multiplicative generator pairs of interval-valued pre-(quasi-)overlap functions as a further extension and supplement to the existing work. Besides, we consider interval-valued -aggregation function, which is a concrete form of -aggregation function on complete lattices [38], to obtain an interval-valued pre-(quasi-)overlap function from existing ones.
The rest of this paper is structured as follows. Section 2 reviews some basic notions on interval mathematics and overlap functions. Section 3 is devoted to the notion of interval-valued pre-(quasi-)overlap functions and several relative properties. Section 4 is devoted to relationship between interval-valued pre-(quasi-)overlap functions and -(quasi-)overlap functions. Section 5 is devoted to a characterization of interval-valued pre-(quasi-)overlap functions. In Section 6, we discuss interval multiplicative generator pairs of interval-valued pre-(quasi-)overlap functions. In Section 7, we consider constructing interval-valued pre-(quasi-)overlap functions from interval-valued -aggregation functions. We end with some conclusions and future work.
Section snippets
Preliminaries
In this section, we recall some notions and results related to -(quasi-)overlap functions and interval mathematics, including interval-valued overlap functions and interval directional monotonicity. Let I denote the real unit interval .
Interval-valued pre-(quasi-)overlap functions
In this section, based on the concept of interval directional monotonicity, we present the notion of interval-valued pre-(quasi-)overlap functions and show some examples and relative properties. At first, we give definition of interval inclusion monotonicity which will be used in the following. Definition 3.1 An n-ary interval-valued operator is said to be -inclusion monotonic if, for any and such that , it holds that
The following proposition is about
Representable interval-valued pre-(quasi-)overlap functions
In this subsection, we focus on the study of a particular family of interval-valued pre-(quasi-)overlap functions: the representable interval-valued functions given in Definition 2.12. From viewpoint of directional monotonicity, this family is relevant since the study of the directions along which such a function increases depends on the directions of increasingness of its component functions. Proposition 4.1 For an -(quasi-)overlap function , its best interval representation is an
A characterization of interval-valued pre-(quasi-)overlap functions
Inspired by the characterization of general interval-valued overlap functions presented in [2], in this section we show that two specific interval-valued binary operators can be transformed into interval-valued pre-(quasi-)overlap functions based on some simple calculation. Lemma 5.1 ([2]) Consider and . The function defined by is Moore-continuous. Lemma 5.2 ([2]) Consider such that and . Thenwhere is the limited sum, defined by
Interval-valued pre-(quasi-)overlap functions based on interval multiplicative generator pairs
In this section, we mainly focus on interval multiplicative generator pairs of interval-valued pre-(quasi-)overlap functions and the relationship between interval multiplicative generator pairs and additive ones. First, we put forward notion of the former. Notice that in the following, we always consider such that . Definition 6.1 Consider unary interval-valued functions , where is increasing and is R-increasing. If the binary interval-valued function defined by
Interval-valued pre-(quasi-)overlap functions based on interval-valued -aggregation functions
In this section, we obtain interval-valued pre-(quasi-)overlap functions based on existing ones and interval-valued -aggregation functions. First, since is a complete lattice, we give the notion of interval-valued -aggregation function, which is actually a concrete form of [[38] Definition 3.2] with . Definition 7.1 An interval-valued -aggregation function is an n-ary interval-valued operator that satisfies the following conditions: () if for any
Concluding remarks
In this work, we focus on interval-valued pre-(quasi-)overlap functions. The main results are listed as follows.
(1) We defined the concepts of interval-valued pre-(quasi-)overlap functions and analyzed several properties and characterizations of such functions. To be specific, we discussed the relationship between a given -(quasi-)overlap function and its best interval representation and gave a weaker form of representation of interval-valued pre-(quasi-)overlap functions. Besides, we gave
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work was supported by grants from the National Natural Science Foundation of China (Grant Nos. 11971365 and 11571010).
References (49)
- et al.
General interval-valued overlap functions and interval-valued overlap indices
Inf. Sci.
(2020) - et al.
Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions
Int. J. Approximate Reasoning
(2017) - et al.
On interval fuzzy Simplications
Inf. Sci.
(2010) On interval fuzzy negations
Fuzzy Sets Syst.
(2010)- et al.
The best interval representations of t-norms and automorphisms
Fuzzy Sets Syst.
(2006) - et al.
Choquet integral optimisation with constraints and the buoyancy property for fuzzy measures
Inf. Sci.
(2021) - et al.
Generation of linear orders for intervals by means of aggregation functions
Fuzzy Sets Syst.
(2013) - et al.
Directional monotonicity of fusion functions
Eur. J. Oper. Res.
(2015) - et al.
Overlap functions
Nonlinear Analysis: Theory, Methods & Applications
(2010) - et al.
On some classes of directionally monotone functions
Fuzzy Sets Syst.
(2020)
A representation of t-norms in interval-valued Lfuzzy set theory
Fuzzy Sets Syst.
Archimedean overlap functions: The ordinal sum and the cancellation, idempotency and limiting properties
Fuzzy Sets Syst.
Interval additive generators of interval t-norms and interval t-conorms
Inf. Sci.
C_Fintegrals: A new family of pre-aggregation functions with application to fuzzy rule-based classification systems
Inf. Sci.
Neuro-inspired edge feature fusion using Choquet integrals
Inf. Sci.
Wavelet-fuzzy power quality diagnosis system with inference method based on overlap functions: Case study in an AC microgrid
Eng. Appl. Artif. Intell.
Lattice-valued overlap and quasi-overlap functions
Inf. Sci.
Overlap and grouping functions on complete lattices
Inf. Sci.
On interval additive generators of interval overlap functions and interval grouping functions
Fuzzy Sets Syst.
On multiplicative generators of overlap and grouping functions
Fuzzy Sets Syst.
On homogeneous, quasi-homogeneous and pseudo-homogeneous overlap and grouping functions
Fuzzy Sets Syst.
Weak and directional monotonicity of functions on Riesz spaces to fuse uncertain data
Fuzzy Sets Syst.
Toward a generalized theory of uncertainty (GTU)–an outline
Inf. Sci.
Migrativity properties of overlap functions over uninorms
Fuzzy Sets Syst.
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