Matrix Approaches for Covering-Based Multigranulation Fuzzy Rough Set Models

Chang, Zaibin; Wei, Junchao

doi:https://doi.org/10.1155/2023/3051168

Journal of Mathematics

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Research Article | Open Access

Volume 2023 | Article ID 3051168 | https://doi.org/10.1155/2023/3051168

Matrix Approaches for Covering-Based Multigranulation Fuzzy Rough Set Models

Zaibin Chang¹and Junchao Wei¹

Academic Editor: Tareq Al-shami

Received08 Oct 2022

Revised05 Dec 2022

Accepted22 Dec 2022

Published04 Jan 2023

Abstract

Multigranulation rough set theory is one of the most effective tools for data analysis and mining in multicriteria information systems. Six types of covering-based multigranulation fuzzy rough set (CMFRS) models have been constructed through fuzzy -neighborhoods or multigranulation fuzzy measures. However, it is often time-consuming to compute these CMFRS models with a large fuzzy covering using set representation approaches. Hence, presenting novel methods to compute them quickly is our motivation for this paper. In this article, we study the matrix representations of CMFRS models to save time in data processing. Firstly, some new matrices and matrix operations are proposed. Then, matrix representations of optimistic CMFRSs are presented. Moreover, matrix approaches for computing pessimistic CMFRSs are also proposed. Finally, some experiments are proposed to illustrate the effectiveness of our approaches.

1. Introduction

As a kind of extension of classical rough sets [1, 2], covering-based rough sets [3, 4] were proposed to manage this kind of covering data. In theory, this data is a family of some subsets of a universe (the set of all objects). Inspired by it, several covering-based rough set models [5, 6] were established to deal with practical problems, such as feature selection [7–9] and rule synthesis [10–13]. As the same with rough set theory, Zadeh’s fuzzy set theory [14] can also manage issues of imperfect information. Therefore, many researchers investigated their connection and distinction [15, 16].

Fuzzy covering rough sets [17–19] were proposed with a special membership of “1.” Then, Ma [20] generalized it to a parameter and constructed fuzzy -covering rough set models. Inspired by Ma’s work, many researches were done. For example, several new fuzzy -covering rough sets models were presented [21–23]; fuzzy -neighborhoods were investigated [24, 25]; fuzzy -covering approximation spaces were used in decision-making [26, 27]. Moreover, many researchers applied matrix methods to deal with fuzzy -covering data, which can simplify calculations in the process of data analysis. For example, Ma [20] gave two types of fuzzy covering rough set models’ matrix representations; Yang and Hu [28] investigated some of the other models by matrices; Huang et al. [29] and Mao et al. [30] proposed matrix methods for computing the reduction of a fuzzy -covering; Wang et al. [31] used matrices to deal with the issues of fuzzy description reduction and group decision-making.

But, all these works above were done in a fuzzy -covering approximation space. Therefore, Zhan et al. [32] generalized a fuzzy -covering to -fuzzy -covering and presented six types of covering-based multigranulation fuzzy rough set (CMFRS) models. Based on Zhan’s work, Zhan et al. [33] presented multigranulation -fuzzy covering rough set models under fuzzy -neighborhoods; Ma et al. [34] constructed three types of multigranulation -fuzzy covering rough set models under complementary fuzzy -neighborhoods; Atef et al. [35] presented soft fuzzy covering rough set models under complementary soft neighborhood; Atef and El Atik [36] proposed some extensions of multigranulation fuzzy covering rough sets from new perspectives. It is important to find a quick way to calculate these models, which will save time for rapid data mining and analysis and is also the motivation of this paper. In this article, we use matrix methods to investigate Zhan’s et al. CMFRS models [32], since Zhan’s CMFRS models are the basis of all these models above.

Inspired by [37], we present matrix representations for computing CMFRS models [32] in this paper. Moreover, the effectiveness of our matrix approaches is given by some experiments. Firstly, several new matrices and corresponding operations are proposed. Then, matrix representations of optimistic CMFRSs are presented. Furthermore, matrix approaches are also used for computing pessimistic CMFRSs. Finally, several experiments are proposed to show the effectiveness of our matrix approaches.

The rest of this article is listed as follows. Section 2 recalls several basic notions about CMFRSs. In Section 3, some novel matrices and corresponding operations are proposed. Furthermore, matrix representations of optimistic CMFRSs are proposed. In Section 4, we present matrix representations of pessimistic CMFRSs. In Section 5, we illustrate the efficiency of our matrix method for computing CMFRSs through several experiments. This article is concluded and further work is indicated in Section 6.

2. Fundamental Definitions

This section shows several fundamental concepts related to CMFRSs. Supposing is a nonempty and finite set called a universe.

To connect covering rough sets with fuzzy sets, Ma [20] proposed the concept of fuzzy -covering approximation space.

Definition 1 (see [20]). Let be a universe and be the fuzzy power set of . For any , we call , with , a fuzzy -covering of , if for all . We also call a fuzzy -covering approximation space.

Several covering-based fuzzy rough set models were established under a fuzzy -covering approximation space. Furthermore, Zhan et al. [32] extended a fuzzy -covering to -fuzzy -coverings. We call a -fuzzy -coverings of , where is a fuzzy -covering of for each . Then, six types of CMFRSs were presented in [32].

Definition 2 (see [32]). Letbe a -fuzzy -coverings of For eachand , we define: (1) the OMGFRLAO and the OMGFRUAO ; (2) the PMGFRLAO and the PMGFRUAO ; (3) the-OMGFLAO and the-OMGFUAO; (4) the-PMGFLAO and the-PMGFUAO; (5) the OMGFLAO and the OMGFUAO; (6) the PMGFLAOand the PMGFUAOas follows:(1) (2) (3) (4) (5) (6) where is defined as the fuzzy -neighborhood of ; and Is defined as a multigranulation fuzzy measure degree of and under .

Example 1. is a 2-fuzzy -coverings of , where and . They are listed in Tables 1 and 2, respectively.
Then, and are shown in Tables 3 and 4.
Hence, we have all and which are shown in Tables 5 and 6.
For and , we have(1), (2), (3), (4), (5), (6),

3. Matrix Approaches for Optimistic CMFRSs

In this section, we present some new matrices and use them to compute approximation operators of optimistic CMFRS models presented in Definition 2. They are OMGFRLAO and OMGFRUAO ; -OMGFLAO and -OMGFUAO ; OMGFLAO and OMGFUAO . In [37], we have given the matrix representations of OMGFRLAO and OMGFRUAO , OMGFLAO , and OMGFUAO . In this section, we investigate them further and give the matrix representations of -OMGFLAO and -OMGFUAO .

In [20], Ma has presented some matrices and corresponding operations to compute a fuzzy covering rough set model. They considered the -fuzzy -coverings of when .

Definition 3 (see [20]). Let be a fuzzy -covering of . We call a matrix representation of and call the Boolean matrix a-matrix representation of , where

Example 2. (Continued from Example 1).Then,

Let and be two matrices. Then and are defined as follows: and , where and . denotes the transpose of . Based on Definition 3, the matrix representations of OMGFRLAO and OMGFRUAO can be obtained as follows when :

Proposition 1 (see [20]). Let be a fuzzy -covering of :

Example 3. (Continued from Example 1).

Inspired by Proposition 1, we give the matrix representations of OMGFRLAO and OMGFRUAO when .

Proposition 2 (see [37]). Let be a -fuzzy -coverings of . Then, for any

Proof. By Definition 2 and Proposition 1, it is immediate.

Example 4. (Continued from Example 1).

At the same time, the matrix representation of has been proposed in [20] as follows:

Lemma 1 (see [20]). Let be a fuzzy -covering of . Then, .

Example 5. (Continued from Example 1). Based on Example 2, we have

On the contrary, we present the matrix representations of -OMGFLAO and -OMGFUAO ; OMGFLAO and OMGFUAO . First, we give the definition of the matrix representation of all multigranulation fuzzy measure degrees .

Definition 4 (see [37]). Let be a fuzzy -covering of and . We call the matrix representation of all multigranulation fuzzy measure degrees , where . We also denote as the -matrix representation of , where

Proposition 3 illustrates that the matrix representation of all multigranulation fuzzy measures degrees is symmetric.

Proposition 3. Let be a fuzzy -covering of . Then, is a symmetric matrix.

Proof. It is immediate, by Definition 4.

Given three matrices , , and . Then, , , and are denoted by: , , and, where and . Hence, all can be computed by matrices.

Proposition 4 (see [37]). Let be a fuzzy -covering of . Then,

Proof. Assume , , and . By Lemma 1, . Therefore, .

Example 6. (Continued from Example 1). Let . Based on Example 5, we haveHence, we have

Let , be two matrices. Then and are defined as follows: and , where and . If is a Boolean matrix and , then we define and as follows:

After that, we investigate the matrix representations of -OMGFLAO and -OMGFUAO . We suppose , i.e., -OMGFLAO and -OMGFUAO should be considered first. Hence, for any

Proposition 5. Let be a fuzzy -covering of and . Then, for any where and are first than and , respectively.

Proof. Suppose, , , and ,whereHence,Therefore,On the contrary, supposeHence,Therefore,

Inspired by Proposition 5, we propose the matrix representations of -OMGFLAO and -OMGFUAO as follows.

Theorem 1. Let be a -fuzzy -coverings of and . Then, for any where and are first than and , respectively.

Proof. It is immediate, by Proposition 5.

Example 7. (Continued from Example 1). By Example 6, we get and . Then,

Finally, we propose the matrix representations of OMGFLAO and OMGFUAO .

Theorem 2 (see [37]). Let be a -fuzzy -coverings of . Then, for any

Proof. By Definition 2 and Proposition 2, it is immediate.

Example 8. (Continued from Example 1). We have

4. Matrix Approaches for Pessimistic MCFRS

Based on Section 3, we mainly use matrix methods to calculate approximation operators of pessimistic MCFRS models presented in Definition 2. They are PMGFRLAO and PMGFRUAO ; -PMGFLAO and -PMGFUAO ; PMGFLAO and PMGFUAO .

Firstly, we consider the matrix representations of PMGFRLAO and PMGFRUAO when .

Proposition 6. .Let be a fuzzy -covering of .

Proof. By Proposition 1, it is immediate.

Example 9. (Continued from Example 1). In Example 5, we haveThen,

Then, we consider the statement of .

Proposition 7. Let be a -fuzzy -coverings of . Then, for any

Proof. By Definition 2 and Proposition 6, it is immediate.

Example 10. (Continued from Example 1)

Inspired by Proposition 5, we present the matrix representations of -PMGFLAO and -PMGFUAO as follows:

Theorem 3. Let be a -fuzzy -coverings of and . Then, for any where and are first than and , respectively.

Proof. By Proposition 5, it is immediate.

Example 11. (Continued from Example 1). In Example 6, we getThen,

Finally, we investigate the matrix representations of OMGFLAO and OMGFUAO .

Theorem 4. Let be a -fuzzy -coverings of . Then, for any

Proof. By Definition 2 and Theorem 2, it is immediate.

Example 12. (Continued from Example 1). We have

5. Applied Examples

To evaluate the efficiency of our matrix method for computing covering-based multigranulation fuzzy rough approximation operators, we construct some fuzzy -covering approximation spaces to compare the matrix approaches with set approaches. In Definition 3, we know that any fuzzy -covering approximation space can be seen as a matrix. The procedure of constructing the fuzzy -covering matrix is as follows: (1) The elements of the matrix are randomly chosen from . (2) For any row of the matrix, if the maximal number of the row is less than , then we denote as its maximal number. Hence, the matrix is the fuzzy -covering matrix, i.e., a fuzzy -covering approximation space. Hence, we consider a fuzzy -covering approximation space only in this section. That is to say, we consider for optimistic CMFRSs and pessimistic CMFRSs. Since optimistic CMFRSs and pessimistic CMFRSs are the same when , we consider optimistic CMFRSs only in this section.

Finally, we compare the computational time of matrix methods and set methods (i.e., traditional methods) with different values of , sizes of a universe and sizes of a fuzzy -covering. All of the experiments were carried out on a personal computer with 64-bit Windows 10, ADM Ryzen 7 3700X 8-Core Processor 3.59 GHz, and 16 GB memory. The programming language was MATLAB r2016a.

To compare the computational time of the matrix method and set method with different values of in Figure 1, we set the size of to 500 and the size of to 500. The value of ranges from 0.1 to 1, gradually increasing by a step of 0.1. In Figure 1, we can see that the computational time of the matrix method and set method decreases with the gradual increase in the value of . The matrix method is more efficient than the set method with different values of , especially with a small value of .

(a)

(b)

To compare the computational time of the matrix method and set method with different sizes of in Figure 2, we set the value of to 0.5 and the size of to 500. The size of ranges from 100 to 1000, gradually increasing by a step of 100. In Figure 2, we can see that the computational time of the matrix method and set method increases with the gradual increase in the size of . The matrix method is more efficient than the set method with different sizes of .

(a)

(b)

To compare the computational time of the matrix method and set method with different sizes of in Figure 3, we set the value of to 0.5 and the size of to 500. The size of ranges from 200 to 1100, gradually increasing by a step of 100. In Figure 3, we can see that the computational time of the matrix method and set method increases with the gradual increase in the size of . The matrix method is more efficient than the set method with different sizes of , especially with large size of .

(a)

(b)

6. Conclusions

In this article, we present the matrix representations of MCFRS models. Hence, through some experiments, calculations can be easily implemented by computers. It gives a way to study the reduction of multigranulation fuzzy covering information systems by matrices in the future. Other multigranulation fuzzy covering rough set models (such as [34–36]) can also be represented by matrices in the future. On the contrary, the recent generalized types of fuzzy sets: (i) (2, 1)-Fuzzy sets: properties, weighted aggregated operators, and their applications to multicriteria decision-making methods [38]; (ii) new generalization of fuzzy soft sets: (a, b)-fuzzy soft sets [39]; (iii) three-way fuzzy sets [40] will be connected with the research content of this paper in further research.

Data Availability

There are no data used in this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the Natural Science Foundation of Shaanxi Province, China, under grant nos. 2022JM-053 and 2021JQ-894.

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Copyright © 2023 Zaibin Chang and Junchao Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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