Information structures in a fuzzy set-valued information system based on granular computing
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:
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 be a FSVIS. Set . , . The distance between and is defined as
Example 3.2 (Continued from Example 2.8) By Definition 3.1, we have
Tolerance relations in a FSVIS
This section discusses the tolerance relation in a FSVIS.
Definition 4.1 Let be a FSVIS. Suppose and . Then can be defined as follows:
Obviously, is a tolerance relation, and .
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
References (58)
- et al.
Sequential covering rule induction algorithm for variable consistency rough set approaches
Inf. Sci.
(2011) - et al.
Non-iterative procedure incorporated into the fuzzy identification on a hybrid method of functional randomization for time series forecasting models
Appl. Soft Comput.
(2019) - et al.
On the connection of hypergraph theory with formal concept analysis and rough set theory
Inf. Sci.
(2016) - et al.
Measures of uncertainty for neighborhood rough sets
Knowl.-Based Syst.
(2017) A graded approach to cardinal theory of finite fuzzy sets, part I: graded equipollence
Fuzzy Sets Syst.
(2016)- et al.
Dynamic variable precision rough set approach for probabilistic set-valued information systems
Knowl.-Based Syst.
(2017) - et al.
Extracting semantic event information from distributed sensing devices using fuzzy sets
Fuzzy Sets Syst.
(2018) - et al.
Information structures in a covering information system
Inf. Sci.
(2020) - et al.
A novel attribute reduction approach for multi-label data based on rough set theory
Inf. Sci.
(2016) - et al.
Three-way decision approaches to conflict analysis using decision-theoretic rough set theory
Inf. Sci.
(2017)
International asset allocation optimization with fuzzy return
Knowl.-Based Syst.
Principles of inclusion and exclusion for interval-valued fuzzy sets and IF-sets
Fuzzy Sets Syst.
Rough sets and Boolean reasoning
Inf. Sci.
Rudiments of rough sets
Inf. Sci.
Knowledge structure, knowledge granulation and knowledge distance in a knowledge base
Int. J. Approx. Reason.
Fuzzy inventory models: a comprehensive review
Appl. Soft Comput.
A rough set approach for approximating differential dependencies
Expert Syst. Appl.
A unified framework for characterizing rough sets with evidence theory in various approximation spaces
Inf. Sci.
The lattice and matroid representations of definable sets in generalized rough sets based on relations
Inf. Sci.
New measures of uncertainty for an interval-valued information system
Inf. Sci.
Fuzzy information granular structures: a further investigation
Int. J. Approx. Reason.
Relational interpretations of neighborhood operators and rough set approximation operators
Inf. Sci.
A characterization of novel rough fuzzy sets of information systems and their application in decision making
Expert Syst. Appl.
A temporal-spatial composite sequential approach of three-way granular computing
Inf. Sci.
Class-specific attribute reducts in rough set theory
Inf. Sci.
Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic
Fuzzy Sets Syst.
Generalized dominance rough set models for the dominance intuitionistic fuzzy information systems
Inf. Sci.
Uncertainty measures for interval set information tables based on interval δ-similarity relation
Inf. Sci.
Information structures and uncertainty measures in a fully fuzzy information system
Int. J. Approx. Reason.
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