Research Article Indiscernibility and Discernibility Relations Attribute Reduction with Variable Precision

Attribute reduction is a popular topic in research on rough sets. In the classical model, much progress has been made in the study of the attribute reduction of indiscernibility and discernibility relations. To enhance the fault tolerance of the model, concepts of both indiscernibility and discernibility relations involving uncertain or imprecise information are proposed in this paper. The attribute reductions of the relative β -indiscernibility relation and relative β -discernibility relation and their algorithms are proposed. When the precision satisfies certain conditions, the reduction of two relation concepts can be converted into a positive region reduction. Therefore, the discernibility matrix is used to construct the reductions of the two relation concepts and the positive region. Furthermore, the corresponding algorithm of the relative β -indiscernibility (discernibility) relation reduction can be optimized when the precision is greater than 0.5, and this is used to develop an optimization algorithm that constructs the discernibility matrix more efficiently. Experiments show the feasibility of the two relation reduction algorithms. More im-portantly, the reduction algorithms of the two relations and the optimization algorithm are compared to demonstrate the feasibility of the optimization algorithm proposed in this paper.


Introduction
e rough set (RS) theory [1,2], proposed by Polish mathematician Zdzislaw Pawlak, is a useful data processing method for dealing with incomplete and inconsistent problems. e investigation of indiscernibility and discernibility relations between objects is an important task in RS theory. In classical RS theory, an indiscernibility relation between objects exists when all of the objects have the same attribute values. A discernibility relation exists between the objects if and only if not every attribute value of the objects is the same. Existing research has studied in-depth indiscernibility and discernibility relations in RSs [3][4][5] in certain research elds. For instance, in political sociology, if one side emphasizes the di erence between countries, this can cause con ict to expand; on the contrary, if one side emphasizes the commonality between countries, this can provide better conditions for negotiation [6]. Because classical RS theory is sensitive to classi cation error, its application is quite limited, and its classi cation ability is poor. To overcome this limitation, the variable precision (VP) model [7,8] is constructed using precision, which processes the data more e ectively, thus advancing the development of RS theory and broadening its application to other elds.
Even more in-depth research has been performed on attribute reduction, which forms the minimum subset of knowledge classi cation by deleting redundant or unrelated attributes according to speci c rules. Attribute reduction has been studied in reference [9][10][11][12][13][14] and has been applied to fault diagnosis, risk assessment, and other elds [15]. In reference [16], a VP model that is more fault-tolerant than the RS model was proposed, and seven forms of reduction in the VP model were discussed. In reference [17][18][19], the concept of attribute reduction in the VP model was proposed, and through a strict proof of the corresponding discernibility matrix, a reduction algorithm based on that matrix was given. Beynon [20] studied the VP reduction by comparing different thresholds. e influence of parameters on classification was studied in the VP model [21]. Chen et al. [22] defined a fuzzy relation on condition attributes, and then, a local attribute reduction algorithm was proposed using the corresponding discernibility matrix. In addition, fuzzy relations [23][24][25] have been studied in [26][27][28]. e aim of most research on attribute reduction is to solve practical problems in different scenarios and has led to various forms of attribute reduction, such as covering reduction [29,30], cost-sensitive reduction [31][32][33], local attribute reduction [22,34,35], incremental attribute reduction [36][37][38], and information entropy reduction [39,40].
Skowron et al. [41,42] were the first to propose using the discernibility matrix to reduce attributes in information systems. is method can determine all the reductions by transforming the function in conjunctive normal form (CNF) into a function in disjunctive normal form (DNF).
is method is called the discernibility matrix-based reduction method and is supported by mathematical proofs. Although its calculation complexity is high, it remains the main method to obtain all reductions.
In this paper, the relative β-indiscernibility relation (β -IR) and relative β-discernibility relation (β −DR) of the VP model are proposed, in which VP model, probabilistic model [43], and decision-theoretic model [44] all extend RS. However, the differences among them lie in the selection of parameters. Because all the results can be obtained using the discernibility matrix, this paper is based on the discernibility matrix. Moreover, the relationships among relative β-IR reduction, relative β-DR reduction, and positive region reduction when the precision is greater than 0.5 are investigated after modifying the decision values of some objects. e structure of this paper is divided into six parts. In Section 2, there are reviews of upper approximation, lower approximation, and β-approximation. In Section 3, the definition and reduction of the relative β-IR are proposed, and then, an algorithm for relative β-IR reduction is given. In Section 4, the definition and reduction of the relative β-DR are proposed, and then, an algorithm for relative β-DR relation reduction is also given. In Section 5, it is shown that when the precision is greater than 0.5, the relative β-IR (relative β-DR) reduction in a decision table is equivalent to the positive region reduction in a new decision table in which some objects' decision values have been updated. en, an optimization algorithm for converting the relative β-IR (relative β-DR) reduction to a positive region reduction is proposed. In Section 6, we used a naive Bayes (NB) classifier, support vector machine (SVM), and decision tree (DT) on UCI datasets to compare the accuracy after reduction and evaluate the effectiveness of the proposed algorithms. Section 7 concludes the paper.

Preliminary
An information table [1,2,5] is also called an information system. It is assumed that the knowledge in an information table is described and represented by a set of rows and a set of columns, in which rows and columns denote objects and attributes, respectively. e information table is a tuple S � (U, At, V a |a ∈ At ), I a |a ∈ At , where U is a universal set, At is a attribute set, V a is nonempty set of values for an attribute a, and I a : U ⟶ V a is a function such that each x ∈ U takes a value I a (x) on attribute a. Given A⊆At, the equivalence relation is defined by R A � (x, y)|x, y ∈ U × U, I a (x) � I a (y) } for each a ∈ A. Moreover, an equivalence class for x on a set of attribute A is denoted as It can be easily seen that this equivalence relation is reflexive, symmetric, and transitive. For A⊆At, the quotient set is denoted by In S � (U, At), if C is a set of condition attributes and D is a set of decision attributes, where At � C ∪ D and C ∩ D � ∅, then an information table is called a decision table. In this paper, a decision table S is written as For the convenience of proof, a set of decision attributes is written as a singleton set, namely Definition 1 (see [1,2]). Let (U, C) be an information system and X be a subset of U, B⊆C. e lower and upper approximations of X are as follows: Definition 2 (see [1,2]). Let (U, C ∪ D) be a decision table in which U/D is a quotient set induced by D. e positive region is defined as follows: Definition 3 (see [19]). Let (U, C ∪ D) be a decision table, where B⊆C. If it satisfies the following conditions,  then B is called the positive region reduction of C.
Given a decision table, its corresponding discernibility matrix [11] of positive region reduction MP � (mp ij ) s×n is as follows: Scientific Programming In a matrix F, s is equal to |Pos C (D)|, n is the cardinality of U, and |Pos C (D)| is the cardinality of the positive region.
Lemma 2 (see [3,4]). For a decision table, suppose that , i is a positive integer, and j is a positive integer.
e proof of eorem 2 is shown in reference [34]. In this section, (μ CD ) β was introduced using the β-cut set of fuzzy set X. In Section 3, the relative β-indiscernibility relation is proposed for decision tables, and then, the corresponding discernibility matrix is proposed.

Relative β-Indiscernibility
Relation Reduction . . , D l induced by the equivalence relation R D , we first define the relative β−indiscernibility relation for β ∈ (0, 1). Definition 6. Given a set B, the relative β-indiscernibility relation is defined as follows: For a binary relation Ind , and hence, it is transitive. Because relation Ind β B (D) satisfies the above three properties, it is an equivalence relation.
In the decision table presented in Table 1, the condition set C contains a 1 , a 2 , a 3 , and D is a decision attribute set. Here, If β � 0.6, then by eorem 2, we have According to Definition 6, we have, for example, Definition 7. Given (U, C ∪ D), if B⊆C satisfies the following two conditions: Given a decision table, a corresponding discernibility matrix M β � (m β ij ) n×n for the relative β-IR is as follows: where n is |U|.
which is the partition of [x] B with respect to R C . We have two cases:  Using Corollary 1, we present the following algorithm for the relative β-lR reduction for (U, C ∪ D).
Considering the decision table given in Table 1, when β � 0.6, the discernibility matrix is constructed as follows: a 3 a 1 , a 3 a 1 , a 3 a 2 , a 3 a 3 a 1 , a 3 a 1 , a 3 a 2 , a 3 a 1   a 1 , a 3 us, the subset a 1 a 3 is the unique attribute reduction of the relative β-IR.

Relative β-Discernibility Relation Reduction
e relative β-indiscernibility relation was proposed in Section 3, and the concept of its complementary relation is proposed in this section. In contrast to the discernibility Scientific Programming matrix of the relative β-IR(β-DR) reduction, in (U, C ∪ D), when the precision is greater than 0.5, the decision values of some objects can be modified, and then, the relative β-discernibility relation is calculated using a positive region reduction.
Definition 8. Let (U, C ∪ D) be a decision table and B be a subset of C. e relative β-discernibility relation is defined as follows: where For Moreover, it is not necessarily transitive.
For different β, the set of Dis β C (D) may also be different. In Table 1  Given a decision table, its corresponding discernibility matrix is M ′ � (m ' ) n×n for the relative β-DR, where n is the cardinality of U. Each element m ' is defined as follows: Proof. e proof is similar to that of Lemma 3 in Section 3.   Because (x i , x j ) ∈ Dis β C (D) is the complement of a set Ind β C (D), it is known from equations (12) and (14) that their discernibility matrices are the same. Using Corollary 2, the algorithm of the relative β-DR reduction for (U, C ∪ D) is proposed as follows.
Considering the decision table given in Table 1, letting β � 0.3, the discernibility matrix is constructed as follows: us, according to Definition 9, a 1 a 3 is the unique result in the decision table.

Optimization of the Reduction Algorithm
Let (U, C ∪ D) be a decision table with D k ∈ U/D. For all x ∈ D k and value d k ∈ V d , let I d (x) � d k , which means that the object x takes the value d k in V d . Obviously, every x in D k has the same decision value d k for the decision attribute. Given precision β ∈ (0.5, 1], for any en, change the values that objects x i have for decision attribute d such that is constructs a new quotient set U/D ′ , denoted by ′ . e decision table that changes the decision values of some objects using this method is called the new decision table and is denoted by (U, C ∪ D ′ ). Indeed, we note that D i ′ may be the empty set for Proof. ∀D k ∈ U/D and for any [x] C satisfying k ∈ U/D', and then, by eorem 2,

Proof. (⇒) If B is the attribute reduction of C, and
us, (μ CD' (x)) 1 � (μ BD' (x)) 1 , and hence, , and there exists x ∈ Pos C (D ′ ) and x ∉ Pos B′ (D ′ ). Hence, From discernibility matrices M β (equations (13) and (15)), the corresponding matrices of the two above concepts are equivalent. For (U, C ∪ D ′ ), B is the reduction of the relative β-IR for β � 1 if and only if B is the reduction of the relative β-DR for β � 1. Proof. By eorem 5 and Proposition 1, Corollary 3 clearly holds. e above shows that the relative β-IR (β-DR) reduction with a precision greater than 0.5 is equivalent to positive region reduction, after modifying the decision values of some objects that satisfy the condition. When the precision of β-IR (β-DR) reduction is greater than 0.5, the decision values of some objects can be modified, and a new decision table can be constructed. In the new decision table, many heuristic algorithms can be used to calculate the positive region (Definition 3) reduction.
According to Corollary 3, an optimization algorithm that converts the relative β-IR (β-DR) reduction to a positive region reduction (CRRPRR) is as follows.
When Scientific Programming space complexity of constructing discernibility matrix is O(|U| 2 ). e time and space complexities of the discernibility function transformation of both algorithms are the same. Obviously, the CRRPRR algorithm is better than the original algorithm.
en, the discernibility matrix is constructed as follows: a 3 a 2 , a 3 a 1 , a 2 a 1 , a 2 us, (a 1 , a 3 ) is the unique attribute reduction of C in (U, C ∪ D ′ ).
It must be explained that the basic condition for the conversion of the relative β -IR (β −DR) reduction into  Scientific Programming positive region reduction is that the precision must be greater than 0.5. When the precision is greater than 0.5, this not only improves the fault tolerance of the model but also ensures its credibility l. If β � 0.5 is given (e.g., as in Table 1), for [x 1 ] C � x 1 , x 2 , because I d (x 1 ) � 0, I d (x 2 ) � 1, the decision values of any objects in x 1 , x 2 cannot be modified according to the conversion method in this paper. Another case occurs when β � 0.8. In this case, none of the decision values of any objects can be modified in Table 1, and the above conversion method is not feasible.

Experimental Analysis
In this section, we evaluate the performances of the proposed algorithms through some comparison experiments. In our experiments, twenty datasets from the UCI were used. All the information is shown in Table 3, in which |U| is the cardinality of U, and |C| and |U/D| are the numbers of condition attributes and decision classes, respectively. Using tenfold cross-validation, three classifiers (kernel NB, fine Gaussian SVM, and a DT) were used to test the classification accuracies of the results after reduction. Kernel NB is a classifier that uses estimated kernel densities. e classification accuracy of fine Gaussian SVM is higher than that of SVM after the Gaussian kernel is introduced. DT is a supervised learning model that learns decision rules. e algorithms were implemented on a MacBook Pro (early 2015) with an Intel(R) Core(TM) i5 CPU at 2.7 GHz and Intel Iris Graphics 6100 GPU. e algorithms in this paper were coded in Python 3.6.8 using scikit-learn 0.20.3.
Because of the requirements of Algorithm 3, the accuracy of the tables in this experiment must be greater than 0.5. e runtime results for different precisions (0.7, 0.8, and 0.9) are shown in Figure 1. In this figure, the yellow line represents the average time for Algorithm 3 to run for each precision. e experimental results show that Algorithm 1, Algorithm 2, and Algorithm 3 obtain the same results, but the runtime of the latter is less than that of the former. Although the cost of converting CNF to DNF is high, the advantage of the Algorithm 3 is obvious when the number of objects is large and the number of attributes is relatively small.
Because the proposed algorithms can obtain all the results, the mean classification accuracy is reported. For all precision values, the classification accuracies obtained by kernel NB, fine Gaussian SVM, and DT are shown in Figures 2-4, respectively.
In Figure 2, in most cases, the classification accuracy of the kernel NB classifier is low for the reduction results obtained from a low precision value. For example, for the HS  For all data sets except for Wine, SH, and TAE, a higher precision value leads to a higher classification accuracy for the fine Gaussian SVM classifier in Figure 3. However, the classification accuracy is relatively low. In most cases, when the precision is high, the reduction results lead to higher classification accuracies for the DT classifier, but the classification accuracy is low. For example, on the HR dataset, given precision values of 0.7, 0.8, and 0.9, the accuracy results are 62.33%, 70.15%, and 76.01%, respectively.
is experiment demonstrates the feasibility of the concept proposed in this paper. When the precision is less than 0.5, its reliability is not high. Moreover, the CRRPRR optimization algorithm and the relative β-IR (β-DR) reduction algorithm could not be linked. erefore, the precision of the algorithms must be greater than 0.5. Figures 2-4 show that when the precision is larger, the classification accuracy is higher when the results are obtained by the proposed algorithms. On the contrary, it is lower when the precision is lower.

Conclusions
is study extends the work of [5,22,34,41] to investigate the relative β-IR (β-DR) for the first time using a discernibility matrix-based method. Under certain conditions, precision β > 0.5, and the relationship between the relative β -IR (β −DR) reduction and positive region reduction was found by modifying some decision values, which is of a certain significance to the study of VP RSs. e corresponding optimization algorithm was then proposed. e discernibility matrix corresponding to the relative β -IR (β −DR) reduction has a high time complexity of O(|U| 2 × |C|), whereas the time complexity for constructing a positive region reduction is much less. erefore, when the precision β > 0.5, although the reduction results are the same, the optimization CRRPRR algorithm reduces the computational complexity. In addition, because the attribute importance   Scientific Programming cannot be calculated in the relative β -IR (β −DR) reduction to obtain the results, we modified some of the decision values that satisfy the condition so that the attribute importance can also be calculated in the new decision table. e feasibility of the algorithm proposed in this paper was demonstrated by experimental analysis. In future, we will attempt to remove the restriction of equivalence relation and further study problems such as precision reduction and the relative β -IR(β −DR) reduction.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.