New Variable Precision Reduction Algorithm for Decision Tables

Variable precision reduction (VPR) and positive region reduction (PRR) are common definitions in attribute reduction. The compacted decision table is an extension of a decision table. In this paper, we propose another extension, called the weighted decision table. In both types of decision tables, VPR is defined, and the corresponding discernibility matrices for the PRR are proposed. Then, algorithms for obtaining the PRR from the discernibility matrices are presented. In both types of decision tables, the relationship between VPR and PRR is established by comparing the corresponding discernibility matrices. If the precision of the VPR meets the given conditions, then the PRR algorithms can be used to obtain the results after modifying some decision values in the decision tables. An analysis of the modification process of the decision values and the compression process of decision tables is used to propose a new algorithm for VPR in decision tables that ensures credibility. The effectiveness of the proposed algorithm was evaluated by an experimental comparison with existing VPR algorithms.


I. INTRODUCTION
As the number of data features increases, the cost of analyzing and processing multi-dimensional data increases, and hence the research on attribute reduction has become a popular topic in recent years. Attribute reduction is an important task in rough sets [1] which were proposed by Pawlak in the 1980s and are used to handle problems involving uncertainty [2]. In recent years, the research on rough sets has combined rough sets with fuzzy sets [3], [4], [5] evidence theory [6], [7] information entropy [8], [9], [10], [11] and other fields, and has made great progress. Attribute reduction removes redundant features and retains the subset with the minimum number of attributes to improve the efficiency of the algorithm. In its initial stages, attribute reduction research mainly focused on The associate editor coordinating the review of this manuscript and approving it for publication was Senthil Kumar . the definitions of positive region reduction (PRR) [12], [13], [14], [15] and absolute reduction [16]. As rough set research has further developed, some concepts such as variable precision reduction (VPR) [17], [18], [19] and assignment reduction [20] have been proposed according to practical needs, and they have been successfully applied in power systems [21] bioinformatics [22] text categorization [23] and industrial applications [24].
In recent years, attribute reduction methods have been used in granular computing [25], [26] formal concept analysis [27], three-way decision analysis [28], [29] and many other fields. Attribute reduction in classical rough sets has been studied thoroughly. The indiscernibility relations between any object have been constructed to study the relationship between the positive region and indiscernibility relation sets in depth [30], [31], [32]. Because some information in information tables will be missing, as the binary relation is equivalence, the reduction method research is limited.
To improve the practicability of the reduction algorithm, the equivalence relation has been extended to non-equivalent binary relations such as tolerance relations [33], [34], [35] and similarity relations [17], [36], [37] and the definition of reduction was extended again after the theory of rough sets was generalized. Reduction methods based on information view [38] or the discernibility matrix [13], [14], [15], [18] and heuristic methods based on attribute importance [37] have been proposed. In addition, considering the dynamic changes of objects in a universe, incremental attribute reduction [39], [40] has also made new progress.
On the basis of decision tables, compacted decision tables [41] are described, and the weighted decision table, which is obtained after the decision table has been compressed, is proposed. In both types of decision table, the relationship between VPR and PRR is studied from the point of view of discernibility matrix construction, and an optimization algorithm for VPR is proposed for each type of decision table. Based on the above research, the construction processes of transforming the decision table into each type  of decision table are compared, and the VPR in the decision  table is transformed into the PRR in the weighted decision  table for calculation, thus improving the efficiency of the VPR algorithm.The core of the VPR algorithm, which is based on a discernibility matrix, is to construct the discernibility matrix. However, the time complexity of constructing the discernibility matrix in existing reduction algorithms is O(n 2 ). The main contribution of this paper is to develop the PRR algorithm for decision tables. The research framework is shown in Figure 1.
The structure of the paper is as follows. Section II introduces the basic concepts of the positive region and variable precision in decision tables. In Section III, the compacted  decision table is introduced, the weighted decision table is proposed, and VPR for both types of decision tables is proposed. In Section IV, in the compacted decision table and the weighted decision table, the PRR corresponding discernibility matrices are proposed, and then the corresponding algorithms are proposed. Based on the discernibility matrix, the relationship between VPR and PRR is analyzed in two types of decision tables. In Section V, the optimization algorithms of VPR are proposed for two types of decision tables, and the optimization algorithm of VPR in decision tables is also proposed. The Section VI verifies the proposed algorithms through experiments. Finally, the conclusion summarizes the paper.

II. PRELIMINARIES
AT is a finite nonempty set of attributes, V a is a nonempty set of values for a ∈ AT , and f (x, a) : U → AT is a function, where f (x, a) takes a value on attribute a.
When A is an nonempty subset of AT , f (x, A) is denoted as a value on attribute set A. An equivalence relation is defined by where f (x, a) and f (y, a) are the attribute values of x and y on a, respectively. Class [x] A is the equivalence class determined with respect to A and is denoted by For the tuple S, if AT = C ∪ D, and C ∩ D = ∅, C is the condition attribute set, D is the decision attribute set. Then, tuple S is called a decision table, briefly written as (U , C ∪D).
Definition 1 ( [1], [2]): Let X ⊆ U , B ⊆ C, and x ∈ U . Then, the lower and upper approximations of X are defined as [2]): Let (U , C ∪D) be a decision table. Then, U /R D = {D 1 , D 2 , . . . , D t } is the quotient set determined by D. The positive region is defined as For a decision table (U , C ∪ D), if Pos C D = U , then the decision table is called consistent; otherwise, it is inconsistent.
Definition 3: Given X ⊆ U , for each x ∈ U , the characteristic function λ X (x) is defined as follows: Lemma 1 [18]: is an equivalence class on the equivalence relation R. Then, W R λ X is as follows: where T denotes the transpose. The Boolean column vector λ X = (λ X (x 1 ), λ X (x 2 ), . . . , λ X (x n )) T for x i ∈ U . In addition, W R is denoted as Definition 4 [18]: Let R be an equivalence relation on U and β ∈ (0, 1]. Then, the β-approximation of X is defined as where Using the β-approximation and the quotient set, a fuzzy matrix can be constructed.
According to the above definition of a fuzzy matrix, VPR can be defined in the decision table as follows.
Definition 5 ( [17], [18]): Let (U , C ∪ D) be a decision table, β ∈ (0, 1], and B ⊆ C. Then, B is called the VPR of C if it satisfies the following conditions: To summarize, the definition of VPR and its discernibility matrix in a decision table were introduced in this section. Objective data can be described by decision tables. However, changes in some data sets may cause the form of the decision tables to also change. In this paper, the research framework is shown in Figure 1.

III. VPR IN COMPACTED AND WEIGHTED DECISION TABLES
On the basis of retaining all decision table information, this section describes how the number of objects (rows) in the decision table is compressed to change the form of the decision table, thus forming two types of decision tables. Compacted decision tables are formed by summing the number of identical decision attribute values in any equivalence class and then adding the number in the decision tables [41]. By contrast, the weighted decision table is formed by determining the decision attribute values of any two objects in any equivalence class and adding weights.
Definition 6 [41]: Let (U , C ∪ D) be a decision table, and . . , D t } is the quotient set determined by decision set D, and the decision values are v d i ∈ V D . Then, Operator |·| is used to denote the cardinality of a set.
The following is an example to illustrate the transformation of a decision table into a compressed decision table. Given (U , C ∪ D) (Table 1), the compacted table U ′ , C ∪ D ′ (Table 2) is obtained by calculating the number of identical decision values in any equivalence class. VOLUME 11, 2023  The process of the weighted decision table construction is as follows. In any equivalence class, for the objects with the same decision value, only one object is retained and the other object(s) are deleted, and the number of identical decision values is used to determine the weight, forming the weighted decision table.
Definition 7: The following is an example to illustrate the transformation of a decision table into a weighted decision table. Given (U , C ∪ D) (Table 1), the weighted decision table is obtained by Definition 7.
In compacted and weighted decision tables, the definitions of VPR are as follows.  Table 2, Definition 9: Let U ′ , C ∪ D ′ be a compacted decision table, where β ∈ (0, 1], and B ⊆ C. Then, B is called the VPR of C if it satisfies the following conditions: Similarly, to define VPR in a weighted decision table, (µ CD ′′ (x)) β is defined in a weighted decision table as follows.
Definition 10: Compacted and weighted decision tables are similar in that both reduce the number of objects (rows) in a decision table. The difference is that the cardinality of any equivalence class in the compacted decision table is 1, whereas an equivalence class may have several objects in the weighted decision table.

IV. ATTRIBUTE REDUCTION IN COMPACTED AND WEIGHTED DECISION TABLES
Because the form of the compacted and weighted decision tables have changed, the method for calculating the positive region changes accordingly. In this section, the corresponding discernibility matrices for the PRR in both types of decision tables are proposed. In addition, these discernibility matrices are derived so that the PPR algorithms for both types of decision tables can be developed.

A. PRR IN COMPACTED DECISION TABLES
For x ∈ Pos C D ′ , x has only one decision value in the compacted decision table. Therefore, the positive region is calculated as follows.
Definition 12: The positive region is defined as follows: ij ) n×n is proposed as follows: where n is the number of objects.
42704 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
∈ R a , and therefore m ′ ij ̸ = ∅. The following theorem can be obtained from Lemma 2.
From Theorem 2, we have the following corollary. According to Corollary 1, for a compacted decision table, the corresponding discernibility matrix for PRR is constructed as described in Algorithm 1. 3: compute Pos C D ′ ; 4: end for 5: for all x in Pos C D ′ do 6: for all x in U ′ do 7: if

Algorithm 1 Discernibility Matrix Construction for PRR
end if 10: end for 11: end for // the discernibility matrix is constructed 12: return M ′ ; The form of the decision table has been changed in the compacted decision table. Hence, the discernibility matrix is constructed to obtain the reduction results by calculat- ing f (x, d).

B. PRR IN WEIGHTED DECISION TABLES
The weighted decision table is another form of decision table, and we calculate the positive region according to the weight. Simultaneously, the weight is used to construct the discernibility matrix. The positive region in the weighted decision table is defined as follows.
Definition 13: The positive region is defined as According to Definition 13, the corresponding discernibility matrix M ′′ = (m ′′ ij ) n×n is expressed as follows: where n is the number of objects.
The proof is similar to that of Lemma 2. The following can be obtained from Lemma 3. According to Corollary 2, the corresponding discernibility matrix of the PPR is constructed for the weighted decision table as described in Algorithm 2.
The method of transforming from the conjunctive normal form (CNF) to the disjunctive normal form (DNF) is an NPhard problem. To improve efficiency, a binary programming algorithm can be used to quickly obtain the result [25]. The pseudocode of this algorithm is given in Algorithm 3.
Algorithm 1 is suitable for the compacted decision table, and Algorithm 2 is suitable for the weighted decision table.
In the process of constructing the discernibility matrix, the former obtains the positive region by calculating the f (x, d), whereas the latter obtains the positive region by calculating the weight.

C. RELATIONSHIP BETWEEN VPR AND PRR
Using Definitions 9 and 11, this section analyzes the relationship between VPR and PRR in decision tables from the perspective of constructing discernibility matrix. Moreover, it VOLUME 11, 2023 3: compute Pos C D ′′ ; 4: end for 5: for all x in Pos C D ′′ do 6: for all x in U ′′ do 7: if The proof is similar to that of Theorem 4. Using the discernibility matrix, Theorems 4 and 5 show that the result of VPR with β = 1 is the result of PRR. The time complexity of VPR for constructing the discernibility matrix is O(|U | 2 |C|), but the time complexity of PRR is O(| Pos C D||U ||U /D||C|), which is lower.

V. PROPOSED VPR ALGORITHMS
When calculating VPR, the decision values or weights of some objects in equivalence classes that meet the condition β > 0.5 are modified. For both types of decision tables, because the relationship between VPR and PRR is established based on the discernibility matrix, the PPR can be used for the calculation to optimize the calculation of VPR. The construction processes of transforming the decision Note that for (U ′ , C ∪ D ′ ), because of the possibility that Proof: There are two cases: VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

Algorithm 4 Optimization Algorithm for VPR
end if 6: end for 7: compute Pos C D ′ ; 8: for all x in Pos C D ′ do 9: for all x in U ′ do 10: if 12: end if 13: end for 14: end for // the discernibility matrix is constructed 15: return M ′ ; 16: run Algorithm 3; 17: return B; 1 . According to Theorem 4, VPR with β = 1 can be transformed into PPR for calculation in a compacted decision table. The new algorithm for constructing the VPR discernibility matrix in the compacted decision table is given in Algorithm 4.
An example based on Table 2 is given to illustrate the feasibility of the proposed Algorithm 4 given β = 0.65. Table 2 Table 4. 2) In Table 4,

1) In
3) The corresponding matrix M ′ is as follows: x j be deleted; //i ̸ = j; 6: end if 7: end for //(U ′′ , C ∪ D ′′ , W ) is updated as (U new′′ , C ∪ D new′′ , W ′′ ) 8: compute Pos C D new′′ ; 9: for all x in Pos C D new′′ do 10: for all x in U ′′ do 11: if 13: end if 14: end for 15: end for // the discernibility matrix is constructed 16: return M ′′ ; 17: run Algorithm 3; 18: return B; Using Theorems 5 and 7, the new algorithm for constructing the VPR discernibility matrix for a weighted decision table is given in Algorithm 5.
An example based on Table 3 Table 3, < β, no decision values in [u 1 ] C are modified. Therefore, the new weighted decision table (Table 5) is updated. 2) In Table 5, Pos C D ′ = {u 5 , u 6 }.
3) The corresponding matrix M ′ is as follows: The main reason for setting β > 0.5 is the high confidence of the reduction. In a VPR process, when the precision is greater than 0.5, the new compacted/weighted decision table is formed after modifying some decision values of the Algorithm 6 New VPR Algorithm for Decision Tables  Input: decision table ( end if 6: end for 7: while |U | > 0 do 8: for all x in U do 9: any information of x i be deleted; //x i = x 10: end for 11  compacted/weighted decision table. It has been proved that VPR with β > 0.5 in the two types of decision tables is equal to the VPR with β = 1. Subsequently, it was proved that VPR with β = 1 is the PRR based on the discernibility matrix. Therefore, after changing the form of the decision table, optimized VPR algorithms were proposed. For VPR in a decision table, considering the number of modified decision values and the number of objects in equivalence class, in contrast to the time complexity of the process of constructing the weighted decision table, the time complexity of constructing d i is O(|U /D|)(U /C) (|[x] C |) when constructing the compacted decision table. Therefore, the method of transforming the decision table into a weighted decision table is adopted after modifying the decision values, and the VPR is calculated using Algorithms 2 and 3. Compared with the existing VPR algorithms [17], [19] the time complexity of discernibility matrix construction is O |U | 2 |C| , whereas the time complexity of Algorithm 6 to construct the discernibility matrix is O | Pos C D ′′ ||U ′′ ||C| , then | Pos C D ′′ ||U ′′ | < |U | 2 . Hence, Algorithm 6 has relatively high computational efficiency.

VI. EXPERIMENTAL ANALYSIS
To evaluate the performance of the algorithms, we selected 10 datasets from the UCI datasets and compared the proposed 42708 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  algorithms with existing algorithms VPR-DM [17] and KNR-DM [19]. Three classifiers including fine Gaussian naïve Bayes (NB), decision trees (DT), and support vector classification (SVC) were used to test the classification accuracy for different reduction results, in which ten-fold cross-validation was used. All experiments were coded in Python 3.7 and were tested on a Lenovo R7000 PC (early 2020s) with an AMD Ryzen 5 CPU at 3.0 GHz and Radeon Graphics 4600H GPU. Table 6 summarizes the details of the selected datasets, in which |U | indicates the number of objects, |C| indicates the number of condition attributes, and |U /D| indicates the number of classes. To ensure high confidence, the precision should be greater than 0.5 in the experiment.
The compacted and weighted decision tables were constructed for each dataset. Figures 2 and 3 show the runtimes for constructing the discernibility matrices using VPR and  the optimized algorithms for both types of decision tables. Figure 2 compares the runtimes for VPR and Algorithm 4 at different precisions for the compacted decision tables. For example, when β = 0.65, the number of objects in compacted decision tables are 131, 691, respectively, for A.S.N. dataset and B.A. dataset. It should be noted that the heuristic VPR algorithms may be efficient, but the focus of this paper is to study the reduction algorithms based on the discernibility matrix. Figure 3 compares the runtimes of the VPR and Algorithm 5 at different precisions on the weighted decision tables. Obviously, the runtimes of Algorithms 4 and 5 are shorter than those of the VPR algorithms for both types of decision tables.
Three algorithms, VPR-DM, KNR-DM, and Algorithm 6, were compared with respect to parameter |U | for different precisions. The results are listed in Table 7. The numbers of rows of the discernability matrices constructed by Algorithm 6 are smaller than those of the other algorithms. The lengths of the reducts of the three algorithms are presented in Table 8. Because all three algorithms are based on the discernibility matrix, the reduct results are obtained by binary programming, and hence the reduction lengths differ.
In terms of runtime (Figure 4), Algorithm 6 is obviously faster than VPR-DM and KNR-DM. For example, on the A.S.N. dataset, when β = 0.65, the runtimes are 12.72 s, 11.28 s, and 1.556 s, respectively, for VPR-DM, KNR-DM, and Algorithm 6. When β = 0.95, the runtimes are 12.72 s, 11.69 s, and 1.48 s, respectively. When the precision is smaller, the positive region is larger. For example, on the Ecoli dataset, when β = 0.65 and | Pos C D ′′ | = 29, size of the constructed discernibility matrix is 29 × 39, and when β = 0.95 and | Pos C D ′′ | = 15, its size is 15 × 55. When the number of modified decision values in the data set is small, Algorithm 6 has a slight performance advantage. However, when the number of modified decision values is large, the performance advantage of Algorithm 6 is clear. Although there is little difference in the lengths of the reducts of the three algorithms, the run time of Algorithm 6 is shorter than that of the other algorithms, which is consistent with the time complexity analysis. We evaluated the reduction quality using the fine Gaussian NB, DT, and SVC classifiers. The training accuracies (as a percentage) and runtimes (in milliseconds) of the reduction results obtained by Algorithm 6 at different precisions are shown Figures. 5-7, in which the solid lines represent the training time and the dashed lines represent classification accuracy. The original datasets are the datasets before reduction. It can be observed that higher precisions lead to higher classification accuracies for the reduction results.
When the precision of Algorithm 6 is higher, the classification accuracy is not much different from that of the original dataset and the training time is reduced. For example, for the DT classifier on the D.I.U. dataset, when β = 0.95, the training accuracy is 87.5%, whereas it is 81.25% on the original dataset. Moreover, the training time on the data after reduction is 3.2 ms, whereas the training time on the original dataset is 4.5 ms.
When the precision is 0.65, the accuracies of the classifiers on the reduction results are lower. For example, on the E.E. dataset, the training accuracy of the fine Gaussian NB is 73.02%, whereas it is 80.51% for the original dataset; the training accuracy of the DT classifier is 69.85%, whereas it is 72.9% for the original dataset; the training accuracy of the SVC classifier is 59.85%, whereas it is 72.89% for the original dataset.    The experiment first compared the VPR algorithms with their optimized algorithms on both types of decision tables, then the proposed VPR algorithm (Algorithm 6) was compared with existing algorithms based on the discernibility matrix. The experimental results show that Algorithm 6 has better operating efficiency, especially when the datasets are large. Moreover, the reduction results of Algorithm 6 at different precisions were compared using different classifiers to verify that the proposed algorithm is feasible.

VII. CONCLUSION
Compacted and weighted decision tables are two extended forms of the decision table. The relationship between VPR and PRR was established for both types of decision table, and the VPR algorithm was optimized by modifying the decision values of objects that satisfy the given condition and then using the PRR algorithm for the calculation. Furthermore, by comparing the modification process of the decision values in both types of decision tables, a new VPR algorithm was proposed that updates the decision table into a weighted decision table and uses PRR to calculate the VPR. Finally, this proposed algorithm was verified by experiments. In future, we will attempt to remove the restriction of equivalence relation and further study problems such as VPR.