Construction of Capacities from Overlap Indexes

In many problems, it is crucial to find a relation between groups of data. Such relation can be expressed, for instance, in terms of an appropriate fuzzy measure or capacity([10, 21]) which reflects the way the different data are connected to each other [20]. In this chapter, taking into account this fact and following the developments in [8],we introduce a method to build capacities ([20, 21]) directly from the data (inputs) of a given problem. In order to do so, we make use of the notions of overlap function and overlap index ([5, 12, 13, 7, 4, 14, 16]) for constructing capacities which reflect how different data are related to each other. This paper is organized as follows: after providing some preliminaries, we analyse, in Section 3, some properties of overlap functions and indexes. In Sections 4 we discuss a method for constructing capacities from overlap functions and overlap indexes. Finally, we present some conclusions and references

Comments on: Interval Type-2 Fuzzy Sets are generalization of Interval-Valued Fuzzy Sets: Towards a Wider view on their relationship

This Letter makes some observations about [2] that further support the distinction between an interval type-2 fuzzy set (IT2 FS) and an interval-valued fuzzy set (IV FS), points out that all operations, methods and systems that have been developed and published about IT2 FSs are, so far, only valid in the special case when IT2 FS = IVFS, and suggests some research opportunities

Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets

AbstractWe begin the paper studying the axioms that the indicators of the grade of inclusion of a fuzzy set in another fuzzy set must satisfy. Next, we present an expression of such indicator, first for fuzzy sets and then for interval-valued fuzzy sets, analyzing in both cases their main properties. Then, we suggest an expression for the similarity measure between interval-valued fuzzy sets. Besides, we study two methods for inference in approximate reasoning based on interval-valued fuzzy sets, the inclusion grade indicator and the similarity measure. Afterwards, we expose some of the most important properties of the methods of inference presented and we compare these methods to Gorzalczany's. Lastly, we use the indicator of the grade of inclusion for interval-valued fuzzy sets as an element that selects from the different methods of inference studied, the one that will be executed in each case

Join and Meet Operations for Type-2 Fuzzy Sets With Nonconvex Secondary Memberships

In this paper, we will present two theorems for the join and meet operations for general type-2 fuzzy sets with arbitrary secondary memberships, which can be nonconvex and/or nonnormal type-1 fuzzy sets. These results will be used to derive the join and meet operations of the more general descriptions of interval type-2 fuzzy sets presented in a paper by Bustince Sola et al. ('Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wider view on their relationship,' IEEE Trans. Fuzzy Syst., vol. 23, pp. 1876-1882, 2015), where the secondary grades can be nonconvex. Hence, this study will help to explore the potential of type-2 fuzzy logic systems which use the general forms of interval type-2 fuzzy sets which are not equivalent to interval-valued fuzzy sets. Several examples for both general type-2 and the more general forms of interval type-2 fuzzy sets are presented

A first attempt on global evolutionary undersampling for imbalanced big data

The design of efficient big data learning models has become a common need in a great number of applications. The massive amounts of available data may hinder the use of traditional data mining techniques, especially when evolutionary algorithms are involved as a key step. Existing solutions typically follow a divide-and-conquer approach in which the data is split into several chunks that are addressed individually. Next, the partial knowledge acquired from every slice of data is aggregated in multiple ways to solve the entire problem. However, these approaches are missing a global view of the data as a whole, which may result in less accurate models. In this work we carry out a first attempt on the design of a global evolutionary undersampling model for imbalanced classification problems. These are characterised by having a highly skewed distribution of classes in which evolutionary models are being used to balance it by selecting only the most relevant data. Using Apache Spark as big data technology, we have introduced a number of variations to the well-known CHC algorithm to work very large chromosomes and reduce the costs associated to fitness evaluation. We discuss some preliminary results, showing the great potential of this new kind of evolutionary big data model

Special construction of Atanassov’s intuitionistic fuzzy S-implications that maintain the Atanassov’s intuitionistic index

In this paper we present a method for the construction of Atanassov’s intuitionistic fuzzy S-implications that satisfy the following property: if in the intuitionistic fuzzy conditional the antecedent is equal to the consequent, then the Atanassov’s intuitionistic fuzzy implication operator has the same Atanassov’s intuitionistic fuzzy index as the antecedent and the consequent

Fuzzy entropy from weak fuzzy subsethood measures

In this paper, we propose a new construction method for fuzzy and weak fuzzy subsethood measures based on the aggregation of implication operators. We study the desired properties of the implication operators in order to construct these measures. We also show the relationship between fuzzy entropy and weak fuzzy subsethood measures constructed by our method

Penalty functions over a cartesian product of lattices

In this work we present the concept of penalty function over a Cartesian product of lattices. To build these mappings, we make use of restricted dissimilarity functions and distances between fuzzy sets. We also present an algorithm that extends the weighted voting method for a fuzzy preference relation

Restricted dissimilarity functions and penalty functions

In this work we introduce the definition of restricted dissimilarity functions and we link it with some other notions, such as metrics. In particular, we also show how restricted dissimilarity functions can be used to build penalty functions

A construction method of Atanassov’s intuitionistic fuzzy sets for image processing

In this work we introduce a new construction method of Atanassov\u27s intuitionistic fuzzy sets (A-IFSs) from fuzzy sets. We use A-IFSs in image processing. We propose a new image magnification algorithm using A-IFSs. This algorithm is characterized by its simplicity and its efficiency