Information structures in a fuzzy set-valued information system based on granular computing

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Abstract

A fuzzy set-valued information system (FSVIS) refers to an information system (IS) whose information values are fuzzy sets. This article investigates information structures (ISts) in a FSVIS based on granular computing (GrC). First, FSVISs and homomorphism between them are introduced. Next, ISts in a FSVIS are described. The dependence and information distance between ISts are discussed, and characterizations of ISts in a FSVIS are acquired. Additionally, as an application of ISts in a FSVIS, the entropy measure of uncertainty for a FSVIS is discussed, and on the basis of this uncertainty measure, the optimal selection of ISts is investigated. The obtained consequence will be conducive to establishing a framework for GrC in a FSVIS. Last, the difference between this paper and the references for ISts in this paper is compared and discussed.

Introduction

Granular computing (GrC), put forward by Zadeh [48], [49], [50], [51], is a discipline that studies the thinking mode, problem solving method, information processing mode and related theories, technologies and tools based on a multilevel granular structure. Because it reflects the global view and approximate solution ability of human beings when dealing with multilevel and multiperspective problems, GrC has gradually become an important theory for solving uncertain problems. Lin [13], [14], [15], Yao [39], [40], [41] and Pedrycz [28], [29], [46] emphasized the importance of GrC.

Rough set theory (RST) was brought forward by Pawlak [24], [26], [27]. RST can be used to deal with imprecision, fuzziness and uncertainty. RST can also process a dataset without additional prior information [5], [7], [10], [18], [20], [33], [34], [35], [45]. This theory is based on a classification mechanism and understands classification as an equivalence relation in a specific space. An equivalence relation involves the division of this space. The main idea of RST is to use a known knowledge base to describe the imprecise or uncertain knowledge with the knowledge of the known knowledge base. RST can effectively deal with the uncertainty of IS. In recent years, RST has attracted the attention of many researchers, and its application is mostly related to ISs [22], [36], [38], [43], [52], [54].

Given an IS, each attribute subset determines an equivalence relation on the object set of this IS. The equivalence relation divides the object set into equivalence classes which are looked upon as information granules [3]. Many research studies on this topic have been carried out. For example, Li et al. [17] proposed information structures (ISts) in a covering IS. Qian et al. [30] investigated the knowledge structure, knowledge granulation and knowledge distance in a knowledge base. Qian et al. [31] studied information granularity in a fuzzy GrC model. Xie et al. [37] investigated fuzzy information granular structures. Yang et al. [44] studied a temporal-spatial composite sequential approach of a three-way GrC. Zhang et al. [53] explored uncertainty measures for interval set information tables based on interval δ-similarity relations. Zhang et al. [55] discussed three-layer granular structures and three-way informational measures of a decision table.

Fuzzy set theory (FST), put forward by Zadeh [47], can describe fuzziness in precise mathematical language. FST and RST as research methods of incomplete knowledge and uncertain problems in an IS have their own advantages and characteristics. They can be combined to study specific problems. FST has been successfully applied to many fields, such as fuzzy services [1], forecast model [4], classification method [9], real economic data [11], healthcare [12], information extraction [16], asset allocation optimization [21], inclusion and exclusion principle [23], linguistic models [28], computational intelligence [29], fuzzy inventory model [32], fuzzy controller [57], neuro-fuzzy architecture [58] and so on. The application of FST in the field of target recognition is mainly reflected in the design of the classifier, which can help the machine to make an accurate judgment in the fuzzy environment and has strong adaptability to the special case of target deformation. Moreover, FST improves the flexibility and practicability of processing fuzzy or uncertain information.

With the forthcoming era in the information age, ISt has become a hot research topic in the information technology world. This means that ISts are becoming increasingly significant. A fuzzy set-valued information system (FSVIS) is a system whose information values are fuzzy sets. However, ISts on the basis of a GrC framework in a FSVIS have not been studied. This paper investigates this subject.

Why do we study ISt in a FSVIS? Because ISt in a FSVIS is conducive to discovering knowledge from a FSVIS. Why do we study homomorphism between FSVISs? Because if homomorphism between FSVISs is constructed, then mapping characterizations of ISts in a FSVIS under homomorphism are obtained, and invariant characterizations of FSVISs under homomorphism can also be obtained. Why do we study the entropy measure of uncertainty for a FSVIS? We do so because a FSVIS has uncertainty. Why do we measure the uncertainty of a FSVIS by using ISts? We do so because it is difficult to compare the size of uncertainty measure values in a FSVIS. Moreover, the size of these measured values can be compared on the basis of the dependence between two ISts. Why do we study ISts optimization on the basis of λ-rough entropy? We do so because in the research of uncertainty measures based on ISts, we always encounter the problem of when the uncertainty measure reaches the maximum and minimum values and how to determine the corresponding optimal ISts.

The work of this article is shown in Fig. 1.

In this part, we discuss the potential values and contributions of this paper. At present, there are more and more research studies on ISs. This paper proposes a new IS, i.e., FSVIS, which extends the category of IS. Related research includes decision-making, reduction, etc. This development will enrich our future research. Because information values of a FSVIS are fuzzy sets, we should combine RST and FST to deal with the problem. These two theories are important and have application value. Additionally, this paper constructs homomorphism between FSVISs. A FSVIS is not only composed of object set and attribute set, but also of the domain on which the information values depend. This kind of homomorphism is more complex than that between general ISs and is an extension. Moreover, this paper proposes and studies the optimal selection of ISts on the basis of λ-rough entropy. It is innovative not only in the study of ISts but also in the study of uncertainty measures in ISs. At the same time, it also closely links ISts and uncertainty measures.

The remainder of this article is structured as follows. Section 2 looks at binary relations and fuzzy sets, and proposes FSVISs and homomorphism between them. Section 3 introduces distances between two information values in a FSVIS. Section 4 examines tolerance relations in a FSVIS. Section 5 presents ISts in a FSVIS, and the dependence and information distance between them. Section 6 discusses group, lattice and mapping characterizations of ISts in a FSVIS. As an application for ISts in a FSVIS, section 7 inquires into entropy measures of uncertainty for a FSVIS by using its ISts, and explores the optimal selection of ISts on account of this uncertainty measure. Section 8 compares this article and some references. Section 9 discusses and concludes this article.

Section snippets

Preliminaries

We first look at binary relations and fuzzy sets, and then introduce FSVISs and homomorphism between them.

Throughout this paper, U, V, AT, BT, S and T signify six finite sets.

The following is given:U={u1,u2,,un},AT={a1,a2,,am},S={s1,s2,,sl}.

Distances between two information values in a FSVIS

To determine the distance between two information values in a FSVIS, a novel distance function should be presented.

Definition 3.1

Let (U,S,AT) be a FSVIS. Set S={s1,s2,,sl}. u,vU, aAT. The distance between a(u) and a(v) is defined asd(a(u),a(v))=(1lk=1l(a(u)(sk)a(v)(sk))2)1/2.

Example 3.2

(Continued from Example 2.8) By Definition 3.1, we haved(L1,L1)=d(L2,L2)=d(L3,L3)=d(L4,L4)=0.0000,d(L1,L2)=d(L2,L1)=0.3912,d(L1,L3)=d(L3,L1)=0.4517,d(L1,L4)=d(L4,L1)=0.3178,d(L2,L3)=d(L3,L2)=0.4254,d(L2,L4)=d(L4,L2)=0.3578,d(L3,L4)

Tolerance relations in a FSVIS

This section discusses the tolerance relation in a FSVIS.

Definition 4.1

Let (U,S,AT) be a FSVIS. Suppose λ(0,1] and AAT. Then RAλU×U can be defined as follows:RAλ={(u,v)U×U:aA,d(a(u),a(v))λ}.

Usually, R{a}λ can be conveniently represented by Raλ.

Obviously, RAλU×U is a tolerance relation, and RAλ=aARaλ.

A tolerance relation is based on a parameter λ. Generally, λ is used in data analysis such as attribute reduction. The value of λ can be chosen according to the accuracy of the reduction results.

By

ISts in a FSVIS

ISts in a FSVIS are considered in this section.

Characterizations of ISts in a FSVIS

This section will introduce group lattice and mapping characterizations of ISts in a FSVIS.

An application

The entropy measure of uncertainty for a FSVIS is explored, and on the basis of this uncertainty measure, the optimal selection of ISts is investigated as an application for the proposed ISts.

Comparison and discussion

In this section, we compare this paper and some related references to see the innovation of this paper more clearly.

1) Li et al. [17] explored ISts in a covering IS. First, they proposed the concept of ISts in a covering IS. Then, they studied the dependence and separation of these two relationships between ISts in a covering IS. They gave properties of ISts in a covering IS. Furthermore, they presented invariant characterizations of covering ISs. Finally, they investigated granularity measures

Conclusions

In this paper, a FSVIS is an IS whose information values are fuzzy sets. The tolerance relation in a FSVIS was given based on the distance between information values. The concept of ISt in a FSVIS was presented on the basis of tolerance relations, properties and invariant characters of these ISts (see Table 7). We examined the dependence and information distance between ISts. This paper displayed group and lattice characterizations of ISts and discussed the entropy measure of uncertainty for a

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.

Acknowledgements

The authors would like to thank the editors and anonymous reviewers for their valuable comments and suggestions, which have helped immensely in improving the quality of this paper. This work was supported by the National Natural Science Foundation of China (11971420), Natural Science Foundation of Guangxi Province (AD19245102, 2018GXNSFDA294003, 2018GXNSFDA281028 and 2018GXNSFAA294134), Guangxi Science and Technology Program (2017AD23056), Guangxi Higher Education Institutions of China

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