Dynamic Variable Precision Attribute Reduction Algorithm

Abstract

Dynamic reduction algorithms have become an important part of attribute reduction research because of their ability to perform dynamic updates without the need to retrain the original model. To enhance the efficiency of variable precision reduction algorithms in processing dynamic data, research has been conducted from the perspective of the construction process of the discernibility matrix. By modifying the decision values of some samples through an absolute majority voting strategy, a connection between variable precision reduction and positive region reduction has been established. Considering the increase and decrease of samples, dynamic variable precision reduction algorithms have been proposed. For four cases of sample increase, four corresponding scenarios have been discussed, and judgment conditions for the construction of the discernibility matrix have been proposed, which has led to the development of a dynamic variable precision reduction algorithm for sample increasing (DVPRA-SI). Simultaneously, for the scenario of sample deletion, three corresponding scenarios have been proposed, and the judgment conditions for the construction of the discernibility matrix have been discussed, which has resulted in the development of a dynamic variable precision reduction algorithm for sample deletion (DVPRA-SD). Finally, the proposed two algorithms and existing dynamic variable precision reduction algorithms were compared in terms of the running time and classification precision, and the experiments demonstrated that both algorithms are feasible and effective.

1. Introduction

The rough set was proposed by Polish scholar Pawlak [1]. The main contribution of attribute reduction is to remove redundant attributes while keeping the classification precision essentially unchanged. However, because of the strictness of classical classification in the rough set, it is often sensitive to noise disturbance or error influence in practical applications. Subsequently, to enhance the applicable robustness, the variable precision rough set (VPRS) [2] was allowed a certain degree of noise to relax the strictness of the classification.

Two main reduction algorithms are used to identify reducts. One is a heuristic algorithm, and the other is a discernibility matrix-based algorithm. Although the discernibility matrix-based algorithm has a high time complexity, it remains the only approach to identify all reducts. The various discernibility matrix based algorithms proposed, such as axisymmetric positive region discernibility matrix and symmetric variable precision discernibility matrix. The invariant reduction studied in [3] generates an discernibility matrix for each invariant. The discernibility matrix for method 1 is axisymmetric, while the discernibility matrices for the other two methods are symmetric.

To meet the extensive application needs of VPRS theory, the academic community extended it, the concept of a general VPRS approximation [4] was proposed, and the effective matrix method for the calculation was provided, thus simplifying the calculation of the VPRS approximation. Simultaneously, two types of attribute reduction for information systems were also provided [5]: $\beta$ reduction and $\beta$ variable precision reduction (VPR). Researchers also found that only the smallest element in the discernibility matrix is sufficient for determining a reduction result [6]. Additionally, the VPRS has been combined with other methods to varying degrees, such as the combination of variable precision and granularity [7,8], and variable precision fuzzy rough set models also exist that are based on coverage [9]. Additionally, various reduction methods exist, such as neighborhood reduction, covering reduction, multi-label reduction, and information entropy reduction [10,11,12,13,14,15,16,17,18,19,20,21,22,23].

Considering the problems of sample set disturbance, attribute set disturbance, and attribute value disturbance, it is particularly important to develop incremental algorithms because of the relatively low efficiency of non-incremental algorithms. To overcome this problem, a feature selection framework based on the discernibility score generated by eliminating redundant samples in the incremental method was proposed [24]. Considering the usefulness of new samples, an active sample selection method [25] was proposed to select new samples dynamically. For the incremental algorithm for attribute value disturbance, a new compound attribute measure method [26] was proposed. The incremental algorithm was proposed to handle missing attribute values for incomplete decision tables [27]. For the simultaneous disturbance of the sample set and attribute set [28], the incremental establishment process was discussed. Additionally, for different application backgrounds, dynamic attribute reduction algorithms [29,30,31,32,33,34,35,36] were proposed, which enhance the adaptability and precision of dynamic samples while maintaining computational efficiency.

In recent years, incremental approximation updates have attracted the interest of scholars. Some approximation update methods require recalculation when new samples are added or existing ones are removed, inevitably leading to substantial redundant computations.

In ongoing research, an incremental matrix algorithm facilitates dynamic reduction while preserving the integrity of the original samples [37,38,39,40,41,42]. Through matrix partitioning, a thorough analysis is conducted on the dynamic interplay between existing and newly introduced samples. Notably, the matrix formed subsequent to the addition of new samples exhibits symmetrical properties, which is a crucial characteristic for algorithmic design [43]. To enhance computational efficiency and minimize computation time, this paper introduces a discernibility matrix-based dynamic approximation update method.

This paper proposes the relationship between positive region reduction (PRR) and variable precision reduction (VPR) from the perspective of constructing discernibility matrix, and proposes a variable precision dynamic update algorithm based on the dynamic changes in the relationship between the two reductions. The main contributions of this paper are as follows:

• At present, there is relatively insufficient discussion on the relationship and dynamic changes between different reduction methods. This paper by modifying the decision values of some samples through an absolute majority voting strategy, the relationship between positive region reduction (PRR) and VPR is established from the perspective of discernibility matrix construction.
• Four scenarios that appear when the number of samples increases are analyzed, and the dynamic relationship between PRR and VPR is discussed for four cases. In the same manner, the analysis and discussion are conducted on three scenarios when samples are deleted.
• Based on the changes to the number of samples in the VPR model, regardless of whether they are an increase or decrease, dynamic VPR algorithms using the PRR model are proposed: the dynamic variable precision reduction algorithm for sample increasing (DVPRA-SI) and dynamic variable precision reduction algorithm for sample reduction (DVPRA-SD).

The remainder of this paper is organized as follows: In Section 2, we review the basic concepts of PRR and VPR. In Section 3, we illustrate the dynamic reduction algorithm when samples are added. In Section 4, we illustrate the dynamic reduction algorithm when samples are deleted. In Section 5, we conduct experiments on the proposed algorithms and illustrate the feasibility of the algorithms. Section 6 contains a conclusion of the paper.

2. Preliminaries

In this section, the related concepts of PRR and VPR are reviewed. To clarify the research content of this paper, the research basis for the VPR algorithm in incremental learning is also introduced.

The quadruple $S=\{U,AT,V_a,f\}$ is an information table [6,23]. U is the universe, $AT$ is a nonempty finite set, $\left\{V_a\right|a\in AT\}$ is a nonempty finite set of values, and $f(x,a):U\to AT$ is a function, where $\left\{f\right(x,a\left)\right|x\in U,a\in AT\}$ denotes the value of object x under the attribute a. The equivalence relation [4,5] is defined as $R_A=\left\{(x,y)\right|(x,y)\in U\times U,f(x,a)=f(y,a),\forall a\in A\}$ and equivalence classes ${\left[x\right]}_A=\left\{y\right|(x,y)\in R_A\}$. Given S, where C is the condition attributes set, D is the decision attributes set, and $C\bigcap D=⌀$, $(U,C\bigcup D)$ is called the decision table.

Definition 1 

([5]). Given $(U,C\bigcup D)$, $U/D=\{D_1,D_2,\cdots ,D_t\}$ is the quotient set by D, and the definition of the positive region is $P{\mathrm{os}}_{C}D={\bigcup }_{i=1}^{\mathrm{t}}\left(\underline{R_{\mathrm{C}}}\left(D_i\right)\right)$. Where $\underline{R_C}\left(x\right)=\{x\mid {\left[x\right]}_C\subseteq D_i\}$.

Definition 2 

([23]). Given $(U,C\bigcup D)$, B is the PRR of C when B satisfies the following conditions:

$\begin{array}{cc}\hfill \left(1\right)& {Pos}_C\left(D\right)={Pos}_B\left(D\right)\hfill \\ \hfill \left(2\right)& {Pos}_C\left(D\right)\ne {Pos}_{B^{\prime }}\left(D\right)\mathit{for}\mathit{all}{B}^{\prime }\ne ⌀\mathit{and}{B}^{\prime }\subset B\hfill \end{array}$

PRR’s corresponding discernibility matrix $M_{Po{s}_{C}D}={\left(m_{Po{s}_{C}D}(x_i,x_j)\right)}_{s\times n}$ is followed as

$$m_{Po{s}_{C}D}(x_i,x_j)=\left\{\begin{array}{c}\{a\in C;(x_i,x_j)\notin R_a\},x_i\in Po{s}_C\left(D\right),(x_i,x_j)\notin R_D\hfill \\ ⌀,otherwise\hfill \end{array}\right.$$

where $s=\mid Po{s}_C\left(D\right)\mid$, $n=\mid U\mid$, and $\mid ·\mid$ denotes the cardinality of the set. Then, each row of the discernibility matrix represents any object in an equivalence class in the positive region and the discernibility matrix can be compressed.

Remark 1. 

If $y\in R_C\left(x\right),$ and $R_C\left(x\right)\subseteq R_D\left(x\right)$, then $M_{Po{s}_{c}D}(x,:)=M_{Po{s}_{c}D}(y,:)$, where $M_{Po{s}_{c}D}(x,:)$ represents the row in which x is located in the discernibility matrix; it is easy to observe that x and y are in the same row in the discernibility matrix.

Definition 3 

([4,5,23]). X is subset of U, and for each $x\in U$, the characteristic function ${\lambda }_X$ of X is defined as ${\lambda }_X\left(x\right)=\left\{\begin{array}{cc}1& x\in X\\ 0& x\notin X\end{array}\right.$

Definition 4 

([4,5,23]). Let $B\subseteq C$, and $\beta \in (0,1]$. For $X\subseteq U$, and ${\left[x_i\right]}_R\subseteq U$, where ${\left[x_i\right]}_R$ is an equivalence class on R, then definition of $W_R{\lambda }_X$ is as follows:

$$W_R{\lambda }_X={[{\displaystyle \frac{|{\left[x_1\right]}_R\bigcap X|}{|{\left[x_1\right]}_R|}},{\displaystyle \frac{|{\left[x_2\right]}_R\bigcap X|}{|{\left[x_2\right]}_R|}},\cdots ,{\displaystyle \frac{|{\left[x_n\right]}_R\bigcap X|}{|{\left[x_n\right]}_R|}}]}^T$$

where T denotes the transpose and column vector ${\lambda }_X={({\lambda }_X\left(x_1\right),{\lambda }_X\left(x_2\right),\cdots ,{\lambda }_X\left(x_n\right))}^T$, with respect to $\forall x_i\in U$. Additionally,

$${\mu }_{CD}=\left(\begin{array}{cccc}p\left(D_1\right|{\left[x_1\right]}_C)& p\left(D_2\right|{\left[x_1\right]}_C)& \cdots & p\left(D_t\right|{\left[x_1\right]}_C)\\ p\left(D_1\right|{\left[x_2\right]}_C)& p\left(D_2\right|{\left[x_2\right]}_C)& \cdots & p\left(D_t\right|{\left[x_2\right]}_C)\\ ⋮& ⋮& & ⋮\\ p\left(D_1\right|{\left[x_n\right]}_C)& p\left(D_2\right|{\left[x_n\right]}_C)& \cdots & p\left(D_t\right|{\left[x_n\right]}_C)\end{array}\right)=\left(\begin{array}{c}{\mu }_{CD}\left(x_1\right)\\ {\mu }_{CD}\left(x_2\right)\\ ⋮\\ {\mu }_{CD}\left(x_n\right)\end{array}\right)$$

where $p(D_j\mid {\left[x_i\right]}_C)={\displaystyle \frac{|{\left[x_i\right]}_C\bigcap D_j\mid }{|{\left[x_i\right]}_C\mid }}(i=1,2\cdots n,j=1,2\cdots t)$.

Definition 5 

([4,5,23]). Given $(U,C\bigcup D)$ and $\beta \in (0,1]$, B is the VPR of C if B satisfies the following conditions:

$\begin{array}{cc}& \left(1\right){\left({\mu }_{CD}\left(x\right)\right)}_{\beta }={\left({\mu }_{BD}\left(x\right)\right)}_{\beta }for\forall x\in U\hfill \\ & \left(2\right)⌀\ne B^{\prime }\subset B,{\left({\mu }_{CD}\left(x\right)\right)}_{\beta }\ne {\left({\mu }_{B^{\prime }D}\left(x\right)\right)}_{\beta }for\exists x\in U\hfill \end{array}$

VPR’s corresponding discernibility matrix, $M_{{\mu }_{CD}}={\left(m_{{\mu }_{CD}}(x_i,x_j)\right)}_{n\times n}$, is followed as

$m_{{\mu }_{CD}}(x_i,x_j)=\left\{\begin{array}{c}\left\{a\right|a\in C,a\left(x_i\right)\ne a\left(x_j\right)\},{\left({\mu }_{CD}\left(x_i\right)\right)}_{\beta }\ne {\left({\mu }_{CD}\left(x_j\right)\right)}_{\beta }\hfill \\ ⌀,\mathrm{otherwise}\hfill \end{array}\right.$

Definition 6 

([4,5,23]). Transform the discernibility function Φ from its Conjunctive Normal Form (CNF) $\mathsf{\Phi }={\wedge }_{i<j}\{\vee m(x_i,x_j)\}$ into the Disjunctive Normal Form (DNF) $\mathsf{\Phi }={\vee }_{u=1}^{\nu }\{\wedge B_u\}$, $(B_u\subseteq C)$, $Red\left(C\right)=\{B_1,B_2,\cdots ,B_{\nu }\}$ and $Core\left(C\right)={\bigcap }_{u=1}^{\nu }{B}_u$.

Lemma 1 

([23]). Given ${\mu }_{CD}\left(x\right)$ for $(U,C\bigcup D)$. If $\beta \in (0.5,1]$ and the updated decision table is $(U,C\bigcup D^{\prime })$, which corresponds to ${\mu }_{C{D}^{\prime }}\left(x\right)$, then ${\left({\mu }_{CD}\left(x\right)\right)}_{\beta }={\left({\mu }_{C{D}^{\prime }}\left(x\right)\right)}_1$ for each x.

According to the discernibility matrix, the results of VPR with $\beta =1$ are the results of PRR, and combined with Lemma 1, the following theorem follows.

Theorem 1 

([23]). Given $(U,C\bigcup D)$, If $\beta \in (0.5,1]$ and the updated decision table is $(U,C\bigcup D^{\prime })$, where $Po{s}_c{D}^{\prime }$ is the positive region, then ${\mathsf{\Phi }}_{{\mu }_{CD}}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$.

Remark 2. 

${\mathsf{\Phi }}_{{\mu }_{CD}}$ denotes the VPR’s discernibility function with $\beta =1$ for $(U,C\bigcup D)$ and ${\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$ denotes the PRR’s discernibility function for $(U,C\bigcup D^{\prime })$. Although the discernibility matrix $M_{{\mu }_{CD}}\ne M_{Po{s}_c{D}^{\prime }}$, ${\mathsf{\Phi }}_{{\mu }_{CD}}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$ for the discernibility function by the proof [22].

Given $\beta \in (0.5,1]$, for ${\displaystyle \frac{|{\left[x_i\right]}_R\bigcap D_i|}{|{\left[x_i\right]}_C|}}$ in $(U,C\bigcup D)$, the new decision table $(U,C\bigcup D^{\prime })$ is obtained using the absolute majority voting strategy in $(U,C\bigcup D)$. The specific modified approach is as follows: if $\exists P\left(D_j\right|{\left[x_i\right]}_C)\ge \beta$ and $f(y,d_k)\ne f(x,d_j)$ for $y\in R_C\left(x\right)$, then $f(y,d_k)=f(x,d_j)$. The quotient set by $D^{\prime }$ in the updated decision table is $U/D^{\prime }=\{D_1^{\prime },D_2^{\prime },\dots ,D_s^{\prime }\}$. It follows from Theorem 1 that VPR with $\beta$ in $(U,C\bigcup D)$ has the same discernibility function and, hence, the same reduction results as PRR in $(U,C\bigcup D^{\prime })$.

Table 1. A decision table.

U		C		D
	${\mathit{a}}_{\mathbf{1}}$	${\mathit{a}}_{\mathbf{2}}$	${\mathit{a}}_{\mathbf{3}}$
$x_1$	1	0	1	0
$x_2$	1	0	1	1
$x_3$	0	1	0	1
$x_4$	0	0	1	2
$x_5$	0	1	0	0
$x_6$	0	1	0	1
$x_7$	0	0	1	2
$x_8$	0	0	1	2

illustrates the relationship between VPR and PRR.

Table 1 obtained $U/C=\{\{x_1,x_2\},\{x_3,x_5,x_6\},\{x_4,x_7,x_8\}\}$, $U/D=\{\{x_1,x_5\}$, $\{x_2,x_3$, $x_6\}$, $\{x_4,x_7,x_8\}\}$. When $\beta =0.55$, ${\left({\mu }_{CD}\right)}_{\beta }=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$. Then, the VPR’s discernibility matrix is

$$M_{{\mu }_{CD}}=\left(\begin{array}{cccccccc}⌀& ⌀& C& a_1& C& C& a_1& a_1\\ C& C& ⌀& a_2{a}_3& ⌀& ⌀& a_2{a}_3& a_2{a}_3\\ a_1& a_1& a_2{a}_3& ⌀& a_2{a}_3& a_2{a}_3& ⌀& ⌀\end{array}\right)$$

discernibility function ${\mathsf{\Phi }}_{{\mu }_{CD}}$ from its CNF ${\mathsf{\Phi }}_{{\mu }_{CD}}=\{a_1\vee a_2\vee a_3\}\wedge \left\{a_1\right\}\wedge \{a_2\vee a_3\}$ into the DNF ${\mathsf{\Phi }}_{{\mu }_{CD}}=\{a_1\wedge a_2\}\vee \{a_1\wedge a_3\}$, and the reduction results are $\left\{a_1{a}_2\right\}$ and $\left\{a_1{a}_3\right\}$.

When $\beta =0.55$, Table 1 is modified to obtain the modified decision table (

Table 2. Modified decision table (change $f(x_5,D)=0$ to $f(x_5,D)=1$).

U		C		D
	${\mathit{a}}_{\mathbf{1}}$	${\mathit{a}}_{\mathbf{2}}$	${\mathit{a}}_{\mathbf{3}}$
$x_1$	1	0	1	0
$x_2$	1	0	1	1
$x_3$	0	1	0	1
$x_4$	0	0	1	2
$x_5$	0	1	0	1
$x_6$	0	1	0	1
$x_7$	0	0	1	2
$x_8$	0	0	1	2

). where the bold display represents the modified decision values.

Because ${\displaystyle \frac{|{\left[x_5\right]}_C\cap D_2|}{|{\left[x_5\right]}_C|}}\ge 0.55$, let $f(x_5,d)=1$. According to Table 2 $Po{s}_C{D}^{\prime }=\{x_3,x_4,x_5$, $x_6,x_7,x_8\}$, and because $M_{Po{s}_C{D}^{\prime }}(x_3,:)=M_{Po{s}_C{D}^{\prime }}(x_5,:)=M_{Po{s}_C{D}^{\prime }}(x_6,:)$, the PRR’s discernibility matrix can be compressed as

$$M_{Po{s}_C{D}^{\prime }}=\left(\begin{array}{cccccccc}C& C& ⌀& a_2{a}_3& ⌀& ⌀& a_2{a}_3& a_2{a}_3\\ a_1& a_1& a_2{a}_3& ⌀& a_2{a}_3& a_2{a}_3& ⌀& ⌀\end{array}\right)$$

discernibility function ${\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$ from its CNF ${\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}=\{a_1\vee a_2\vee a_3\}\wedge \{a_2\vee a_3\}\wedge \left\{a_1\right\}$ into the DNF ${\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}=\{a_1\wedge a_2\}\vee \{a_1\wedge a_3\}$, and the reduction results are $\left\{a_1{a}_2\right\}$ and $\left\{a_1{a}_3\right\}$.

It is worth repeating that, although the discernibility matrix of VPR in $(U,C\bigcup D)$ and the discernibility matrix of PRR in $(U,C\bigcup D^{\prime })$ are different, the results obtained by both are the same. The purpose of this paper is to study the process of obtaining reduction results from the perspective of discernibility matrix construction considering the dynamic change of the universe.

3. Incremental Mechanism of Dynamic Data Reduction

When incremental learning encounters new samples, it does not require the retraining of the original model and enables dynamic updates, which is crucial for adapting to changing environments in a timely manner. Relatively many examples exist where the sample is frequently perturbed, such as stock prices in financial markets and traffic flow monitoring systems. In this section, the dynamic reduction algorithm for increasing samples is proposed. The following four scenarios in which new samples are added are proposed in

Table 3. Scenarios for increasing the sample.

Original Sample	New Sample	Condition Attribute Value	Decision Attribute Value	Scenario
$\exists x\in U$	$y\in U^{+}-U$	$\forall a,f(x,a)=f(y,a)$	$f(x,d)=f(y,d)$	1
$\exists x\in U$	$y\in U^{+}-U$	$\forall a,f(x,a)=f(y,a)$	$f(x,d)\ne f(y,d)$	2
$\forall x_i\in \mathrm{U},\exists x_j\in U$	$y\in U^{+}-U$	$\exists a,f(x_i,a)\ne f(y,a)$	$f(x,d)=f(y,d)$	3
$\forall x\in U$	$y\in U^{+}-U$	$\exists a,f(x_i,a)\ne f(y,a)$	$f(x,d)\ne f(y,d)$	4

. The four frameworks for increasing sample scenarios are shown in Figure 1.

The set $Y=\{y_1,y_2,\cdots ,y_p\}$ represents adding new samples based on $(U,C\bigcup D)$. For the value of ${\displaystyle \frac{|{\left[x_i\right]}_C\cap D_i|}{|{\left[x_i\right]}_C|}}$, and adopting the absolute majority voting strategy, the modified decision table is denoted by $(U^{+},C\cup D^{″})$, where $U^{+}=U\cup Y$.

Definition 7. 

Given $(U,C\bigcup D^{\prime })$, its PRR’s discernibility matrix is $M_{Po{s}_c{D}^{\prime }}$. The modified table is $(U^{+},C\cup D^{″})$ as new samples $Y=\{y_1,y_2,\cdots ,y_p\}$ are added and corresponding PRR’s discernibility matrix is denoted by $M_{Po{s}_C{D}^{″}}^{Add}$.

The corresponding discernibility function for $M_{Po{s}_c{D}^{\prime }}$ is ${\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$, and the corresponding discernibility function for $M_{Po{s}_C{D}^{″}}^{Add}$ is ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}$.

Definition 8. 

Given $(U,C\bigcup D)$, its VPR’s discernibility matrix is $M_{{\mu }_{CD}}$, and the corresponding VPR’s discernibility matrix is denoted by $M_{{\mu }_{CD}}^{Add}$ for $(U^{+},C\cup D)$, when new samples $Y=\{y_1,y_2,\cdots ,y_p\}$ are added.

The corresponding discernibility function for $M_{{\mu }_{CD}}$ and $M_{{\mu }_{CD}}^{Add}$ are ${\mathsf{\Phi }}_{{\mu }_{CD}}$ and ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}$, respectively.

Remark 3. 

According to Definitions 7–8 and Theorem 1, the discernibility function ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}$ in $(U^{+},C\cup D)$ in $(U^{+},C\cup D)$ and the discernibility function ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}$ in $(U^{+},C\cup D^{″})$ are equivalent.

The positive region for $(U,C\cup D^{\prime })$ is denoted by $Po{s}_c{D}^{\prime }$ and the corresponding positive region for $(U^{+},C\cup D^{″})$ is denoted by $Po{s}_c{D}^{″}$. Upon the introduction of new samples, four distinct scenarios may emerge, each potentially encompassing one of the following four cases (Figure 1).

$\begin{array}{cc}& Case1:x_i\in Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″}\hfill \\ & Case2:x_i\in Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″}\hfill \\ & Case3:x_i\notin Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″}\hfill \\ & Case4:x_i\notin Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″}\hfill \end{array}$

Four cases under Scenario 1 (Table 3) are discussed as follows. Given $(U,C\bigcup D)$, the modified decision table is $(U^{+},C\cup D^{″})$, and the first case is discussed.

Theorem 2. 

Let y be a new sample. If $x_i\in Po{s}_C{D}^{\prime }$ and $x_i\in Po{s}_C{D}^{″}$ for $y\in {\left[x_i\right]}_C$, $y\in R_D\left(x_i\right)$, then ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_c{D}^{\prime }}$.

Proof. 

$m_{Po{s}_C{D}^{\prime }}(x,x_k)=m_{Po{s}_C{D}^{″}}^{Add}(x,x_k)$, where $x\in Po{s}_C{D}^{\prime }$, and $x_k\in U$. For $x_i\in Po{s}_C{D}^{\prime }$, there exist $m_{Po{s}_C{D}^{\prime }}(x_i,x_j)\in {\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$. For matrix row changes: For each $x_j\in U$, if $x_i\in Po{s}_C{D}^{″}$, and $y\in {\left[x_i\right]}_C$, then $m_{Po{s}_C{D}^{\prime }}(x_i,x_j)=m_{Po{s}_C{D}^{″}}^{Add}(y,x_j)$. For matrix column changes: If each $x_j\in Po{s}_C{D}^{\prime }$, then $m_{Po{s}_C{D}^{\prime }}(x_j,x_i)=m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)$, and if each $x_j\notin Po{s}_C{D}^{\prime }$ then $m_{Po{s}_C{D}^{\prime }}(x_j,x_i)=m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)=⌀$. Additionally if $y\in {\left[x_i\right]}_C$, $y\in R_D\left(x_i\right)$ for $x_i\in Po{s}_C{D}^{″}$, then $m_{Po{s}_C{D}^{″}}^{Add}(x_i,y)=⌀$. Therefore, ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_c{D}^{\prime }}$. □

The following corollary follows from Theorems 1 and 2.

Corollary 1. 

Let y be a new sample, If $x_i\in Po{s}_C{D}^{\prime }$ and $x_i\in Po{s}_C{D}^{″}$ for $y\in {\left[x_i\right]}_C$, $y\in R_D\left(x_i\right)$, then ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{{\mu }_{CD}}$.

As a continuation of Table 1, given a table $(U^{+},C\cup D)$, which contains new samples $Y=\{y_1,y_2\}$, $y_1\in {\left[x_3\right]}_C$, $y_1\in R_D\left(x_3\right)$ and $y_2\in {\left[x_4\right]}_C$, $y_2\in R_D\left(x_4\right)$, it is easy to observe that $x_3,x_4\in Po{s}_C{D}^{\prime }$ for Table 2 when samples are added. The modified samples are shown in

Table 4. Decision table after increasing the sample.

U		C		D
	${\mathit{a}}_{\mathbf{1}}$	${\mathit{a}}_{\mathbf{2}}$	${\mathit{a}}_{\mathbf{3}}$
$x_1$	1	0	1	0
$x_2$	1	0	1	1
$x_3$	0	1	0	1
$x_4$	0	0	1	2
$x_5$	0	1	0	1
$x_6$	0	1	0	1
$x_7$	0	0	1	2
$x_8$	0	0	1	2
$y_1$	0	1	0	1
$y_2$	0	0	1	2

. Then, $x_3,x_4\in Po{s}_C{D}^{″}$.

It follows that $M_{Po{s}_C{D}^{″}},=\left(\begin{array}{cccccccccc}C& C& ⌀& a_2{a}_3& ⌀& ⌀& a_2{a}_3& a_2{a}_3& ⌀& a_2{a}_3\\ a_1& a_1& a_2{a}_3& ⌀& a_2{a}_3& a_2{a}_3& ⌀& ⌀& a_2{a}_3& ⌀\end{array}\right)$. Clearly, ${\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_c{D}^{\prime }}$, and then ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{{\mu }_{CD}}$ from Theorem 1.

According to Definition 2, upon the addition of the sample set $Y=\{y_1,y_2\}$, the static algorithm necessitates a recalculation of $U/C$, $U/D$, $M_{Po{s}_C{D}^{″}}$, and ${\mathsf{\Phi }}_{Po{s}_C{D}^{″}}$. The preceding analysis has demonstrated that the reduction outcomes following the incorporation of $y_1$ and $y_2$ are consistent. The dynamic reduction algorithm, in contrast, is capable of directly calculating and reflecting the impact of new samples on the existing original sample. In order to effectively address the issue of redundant calculations inherent in static algorithms, this study will further investigate the effects of four distinct scenarios involving the increasing of samples on the reduction process.

Definition 9. 

${\delta }^{row}\left(x_i\right)=\left\{\begin{array}{c}\{(x_i,x_j)\notin R_a\mid a\in C,x_j\in U\},x_i\in Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″}\hfill \\ \{(x_i,x_j)\notin R_a\mid a\in C,x_j\in U^{+}\},x_i\notin Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″}\hfill \end{array}\right.$, and the whole row elements corresponding to $x_i$ of the positive region in the discernibility matrix are denoted by ${\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)$.

The whole row and column element corresponding to the newly added sample y are defined as ${\delta }^{row}\left(y\right)=\{(y,x_j)\notin R_a\mid \forall x_j\in U\}$, ${\delta }^{col}\left(y\right)=\{(x_j,y)\notin R_a\mid for\forall x_j\in Po{s}_C{D}^{″}\}$, and the whole row and column elements corresponding to the positive region in the discernibility matrix are denoted by ${\delta }_{Po{s}_{C^D}}^{row}\left(y\right)$, ${\delta }_{Po{s}_{C^D}}^{col}\left(y\right)$.

Case 2 in Scenario 1 (Table 3) is discussed next.

Theorem 3. 

Let y be a new sample. If $x_i\in Po{s}_C{D}^{\prime }$ and $x_i\notin Po{s}_C{D}^{″}$, for $y\in {\left[x_i\right]}_C$, $y\in R_D\left(x_i\right)$, then ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{\prime }}={\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}\wedge {\delta }_{Po{s}_{\mathrm{C}}D}^{row}\left(x_i\right)$.

Proof. 

For $m_{Po{s}_C{D}^{\prime }}(x_i,x_j)\in {\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$, because $x_i\in Po{s}_C{D}^{\prime }$, and $x_i\notin Po{s}_C{D}^{″}$, $m_{Po{s}_C{D}^{\prime }}(x,x_k)$ = $m_{Po{s}_C{D}^{″}}^{Add}(x,x_k)$, where $x\in Po{s}_C{D}^{\prime }$ and $x\notin {\left[x_i\right]}_C$. For matrix row changes: For each $x_j\in U$, if $x_i\notin Po{s}_C{D}^{″}$, then $m_{Po{s}_c{D}^{″}}^{Add}(x_i,x_j)=⌀$, where $m_{Po{s}_c{D}^{\prime }}(x_i,x_j)=\{a\mid (x_i,x_j)\notin R_{a}and(x_i,x_j)\notin R_{D^{\prime }}\}$. For matrix column changes: For each $x_j\in Po{s}_C{D}^{\prime }$, and $y\in {\left[x_i\right]}_C$, $y\in R_D\left(x_i\right)$, $m_{Po{s}_C{D}^{\prime }}(x_j,x_i)=m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)$, and for each $x_j\notin Po{s}_C{D}^{\prime }$, $m_{Po{s}_C{D}^{\prime }}(x_j,x_i)=m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)=⌀$. Because $x_i\notin Po{s}_C{D}^{″}$, $m_{Po{s}_c{D}^{″}}^{Add}(x_i,y)=⌀$. Therefore, ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{\prime }}={\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}\wedge {\delta }_{Po{s}_{\mathrm{C}}D}^{row}\left(x_i\right)$. □

Corollary 2. 

Let y be a new sample. For $x_i\in Po{s}_C{D}^{\prime }$, $y\in {\left[x_i\right]}_C$ and $y\in R_D\left(x_i\right)$, if $x_i\notin Po{s}_C{D}^{″}$, then ${\mathsf{\Phi }}_{{\mu }_{CD}}={\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)$.

$x_i\in Po{s}_C{D}^{\prime }$ was discussed in the above two cases, and now two cases of $x_i\notin Po{s}_C{D}^{\prime }$ are discussed. Case 3 is discussed as follows.

Theorem 4. 

Let y be a new sample. If $x_i\notin Po{s}_C{D}^{\prime }$ and $x_i\in Po{s}_C{D}^{″}$, for $y\in {\left[x_i\right]}_C$ and $y\in R_D\left(x_i\right)$, then ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{\prime }}\wedge {\delta }_{Po{s}_{\mathrm{C}}\mathrm{D}}^{row}\left(x_i\right)$.

Proof. 

$m_{Po{s}_C{D}^{\prime }}(x,x_k)=m_{Po{s}_C{D}^{″}}^{Add}(x,x_k)$, where $x\in Po{s}_C{D}^{\prime }$, and $x_k\in U$. For $x_i\in Po{s}_C{D}^{″}$, there exist $m_{Po{s}_C{D}^{″}}^{Add}(x_i,x_j)\in {\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}$. For matrix row changes, for each $x_j\in U$, if $x_i\notin Po{s}_C{D}^{\prime }$, then $m_{Po{s}_c{D}^{\prime }}(x_i,x_j)=⌀$, $m_{Po{s}_c{D}^{″}}^{Add}(x_i,x_j)=\{a\mid (x_i,x_j)\notin R_{a}and(x_i,x_j)\notin R_{D^{″}}\}$. For matrix column changes, for each $x_j\in Po{s}_C{D}^{\prime }$, $m_{Po{s}_C{D}^{\prime }}(x_j,x_i)=m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)$, and for each $x_j\notin Po{s}_C{D}^{\prime }$, $m_{Po{s}_C{D}^{\prime }}(x_j,x_i)=m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)=⌀$. Additionally, $y\in {\left[x_i\right]}_C$$y\in R_D\left(x_i\right)$, for $x_i\in Po{s}_C{D}^{″}$, $m_{Po{s}_c{D}^{″}}^{Add}(x_i,y)=⌀$. Therefore, ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{\prime }}\wedge {\delta }_{Po{s}_{\mathrm{C}}\mathrm{D}}^{row}\left(x_i\right)$. □

Corollary 3. 

Let y be a new sample, for $x_i\notin Po{s}_C{D}^{\prime }$, $y\in {\left[x_i\right]}_C$ and $y\in R_D\left(x_i\right)$. If $x_i\in Po{s}_C{D}^{″}$, then ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)$

Corollary 3 is obtained according to Theorem 4. The final case within Scenario 1 (Table 3) is now being discussed.

Theorem 5. 

Let y be a new sample. For $x_i\notin Po{s}_C{D}^{\prime }$, $y\in {\left[x_i\right]}_C$ and $y\in R_D\left(x_i\right)$, if $x_i\notin Po{s}_C{D}^{″}$, then ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_c{D}^{\prime }}$.

Proof. 

For $x_i\in Po{s}_C{D}^{\prime }$, there exist $m_{Po{s}_C{D}^{\prime }}(x_i,x_j)\in {\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$. $m_{Po{s}_C{D}^{\prime }}(x,x_k)$ = $m_{Po{s}_C{D}^{″}}^{Add}(x,x_k)$, where $x\in Po{s}_C{D}^{\prime }$, $x_k\in U$. For matrix column changes, for each $x_j\in Po{s}_C{D}^{\prime }$, $m_{Po{s}_C{D}^{\prime }}(x_j,x_i)$ = $m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)$, and for each $x_j\notin Po{s}_C{D}^{\prime }$, $m_{Po{s}_C{D}^{\prime }}(x_j,x_i)=m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)=⌀$. Therefore, ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_c{D}^{\prime }}$. □

Corollary 4. 

Let y be a new sample. For $x_i\notin Po{s}_C{D}^{\prime }$, $y\in {\left[x_i\right]}_C$ and $y\in R_D\left(x_i\right)$, if $x_i\notin Po{s}_C{D}^{″}$, then ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{{\mu }_{CD}}$.

Theorems 2–5 have detailed the four cases encompassed within Scenario 1 (Table 3); the ensuing discussion is now directed towards the cases presented in Scenario 2 (Table 3). Because of the different decision values for x and $y_i$, some corresponding changes occur. $(U,C\cup D)$ is given and the modified table is $(U^{+},C\cup D^{″})$ when new samples are added.

Theorem 6. 

Let y be a new sample. If $y\in {\left[x_i\right]}_C$, $y\notin R_D\left(x_i\right)$, the following conclusions holds:

$\begin{array}{cc}& \left(1\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\hfill \\ & \left(2\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}={\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(3\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(4\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\hfill \end{array}$

Proof. 

The proofs for Theorem 6 are similar to the processes for Theorems 2–5. □

Remark 4. 

From conclusion (1) of Theorem 6, if $x_i\in Po{s}_C{D}^{\prime }$, then $R_C\left(x_i\right)\subseteq R_{D^{\prime }}\left(x_i\right)$. For $y\in U^{+}-U$, because $y\in {\left[x_i\right]}_C$ and $y\notin R_D\left(x_i\right)$, $R_C\left(x_i\right)⫋R_D\left(x_i\right)$. For ${\displaystyle \frac{|{\left[x_i\right]}_C\cap D_i|}{|{\left[x_i\right]}_C}}|$, let $f(y,d)=f(x_i,d)$, by the absolute majority voting strategy. Then, $R_C\left(x_i\right)\subseteq R_{D^{″}}\left(x_i\right)$. Therefore, $x_i\in Po{s}_C{D}^{\prime }$, and $x_i\in Po{s}_C{D}^{″}$.

Corollary 5. 

Let y be a new sample. If $y\in {\left[x_i\right]}_C$, $y\notin R_D\left(x_i\right)$, the following conclusions holds:

$\begin{array}{cc}& \left(1\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{{\mu }_{CD}}\hfill \\ & \left(2\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{{\mu }_{CD}}={\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(3\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(4\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{{\mu }_{CD}}\hfill \end{array}$

In scenario 3 (Table 3), the new samples have different condition values from any $x_i$ in $(U,C\bigcup D)$; hence, the new samples belong to the positive region in $(U^{+},C\cup D^{″})$. Scenarios 3 and 4 exhibit more distinctive cases; discussions pertaining to these cases are detailed in Theorems 7 and 8.

Theorem 7. 

Let y be a new sample, $y\notin {\left[x_i\right]}_C$ for each $x_i\in U$. If there exist $x_j\in U$ and $y\in R_D\left(x_j\right)$, then ${\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(y\right)\wedge {\delta }_{Po{s}_{C}D}^{col}\left(y\right)$.

Proof. 

There exist $m_{Po{s}_C{D}^{\prime }}(x_i,x_j)\in {\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}$, for $x_i\in Po{s}_C{D}^{\prime }$ and $m_{Po{s}_C{D}^{\prime }}(x,x_k)$ = $m_{Po{s}_C{D}^{″}}^{Add}$$(x,x_k)$ for $x\in Po{s}_C{D}^{\prime }$, $x_k\in U$. For each $x_i$, $y\notin R_C\left(x_j\right)$ and ${\left[y\right]}_C=\left\{y\right\}$, clearly $R_C\left(y\right)\subseteq R_D\left(y\right)$, i.e., $y\in Po{s}_C{D}^{″}$. For matrix row changes, because ${\left[y\right]}_C=\left\{y\right\}$, $m_{Po{s}_C{D}^{″}}^{Add}(y,x_j)\in {\delta }_{Po{s}_{C}D}^{row}\left(y\right)$, for each $x_j\in U$. For matrix column changes, $m_{Po{s}_C{D}^{″}}^{Add}(x_j,y)\in {\delta }_{Po{s}_{C}D}^{col}\left(y\right)$, for each $x_j\in Po{s}_C{D}^{\prime }$. Because $y\in Po{s}_C{D}^{″}$, $m_{Po{s}_C{D}^{″}}^{Add}(y,y)=⌀$. Therefore, ${\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(y\right)\wedge {\delta }_{Po{s}_{C}D}^{col}\left(y\right)$. □

Corollary 6. 

Let y be a new sample. For each $x_i\in U$, if $y\notin {\left[x_i\right]}_C$, there exist $x_j\in U$, $y\in R_D\left(x_j\right)$, then ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(y\right)\wedge {\delta }_{Po{s}_{C}D}^{col}\left(y\right)$.

Theorem 8. 

Let y be a new sample. For each $x_i\in U$, $y\notin {\left[x_i\right]}_C$, $y\notin R_D\left(x_i\right)$, ${\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(y\right)\wedge {\delta }_{Po{s}_{C}D}^{col}\left(y\right)$.

Proof. 

The proof is similar to that of Theorem 7. □

Corollary 7. 

Let y be a new sample. If $y\notin {\left[x_i\right]}_C$ for each $x_i\in U$ and $y\notin R_D\left(x_i\right)$, then ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(y\right)\wedge {\delta }_{Po{s}_{C}D}^{col}\left(y\right)$.

According to the discussion of the four scenarios, in this paper, a dynamic reduction algorithm, DVPRA-SI is proposed when samples are increased. Let ${\mathrm{M}}_S^D={\left\{{\mathrm{M}}_i^D\right\}}_{n\times 1}={({\mathrm{M}}_1^D,{\mathrm{M}}_2^D,\cdots ,{\mathrm{M}}_n^D)}^T$, where ${\mathrm{M}}_i^D$ is the label of each row in the discernibility matrix.

The time complexity of the algorithm is $O\left(\right|Y|\times |U{|}^2)$, where $\left|Y\right|$ is the cardinality of the newly added sample set Y. However, the above worst case rarely occurs, and the overall algorithm time is far less than $O\left(\right|Y|\times |U{|}^2)$. Theoretically, Algorithm 1 DVPRA-SI is feasible.

Algorithm 1: Dynamic VPR algorithm for sample increasing (DVPRA-SI)

Input: $(U,C\bigcup D)$, newly added sample set $Y=\{y_1,y_2,\cdots ,y_l\}$

Output: Reduction Results // discernibility function $\mathsf{\Phi }$ from its CNF into the DNF

1: Begin

2: Compute $U/C=\{C_1,C_2,\dots ,C_q\}$, $U/D=\{D_1,D_2,\dots ,D_t\}$ and $Po{s}_C\left(D^{\prime }\right)$

3: Modify the decision table

4: Compute $Po{s}_C\left(D^{″}\right)$

5: for $i=1$ to l do

6:   Compare $y_i$ with label $M_i^D$.

7:      If ${\left[y_i\right]}_C={\left[M_i^D\right]}_C$ then

8:         assign ∅ to all $M_{Po{s}_C{D}^{″}}(M_i^D,y_i)$ columns where $y_i$ is located

9:      else

10:         according to $a\left(x_i\right)\ne a\left(x_j\right)$, the elements are added to $M_{Po{s}_C{D}^{″}}(M_i^D,y_i)$

11:      for $j=1$ to $M_s^D$ do

12:        If $M_i^D\in Po{s}_C\left(D^{″}\right)$ then

13:          rows of the discernibility matrix are not modified

14:        If $M_i^D\notin Po{s}_C\left(D^{″}\right)$ then

15:          delete $M_i^D$

16:      end for

17:      When $y_i\in Po{s}_C\left(D^{″}\right)$

18:        If ${\left[y_i\right]}_C={\left[M_i^D\right]}_C$ and $M_i^D\in M_s^D$ then

19:          without modifying it

20:        If ${\left[y_i\right]}_C={\left[M_i^D\right]}_C$ and $M_i^D\notin M_s^D$ then

21:          according to $a\left(x_i\right)\ne a\left(x_j\right)$, the elements are added to the discernibility.

22:        If $y_i\notin Po{s}_C\left(D^{″}\right)$ then

23:          $M_i^D$ and $M_{Po{s}_c{D}^{″}}$ are not modified

24: end for

24: Compute ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Add}$

25: end

4. Deleting Mechanism of Dynamic Data Reduction

In this section, we focus on the cases of deleting samples dynamically. For the case of deleting samples, a dynamic VPR algorithm for sample deletion is proposed. The three frameworks for deleting sample scenarios are shown in Figure 2. Deleting samples leads to three scenarios (

Table 5. Scenarios for deleting the sample.

Original Sample	New Sample	Condition Attribute Value	Decision Attribute Value	Scenario
$\exists x\in U$	$z\in U-U^{-}$	$\forall a,f(x,a)=f(z,a)$	$f(x,d)=f(y,d)$	1
$\exists x\in U$	$z\in U-U^{-}$	$\forall a,f(x,a)=f(z,a)$	$f(x,d)\ne f(y,d)$	2
$\nexists x\in U$	$z\in U-U^{-}$	$\exists a,f(x,a)=f(z,a)$	“-”	3

), in which $U-U^{-}$ denotes deleting samples $Z=\{z_1,z_2,\cdots ,z_q\}$ for U. For Scenario 3 (Table 5), no sample x exists that satisfies $z\in {\left[x\right]}_C$, it is easy to observe that all equivalence classes of ${\left[x\right]}_C$ have been deleted in the discernibility matrix; hence, the decision value of x will not affect the discernibility matrix for $(U^{-},C\cup D^{″})$. Therefore, the section is confined to a discussion of the cases pertinent to Scenarios 1–2.

In $(U,C\bigcup D)$, and adopting the absolute majority voting strategy for ${\displaystyle \frac{|{\left[x_i\right]}_C\cap D_i|}{|{\left[x_i\right]}_C|}}$, the modified decision table is denoted by $(U^{-},C\cup D^{″})$, where $U^{-}=U-Z$.

Definition 10. 

Given $(U,C\bigcup D)$, the modified decision table is $(U^{-},C\cup D^{″})$ when sample z is deleted. The positive region and variable precision discernibility matrix are denoted by $M_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Del}$ and $M_{{\mu }_{CD}}^{Del}$, respectively.

The corresponding discernibility functions for $M_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Del}$ and $M_{{\mu }_{CD}}^{Del}$ are ${\mathsf{\Phi }}_{Po{s}_{\mathrm{C}}{\mathrm{D}}^{″}}^{Del}$, ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Del}$, respectively.

Provide an illustration for Scenario 3 (Table 5). When the sample $Z=\{x_3,x_4,x_5\}$ is deleted, namely $\nexists x\in U$, $z\in {\left[x\right]}_C$. Calculate $U/C=\{\{x_1,x_2\},\{x_4,x_7,x_8\}\}$, $U/D^{″}=\{\left\{x_1\right\}$, $\left\{x_2\right\},\{x_4,x_7,x_8\}\}$, which then results in $Po{s}_C{D}^{″}=\{x_4,x_7,x_8\}$. As indicated in Remark 1, the number of rows in the discernibility matrix is $|Po{s}_C{D}^{″}/{\left[x\right]}_C|=1$, i.e., $M_{Po{s}_C{D}^{\prime }}$ = $\left(\begin{array}{ccccc}{a}_1& a_1& a_1& ⌀& ⌀\end{array}\right)$, thus ${\mathsf{\Phi }}_{Po{s}_C{D}^{″}}=\left\{a_1\right\}$. The decision value of the deletion sample, which is either $z\in {\left[x\right]}_D$ or $z\notin {\left[x\right]}_D$, results in the removal of the corresponding row from the discernibility matrix. Consequently, the decision attribute values for Scenario 3 are filled with “-”, and the processing for Scenario 3 has also been executed in line 18 of the DVPRA-SD algorithm.

When samples are deleted, the positive region may either increase or decrease. Similar to the analysis of the four cases of the discernibility matrix for adding samples, it is necessary to discuss the four cases of the discernibility matrix when deleting samples. Four cases under Scenario 1 (Table 5) are discussed as follows.

Theorem 9. 

Given $(U,C\bigcup D)$, let z be a deleted sample, $z\in {\left[x_i\right]}_C$ and $z\in R_D\left(x_i\right)$. Then, the following holds:

$\begin{array}{cc}& \left(1\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\hfill \\ & \left(2\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}={\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(3\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(4\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\hfill \end{array}$

Proof. 

The proofs are similar to those of Theorems 2–5, respectively. □

According to Scenario 1 (Table 5), the following corollary can be obtained from Theorems 1 and 9.

Corollary 8. 

Let z be a delete sample, $z\in {\left[x_i\right]}_C$ and $z\in R_D\left(x_i\right)$. Then, the following corollary holds:

$\begin{array}{cc}& \left(1\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{{\mu }_{CD}}^{Del}={\mathsf{\Phi }}_{{\mu }_{CD}}\hfill \\ & \left(2\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{{\mu }_{CD}}={\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(3\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{{\mu }_{CD}}^{Del}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(4\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{{\mu }_{CD}}^{Del}={\mathsf{\Phi }}_{{\mu }_{CD}}\hfill \end{array}$

For ${\displaystyle \frac{|{\left[x_i\right]}_c\cap D_i|}{|{\left[x_i\right]}_c|}}$ in $(U,C\cup D)$, due to the decision values of modifying some samples reaching $(U,C\cup D^{\prime })$, in which $z\in {\left[x_i\right]}_C$, if sample z is deleted in $(U,C\cup D)$, then ${\left[x_i\right]}_C$ may not meet the conditions for modifying decision values for $(U^{-},C\cup D)$, thus, ${\left[x_i\right]}_C$ contains different decision values for Scenario 2 (Table 5). If the deleted sample belongs to Scenario 2, the above changes are detailed in the following theorems.

Theorem 10. 

Let z be a deleted sample. If $z\in {\left[x_i\right]}_C$, $z\notin R_D\left(x_i\right)$, then the following holds:

$\begin{array}{cc}& \left(1\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\hfill \\ & \left(2\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}={\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(3\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(4\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\hfill \end{array}$

Proof. 

The proofs are similar to those of Theorems 2–5, respectively. □

Remark 5. 

If $x_i\in Po{s}_C{D}^{\prime }$, then $P\left(D_j^{\prime }\right|{\left[x_i\right]}_C)=1\ge \beta$, if and only if $R_C\left(x_i\right)\subseteq R_{D^{\prime }}\left(x_i\right)$. After the sample is deleted, because $z\in {\left[x_i\right]}_C$, $z\notin R_D\left(x_i\right)$ and $x_i\in Po{s}_C{D}^{″}$. Then, there exist $P\left(D_j^{″}\right|{\left[x_i\right]}_C)=1\ge \beta$, hence, $R_C\left(x_i\right)\subseteq R_{D^{″}}\left(x_i\right)$.

Corollary 9. 

Let z be a deleted sample. If $z\in {\left[x_i\right]}_C$, $z\notin R_D\left(x_i\right)$, then the following corollary holds:

$\begin{array}{cc}& \left(1\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{{\mu }_{CD}}^{Del}={\mathsf{\Phi }}_{{\mu }_{CD}}\hfill \\ & \left(2\right)If{x}_i\in Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{{\mu }_{CD}}={\mathsf{\Phi }}_{Po{s}_C{D}^{″}}^{Del}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(3\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\in Po{s}_C{D}^{″},Then{\mathsf{\Phi }}_{{\mu }_{CD}}^{Del}={\mathsf{\Phi }}_{Po{s}_C{D}^{\prime }}\wedge {\delta }_{Po{s}_{C}D}^{row}\left(x_i\right)\hfill \\ & \left(4\right)If{x}_i\notin Po{s}_C{D}^{\prime },x_i\notin Po{s}_C{D}^{″},Then,{\mathsf{\Phi }}_{{\mu }_{CD}}^{Del}={\mathsf{\Phi }}_{{\mu }_{CD}}\hfill \end{array}$

The theorem above explains two scenarios for deleting samples, from which the dynamic reduction algorithm for deleted samples is proposed as follows.

Many examples exist of sample disturbance in many applications, such as samples increasing or decreasing. Compared with the four scenarios considered for samples increasing in Section 3, sample deletion considers three scenarios, for which Algorithm 2 DVPRA-SD was proposed in the section. Both Algorithm 2 DVPRA-SD and Algorithm 1 DVPRA-SI consider the issue of equivalence class changes caused by the decrease (increase) of samples, and the corresponding reduction algorithms based on the discernibility matrices are proposed.

Algorithm 2: Dynamic VPR algorithm for sample deletion (DVPRA-SD)

Input: $(U,C\bigcup D)$, newly added sample set $Z=\{z_1,z_2,\cdots ,z_l\}$

Output: Reduction Results // discernibility function $\mathsf{\Phi }$ from its CNF into the DNF

1: Begin

2: Compute $U/C=\{C_1,C_2,\dots ,C_q\}$, $U/D=\{D_1,D_2,\dots ,D_t\}$ and $Po{s}_C\left(D^{\prime }\right)$

3: Modify the decision table

4: Compute $Po{s}_C\left(D^{″}\right)$

5: for $i=1$ to l do

6:   for $i=1$ to $M_i^D$ do

7:      Delete $M_{{Pos}_C{D}^{″}}(M_i^D,z)$

8:    end for

9: end for

10: for $i=1$ to $|U^{-}|$ do

11:      When $x_i\in Po{s}_C\left(D^{″}\right)$

12:        If ${\left[x_i\right]}_C={\left[M_i^D\right]}_C$ and $\exists M_i^D\in M_s^D$ then

13:           the discernibility matrix is not modified

14:        if $\exists M_i^D\notin M_s^D$ then

15:          according to $a\left(x_i\right)\ne a\left(x_j\right)$, and then add the elements to $M_s^D$.

16:      When $x_i\notin Po{s}_C\left(D^{″}\right)$

17:        if ${\left[x_i\right]}_C={\left[M_i^D\right]}_C$ and $\exists M_i^D\in M_s^D$ then

18:           delete the row where label $M_s^D$ is located

19:        if $\exists M_i^D\notin M_s^D$ then

20:           because the discernibility matrix conditions are not met, the discernibility matrix is not modified

21: end for

22: Compute ${\mathsf{\Phi }}_{{\mu }_{CD}}^{Del}$

23: end

5. Experimental Analysis

Experiments were conducted in which eight datasets (

Table 6. Experimental datasets.

No	DataSets	Abbr.	$\left|\mathit{U}\right|$	$\left|\mathit{C}\right|$	$|\mathit{U}/\mathit{D}|$	Missing Values
1	Iris	-	150	4	3	No
2	Wine	-	178	12	3	No
3	Seeds	-	210	7	3	No
4	Forest_Fire	F.F.	244	13	2	No
5	Credit Approval	C.A.	690	15	2	Yes
6	Raisin	-	900	7	2	No
7	Wine Quality	W.Q.	1599	11	4	No
8	Rice (Cammeo and Osmancik)	Rice	3810	7	2	No

) were selected to verify the effectiveness of the proposed algorithms, DVPRA-SI and DVPRA-SD, which were compared with existing algorithms [24,36], where $\left|U\right|$ represents the number of objects, $\left|C\right|$ represents the number of condition attributes, and $|U/D|$ represents the number of classifications. For some datasets with missing values, the average value of the column was selected at the missing position in the columns.

The experimental environment utilized a computing environment with an Intel Core i5-9300H processor, 24 GB of RAM, and the Windows 11 operating system. The algorithms were implemented in Python. Conduct pre-processing on eight datasets from the UCI Repository, transforming all numerical data into integer format.

Running time and classification accuracy were the main evaluation criteria in the experiment. Algorithm DVPRA-SI was compared with two incremental reduction algorithms: IFS-SSFA [24] and IFSA [36]. For classification accuracy, the reduction results with different precisions of DVPRA-SI were compared with that of the original datasets on the classifiers support vector machine (SVM), K-nearest neighbor (KNN), and random forest (RF).

Algorithm DVPRA-SI was compared with Algorithms IFS-SSFA and IFSA in terms of the running time for the datasets in this section. For Algorithms DVPRA-SI and DVPRA-SD, the reduction results were calculated using binary integer programming. For the running time, Algorithms IFS-SSFA and IFSA were compared with Algorithm DVPRA-SI, where precision was set to 0.65, 0.75, and 0.85, respectively. To ensure the condition for comparison, the first 75% of the samples was selected as the raw dataset and the remaining 25% as the newly added samples. The results for the running time are shown in

Table 7. Running time for the algorithms.

Running Time (s)		DVPRA-SI		IFS-SSFA	IFSA
	β = 0.65	β = 0.75	β = 0.85
Iris	0.0156	0.0156	0.0156	0.4023	0.2752
Wine	0.0469	0.0312	0.0481	0.2363	2.4654
Seeds	0.0312	0.0156	0.0156	0.1366	0.8966
F.F.	0.0781	0.0625	0.0625	0.5153	3.5778
C.A.	0.5949	0.5923	0.6866	7.3291	28.7329
Raisin	0.6880	0.6536	0.6733	1.4084	13.05
W.Q.	2.4111	2.1247	2.1248	37.61	94.901
Rice	11.8424	11.4422	11.4310	54.5942	403.0925

, where it can be clearly observed that Algorithm DVPRA-SI had an obvious advantage over the other algorithms. As the size of the dataset expanded, the advantage of Algorithm DVPRA-SI became more apparent. Taking the Iris dataset as an example, although the running time of all three algorithms remained relatively short, the advantages of Algorithm DVPRA-SI became particularly prominent when shifting to larger datasets. For example, on the Rice dataset, DVPRA-SI quickly obtained simplified results 11.84, 11.44, and 11.43 when precision was 0.65, 0.75, and 0.85, respectively.

For the running time of Algorithm DVPRA-SD (Figure 3), the precision was also set to 0.65, 0.75, and 0.85, respectively. For Algorithm DVPRA-SD, all of the datasets were raw datasets, and then 25% of the samples was selected randomly as deleted samples. The running time of Algorithm DVPRA-SD (Figure 3) remained at the same scale as the running time of DVPRA-SI (Table 7). It is noted that to more intuitively observe the performance of the running time of algorithms, six datasets were selected from eight datasets for comparison in Figure 3 based on the scale of the datasets. Their execution efficiency and stability met the expected standards.

Algorithm DVPRA-SI with different precisions was compared on the classifiers SVM, KNN, and RF in Figure 4, Figure 5 and Figure 6, in which the performance of Algorithm DVPRA-SI varied on eight datasets. For all datasets in the comparative experiments, the result of Algorithm DVPRA-SI was the testing set, whose proportions were set to 60%, 70%, 80%, and 90%, respectively. When precision was 0.65, eight datasets were used for classification accuracies on three classifiers, for example, SVM-D.A.-F.F. represents the classification accuracies of D.A., F.F., and those of Algorithm DVPRA-SI on classifier SVM in Figure 4. The result of Algorithm DVPRA-SI exhibited higher classification accuracy than that of the original datasets on classifier KNN. For example, given precision 0.65, for the proportions 60%, 70%, 80%, and 90%, the classification accuracies obtained by Algorithm DVPRA-SI were 49.53%, 48.54%, 50.31%, and 44.37% when the classification accuracies of the corresponding original W.Q. dataset were 49.53%, 48.13%, 50.31%, and 44.37%, respectively. The classification accuracy of Algorithm DVPRA-SI on the R.F. classifier was better than that of the corresponding original dataset; for example, on the Wine dataset, precision was 0.65, the classification accuracies of the corresponding original dataset were 97.18%, 94.44%, 91.67%, and 100% when the classification accuracies obtained by Algorithm DVPRA-SI were 98.59%, 96.30%, 94.44%, and 100%, respectively.

According to the overall analysis above, the results of the classification accuracies demonstrate Algorithm DVPRA-SI’s robustness and effectiveness on different datasets. Based on the analysis of the above figures and tables, our proposed algorithms have been verified, and have advantages, to some extent.

6. Conclusions

The proposed algorithms can quickly respond to changes in data when dealing with sample set disturbance. When the proposed algorithms encounter scenarios such as data noise and outliers, they can reduce the impact of these factors on the processing results while ensuring the quality of processing, thus improving processing efficiency and robustness. PRR ensures that classification is not affected by sample disturbance while reducing redundant attributes, whereas VPR reduces redundant attributes for different precision requirements while maintaining classification capability. Based on the discernibility matrix construction of PRR and VPR, the process of dynamic VPR was discussed with the aid of PRR, and corresponding algorithms were proposed. In the future, we will continue to explore more dynamic variable-precision simplification problems and expand the application scope of dynamic reduction algorithms.