Rough set theory for the interval-valued fuzzy information systems

The notion of a rough set was originally proposed by Pawlak [Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (5) (1982) 341-356]. Later on, Dubois and Prade [D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System 17 (2-3) (1990) 191-209] introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets. This paper deals with an interval-valued fuzzy information system by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory and discusses the basic rough set theory for the interval-valued fuzzy information systems. In this paper we firstly define the rough approximation of an interval-valued fuzzy set on the universe U in the classical Pawlak approximation space and the generalized approximation space respectively, i.e., the space on which the interval-valued rough fuzzy set model is built. Secondly several interesting properties of the approximation operators are examined, and the interrelationships of the interval-valued rough fuzzy set models in the classical Pawlak approximation space and the generalized approximation space are investigated. Thirdly we discuss the attribute reduction of the interval-valued fuzzy information systems. Finally, the methods of the knowledge discovery for the interval-valued fuzzy information systems are presented with an example.     

Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators.     

Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning

Among the computational intelligence techniques employed to solve classification problems, Fuzzy Rule-Based Classification Systems (FRBCSs) are a popular tool because of their interpretable models based on linguistic variables, which are easier to understand for the experts or end-users.The aim of this paper is to enhance the performance of FRBCSs by extending the Knowledge Base with the application of the concept of Interval-Valued Fuzzy Sets (IVFSs). We consider a post-processing genetic tuning step that adjusts the amplitude of the upper bound of the IVFS to contextualize the fuzzy partitions and to obtain a most accurate solution to the problem.We analyze the goodness of this approach using two basic and well-known fuzzy rule learning algorithms, the Chi et al.’s method and the fuzzy hybrid genetics-based machine learning algorithm. We show the improvement achieved by this model through an extensive empirical study with a large collection of data-sets.     

Learning fuzzy rules from fuzzy samples based on rough set technique

Although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness, its performance in dealing with initial fuzzy data is usually poor. This paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment. The extension is twofold. One is knowledge representation and the other is knowledge reduction. First, we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects. Second, we extend a number of underlying concepts of knowledge reduction (such as the reduct and core) to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data. Finally, we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm. One of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension.     

Polynomial regression interval-valued fuzzy systems

In recent years, the type-2 fuzzy sets theory has been used to model and minimize the effects of uncertainties in rule-base fuzzy logic system (FLS). In order to make the type-2 FLS reasonable and reliable, a new simple and novel statistical method to decide interval-valued fuzzy membership functions and probability type reduce reasoning method for the interval-valued FLS are developed. We have implemented the proposed non-linear (polynomial regression) statistical interval-valued type-2 FLS to perform smart washing machine control. The results show that our quadratic statistical method is more robust to design a reliable type-2 FLS and also can be extend to polynomial model.     

Robustness of interval-valued fuzzy inference

Since interval-valued fuzzy set intuitively addresses not only vagueness (lack of sharp class boundaries) but also a feature of uncertainty (lack of information), interval-valued fuzzy reasoning plays a vital role in intelligent systems including fuzzy control, classification, expert systems, and so on. To utilize interval-valued fuzzy inference better, it is very important to study the fundamental properties of interval-valued fuzzy inference such as robustness. In this paper, we first discuss the robustness of interval-valued fuzzy connectives. And then investigate the robustness of interval-valued fuzzy reasoning in terms of the sensitivity of interval-valued fuzzy connectives and maximum perturbation of interval-valued fuzzy sets. These results reveal that the robustness of interval-valued fuzzy reasoning is directly linked to the selection of interval-valued fuzzy connectives.     

Knowledge modeling based on interval-valued fuzzy rough set and similarity inference: prediction of welding distortion

Knowledge-based modeling is a trend in complex system modeling technology. To extract the process knowledge from an information system, an approach of knowledge modeling based on interval-valued fuzzy rough set is presented in this paper, in which attribute reduction is a key to obtain the simplified knowledge model. Through defining dependency and inclusion functions, algorithms for attribute reduction and rule extraction are obtained. The approximation inference plays an important role in the development of the fuzzy system. To improve the inference mechanism, we provide a method of similaritybased inference in an interval-valued fuzzy environment. larity based approximate reasoning, an inference result is Combining the conventional compositional rule of inference with simideduced via rule translation, similarity matching, relation modification, and projection operation. This approach is applied to the problem of predicting welding distortion in marine structures, and the experimental results validate the effectiveness of the proposed methods of knowledge modeling and similarity-based inference.     

Granulation of a fuzzy set: Nonspecificity

Granulation (decomposition) of a fuzzy set A defined on a finite set of objects X is studied. Two types of decomposition are considered: external granulation determined by a given equivalence relation on X and internal granulation created by clusters of elements from X with similar membership grades in A. Axiomatic definitions of measures of granular nonspecificity and granular specificity are proposed. Some general approaches to the construction of measures of granular nonspecificity (specificity) are suggested. Relationship between granular nonspecificity, roughness and nonspecificity of a fuzzy set is discussed.     

Positive approximation and converse approximation in interval-valued fuzzy rough sets

Methods of fuzzy rule extraction based on rough set theory are rarely reported in incomplete interval-valued fuzzy information systems. Thus, this paper deals with such systems. Instead of obtaining rules by attribute reduction, which may have a negative effect on inducting good rules, the objective of this paper is to extract rules without computing attribute reducts. The data completeness of missing attribute values is first presented. Positive and converse approximations in interval-valued fuzzy rough sets are then defined, and their important properties are discussed. Two algorithms based on positive and converse approximations, namely, mine rules based on the positive approximation (MRBPA) and mine rules based on the converse approximation (MRBCA), are proposed for rule extraction. The two algorithms are evaluated by several data sets from the UC Irvine Machine Learning Repository. The experimental results show that MRBPA and MRBCA achieve better classification performances than the method based on attribute reduction.     

Dominance-based rough set approach to incomplete interval-valued information system

Since preference order is a crucial feature of data concerning decision situations, the classical rough set model has been generalized by replacing the indiscernibility relation with a dominance relation. The purpose of this paper is to further investigate the dominance-based rough set in incomplete interval-valued information system, which contains both incomplete and imprecise evaluations of objects. By considering three types of unknown values in the incomplete interval-valued information system, a data complement method is used to transform the incomplete interval-valued information system into a traditional one. To generate the optimal decision rules from the incomplete interval-valued decision system, six types of relative reducts are proposed. Not only the relationships between these reducts but also the practical approaches to compute these reducts are then investigated. Some numerical examples are employed to substantiate the conceptual arguments.     

A soft computing method of performance evaluation with MCDM based on interval-valued fuzzy numbers

This study presented a new performance evaluation method for tackling fuzzy multicriteria decision-making (MCDM) problems based on combining VIKOR and interval-valued fuzzy sets. The performance evaluation problem often exists in complex administrative processes in which multiple evaluation criteria, subjective/objective assessments and fuzzy conditions have to be taken into consideration simultaneously in management. Here, the subjective, imprecise, inexact and uncertain evaluation processes are modeled as fuzzy numbers by means of linguistic terms, as fuzzy theory can provide an appropriate tool to deal with such uncertainties. However, the presentation of linguistic expressions in the form of ordinary fuzzy sets is not clear enough [15] and [21]. Interval-valued fuzzy sets can provide more flexibility [4] and [14] to represent the imprecise/vague information that results, and it can also provide a more accurate modeling. This paper presents the interval-valued fuzzy VIKOR, which aims to solve MCDM problems in which the weights and performances of criteria are unequal by using the concepts of interval-valued fuzzy sets. A case study for evaluating the performances of three major intercity bus companies from an intercity public transport system is conducted to illustrate the effectiveness of the method.     

On generalized intuitionistic fuzzy rough approximation operators

In rough set theory, the lower and upper approximation operators defined by binary relations satisfy many interesting properties. Various generalizations of Pawlak’s rough approximations have been made in the literature over the years. This paper proposes a general framework for the study of relation-based intuitionistic fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper intuitionistic fuzzy rough approximation operators induced from an arbitrary intuitionistic fuzzy relation are defined. Basic properties of the intuitionistic fuzzy rough approximation operators are then examined. By introducing cut sets of intuitionistic fuzzy sets, classical representations of intuitionistic fuzzy rough approximation operators are presented. The connections between special intuitionistic fuzzy relations and intuitionistic fuzzy rough approximation operators are further established. Finally, an operator-oriented characterization of intuitionistic fuzzy rough sets is proposed, that is, intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of lower and upper intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of intuitionistic fuzzy relations which produce the same operators.     

基于直觉模糊粗糙集的属性约简

针对Jensen下近似定义的局限性,提出一种新的等价类形式的近似算子表示,并将其推广到直觉模糊环境.在此基础上,将相对正域、相对约简、相对核等粗糙集的知识约简概念推广到直觉模糊环境,提出一种直觉模糊信息系统的启发式属性约筒算法.实例计算表明.该方法比Jensen的属性约简方法更为合理有效.     

Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure

This article proposes a new axiomatic definition of entropy of interval-valued fuzzy sets (IVFSs) and discusses its relation with similarity measure. First, we propose an axiomatic definition of entropy for IVFS based on distance which is consistent with the axiomatic definition of entropy of a fuzzy set introduced by De Luca, Termini and Liu. Next, some formulae are derived to calculate this kind of entropy. Furthermore we investigate the relationship between entropy and similarity measure of IVFSs and prove that similarity measure can be transformed by entropy. Finally, a numerical example is given to show that the proposed entropy measures are more reasonable and reliable for representing the degree of fuzziness of an IVFS.     

变精度模糊粗糙集的一种定义

模糊粗糙集模型同经典粗糙集模型类似,容易受到噪音数据的影响.针对该问题,受变精度粗糙集模型的启发,提出了变精度模糊粗糙集的概念.针对现有变精度模糊粗糙集模型尚不能满足一些基本性质的缺陷,重新定义了模糊近似空间中某一模糊集的β-下近似和β-上近似,该定义方式能够满足上述的基本性质.     

Some information measures for interval-valued intuitionistic fuzzy sets

A new information entropy measure of interval-valued intuitionistic fuzzy set (IvIFS) is proposed by using membership interval and non-membership interval of IvIFS, which complies with the extended form of Deluca-Termini axioms for fuzzy entropy. Then the cross-entropy of IvIFSs is presented and the relationship between the proposed entropy measures and the existing information measures of IvIFSs is discussed. Additionally, some numerical examples are given to illustrate the applications of the proposed entropy and cross-entropy of IvIFSs to pattern recognition and decision-making.     

Induction of multiple fuzzy decision trees based on rough set technique

The integration of fuzzy sets and rough sets can lead to a hybrid soft-computing technique which has been applied successfully to many fields such as machine learning, pattern recognition and image processing. The key to this soft-computing technique is how to set up and make use of the fuzzy attribute reduct in fuzzy rough set theory. Given a fuzzy information system, we may find many fuzzy attribute reducts and each of them can have different contributions to decision-making. If only one of the fuzzy attribute reducts, which may be the most important one, is selected to induce decision rules, some useful information hidden in the other reducts for the decision-making will be losing unavoidably. To sufficiently make use of the information provided by every individual fuzzy attribute reduct in a fuzzy information system, this paper presents a novel induction of multiple fuzzy decision trees based on rough set technique. The induction consists of three stages. First several fuzzy attribute reducts are found by a similarity based approach, and then a fuzzy decision tree for each fuzzy attribute reduct is generated according to the fuzzy ID3 algorithm. The fuzzy integral is finally considered as a fusion tool to integrate the generated decision trees, which combines together all outputs of the multiple fuzzy decision trees and forms the final decision result. An illustration is given to show the proposed fusion scheme. A numerical experiment on real data indicates that the proposed multiple tree induction is superior to the single tree induction based on the individual reduct or on the entire feature set for learning problems with many attributes.     

A general approach to attribute reduction in rough set theory

The concept of a consistent approximation representation space is introduced. Many types of information systems can be treated and unified as consistent ap- proximation representation spaces. At the same time, under the framework of this space, the judgment theorem for determining consistent attribute set is established, from which we can obtain the approach to attribute reductions in information systems. Also, the characterizations of three important types of attribute sets (the core attribute set, the relative necessary attribute set and the unnecessary attribute set) are examined.