Shadowed c-means: Integrating fuzzy and rough clustering

A new method of partitive clustering is developed in the framework of shadowed sets. The core and exclusion regions of the generated shadowed partitions result in a reduction in computations as compared to conventional fuzzy clustering. Unlike rough clustering, here the choice of threshold parameter is fully automated. The number of clusters is optimized in terms of various validity indices. It is observed that shadowed clustering can efficiently handle overlapping among clusters as well as model uncertainty in class boundaries. The algorithm is robust in the presence of outliers. A comparative study is made with related partitive approaches. Experimental results on synthetic as well as real data sets demonstrate the superiority of the proposed approach.     

Shadowed sets of type‐II: Representing and computing shadowiness in shadowed sets

Shadowed sets were developed to interpret and determine the required pair of thresholds, which resolves the issue with Zadeh's proposal about a three‐way approximation of fuzzy sets. However, a serious shortcoming of shadowed sets is that they are not capable of coming to grips with the degree of shadowiness (or doubt) associated with the elements of a shadowed set, especially the elements in the doubtful zone (i.e., shadow region) and as a result, all the elements in the shadow region are shown no commitment during the process of decision making. This paper proposes an alternative construct that keeps record of the degree of doubt associated with the elements in a shadowed set. It aims at introducing, discussing the concepts of shadowed sets of type‐II, and studying their related operations. Shadowed sets of type‐II provide a capable framework that despite localizing the uncertainty associated with fuzzy sets; adheres to the gradual transition of membership grades of the elements in the doubtful zone.     

Soft rough fuzzy sets and soft fuzzy rough sets

Fuzzy set theory, soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. Feng et al. introduced the notions of rough soft set, soft rough set and soft rough fuzzy set by combining fuzzy set, rough set and soft set all together. This paper is devoted to the further discussion of the combinations of fuzzy set, rough set and soft set. A new soft rough set model is proposed and its properties are derived. Furthermore, fuzzy soft set is employed to granulate the universe of discourse and a more general model called soft fuzzy rough set is established. The lower and upper approximation operators are presented and their related properties are surveyed.     

Sine-square embedded fuzzy sets versus type-2 fuzzy sets

The uncertainty is an inherent part of real-world applications. Type-2 fuzzy sets minimize the effects of uncertainties that cannot be modeled using type-1 fuzzy sets. However, the computational complexity of the type-2 fuzzy sets is very high and it is more difficult than type-1 fuzzy sets to use and understand. This paper proposes sine-square embedded fuzzy sets and gives a comparison with type-2 and nonstationary fuzzy sets. The sine-square embedded fuzzy sets consist of type-1 fuzzy sets and the sine function. The footprint of uncertainty in the type-2 fuzzy sets is provided with amplitude and frequency of sine-square function in the proposed algorithm. The proposed sine-square embedded fuzzy sets are much simpler than the type-2 fuzzy sets and the nonstationary fuzzy sets. Two control applications that are chosen as position control of a dc motor and simulation of human lifting motion using five-segment human model are carried out to demonstrate the effectiveness of the proposed approach.     

Soft sets combined with fuzzy sets and rough sets: a tentative approach

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft–rough fuzzy sets, which extends Dubois and Prade’s rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.     

Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology

Mathematical morphology is based on the algebraic framework of complete lattices and adjunctions, which endows it with strong properties and allows for multiple extensions. In particular, extensions to fuzzy sets of the main morphological operators, such as dilation and erosion, can be done while preserving all properties of these operators. Another extension concerns bipolar fuzzy sets, where both positive information and negative information are handled, along with their imprecision. We detail these extensions from the point of view of the underlying lattice structure. In the case of bipolarity, its two-components nature raises the question of defining a proper partial ordering. In this paper, we consider Pareto (component-wise) and lexicographic orderings.     

Switching between type-2 fuzzy sets and intuitionistic fuzzy sets: an application in medical diagnosis

When dealing with vagueness, there are situations when there is insufficient information available, making it impossible to satisfactorily evaluate membership. The intuitionistic fuzzy set theory is more suitable than fuzzy sets to deal with such problem. In 1996, Atanassov proposed the mapping from intuitionistic fuzzy sets to fuzzy sets. Furthermore, intuitionistic fuzzy sets are isomorphic to interval valued fuzzy sets, and interval valued fuzzy sets are regarded as the special cases of type-2 fuzzy sets in recently studies. However, their discussions are not only hardly comprehending but also lacking the reliable applications. In this study, the advantage of type-2 fuzzy sets is employed, and the switching relation between type-2 fuzzy sets and intuitionistic fuzzy sets is defined axiomatically. The switching results are applied to show the usefulness of the proposed method in pattern recognition and medical diagnosis reasoning.     

Superharmonic fuzzy sets on recurrent sets in dynamic fuzzy systems

We analyse P-superharmonic fuzzy sets on recurrent sets in dynamic fuzzy systems and we derive a simple estimation for the fuzzy sets. This paper presents a method to calculate an optimal value for Snell's optimal stopping problem.     

A new extension of fuzzy sets using rough sets: R-fuzzy sets

This paper presents a new extension of fuzzy sets: R-fuzzy sets. The membership of an element of a R-fuzzy set is represented as a rough set. This new extension facilitates the representation of an uncertain fuzzy membership with a rough approximation. Based on our definition of R-fuzzy sets and their operations, the relationships between R-fuzzy sets and other fuzzy sets are discussed and some examples are provided.     

Hesitant fuzzy sets

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.     

Balanced fuzzy sets

The paper presents a new approach to fuzzy sets and uncertain information based on an observation of asymmetry of classical fuzzy operators. Parallel is drawn between symmetry and negativity of uncertain information. The hypothesis is raised that classical theory of fuzzy sets concentrates the whole negative information in the value 0 of membership function, what makes fuzzy operators asymmetrical. This hypothesis could be seen as a contribution to a broad range discussion on unification of aggregating operators and uncertain information processing rather than an opposition to other approaches. The new approach “spreads” negative information from the point 0 into the interval [−1, 0] making scale and operators symmetrical. The balanced counterparts of classical operators are introduced. Relations between classical and balanced operators are discussed and then developed to the hierarchies of balanced operators of higher ranks. The relation between balanced norms, on one hand, and uninorms and nullnorms, on the other, are quite close: balanced norms are related to equivalence classes of some equivalence relation build on linear dependency in the spaces of uninorms and nullnorms. It is worth to stress that this similarity is raised by two entirely different approaches to generalization of fuzzy operators. This observation validates the generalized hierarchy of fuzzy operators to which both approaches converge. The discussion in this paper is aimed at presenting the idea and does not aspire to detailed exploration of all related aspects of uncertainty and information processing.     

Ordinal fuzzy sets

Fuzzy set theory has been used as a framework for interpreting imprecise linguistic expressions. In general, a linguistic term is described by the compatibility ordering induced in some universe of discourse (UoD). A membership function in fuzzy set theory serves to reflect this ordering by assignment of values in [0, 1] for objects in UoD. When we compute the meaning of a linguistic expression such as "young and tall" using fuzzy membership functions, two implicit assumptions are made. First, we assume the membership values have quantitative meaning so that they can be quantitatively manipulated, for example, by adding or subtracting (the extensive scale assumption). Second, we assume that the scales of the membership values used in describing the different linguistic terms are comparable and the same (the common scale assumption). In many cases, these assumptions cannot be justified. Some proposals have been made to address the first issue by using ordinal scale in defining fuzzy membership functions. However, the second issue has not been properly investigated. In this paper, we propose a framework that does not depend on both of these assumptions. Such framework will facilitate our understanding and investigation of qualitative reasoning without the extensive scale and common scale assumptions.     

Complex fuzzy sets

The objective of this paper is to investigate the innovative concept of complex fuzzy sets. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0, 1], but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. The inherent difficulty in acquiring intuition for the concept of complex-valued membership presents a significant obstacle to the realization of its full potential. Consequently, a major part of this work is dedicated to a discussion of the intuitive interpretation of complex-valued grades of membership. Examples of possible applications, which demonstrate the new concept, include a complex fuzzy representation of solar activity (via measurements of the sunspot number), and a signal processing application. A comprehensive study of the mathematical properties of the complex fuzzy set is presented. Basic set theoretic operations on complex fuzzy sets, such as complex fuzzy complement, union, and intersection, are discussed at length. Two novel operations, namely set rotation and set reflection, are introduced. Complex fuzzy relations are also considered. Index Terms-Complex fuzzy intersection, complex fuzzy relations, complex fuzzy sets, complex fuzzy union, complex-valued grades of membership, fuzzy complex numbers     

The three-dimensional fuzzy sets and their cut sets

In this paper, a new kind of L-fuzzy set is introduced which is called the three-dimensional fuzzy set. We first put forward four kinds of cut sets on the three-dimensional fuzzy sets which are defined by the 4-valued fuzzy sets. Then, the definitions of 4-valued order nested sets and 4-valued inverse order nested sets are given. Based on them, the decomposition theorems and representation theorems are obtained. Furthermore, the left interval-valued intuitionistic fuzzy sets and the right interval-valued intuitionistic fuzzy sets are introduced. We show that the lattices constructed by these two special L-fuzzy sets are not equivalent to sublattices of lattice constructed by the interval-valued intuitionistic fuzzy sets. Finally, we show that the three-dimensional fuzzy set is equivalent to the left interval-valued intuitionistic fuzzy set or the right interval-valued intuitionistic fuzzy set.     

一般图中的最小概要表示集问题

在一般图中,通常基于图的拓扑结构来刻画任意2个节点之间的相似度。基于节点相似度提出概要表示集SRS的概念,从图中寻找最少节点数的概要表示集称为最小概要表示集问题。证明了在一般图中求解最小概要表示集问题是NP (非确定性多项式)难的,不太可能存在多项式时间复杂度的精确算法。基于次模函数提出了多项式时间复杂度的贪心近似算法,用于求解最小概要表示集问题,得出近似比结果。     

The cut sets,decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets

In this paper,the cut sets,decomposition theorems and representation theorems of intuitionistic fuzzy sets and interval valued fuzzy sets are researched indail.First,new definitions of four kinds of cut sets on intuitionistic fuzzy sets are introduced,which are generalizations of cut sets on Zadeh fuzzy sets and have the same properties as that of Zadeh fuzzy sets.Second,based on these new cut sets,the decomposition theorems and representation theorems on intuitionistic fuzzy sets are established.Each kind ...