# Uncertainty Measurement and Attribute Reduction Algorithm Based on Kernel Similarity Rough Set Model

Authors: Baoguo Chen; Lei Chen; Ming Deng
Journal: Journal of Mathematics
 (2022)

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## Abstract

Attribute reduction is the core research content in rough set theory. At present, the attribute reduction of numerical information system mostly adopts the neighborhood rough set method. In order to further improve the similarity measurement effect between data objects, kernel function method is used to construct a new rough set model in numerical information system, and an uncertainty measurement method and attribute reduction method are proposed. Firstly, the similarity between objects of numerical information system is calculated by kernel function, and a granular structure model and rough set model based on kernel similarity relation are proposed. Then, from the perspective of kernel similarity rough approximation, an information system uncertainty measurement method called kernel approximation precision and kernel approximation roughness is proposed. Because these two measurement methods do not meet the strict monotonicity of information granulation, the concept of kernel knowledge granularity based on kernel similarity granular structure is further proposed in this paper. By combining kernel approximation precision and kernel approximation roughness with kernel knowledge granularity, an uncertainty measurement method of kernel similarity combination measurement is proposed. Finally, using the strict monotonicity of kernel similarity combination measurement, an attribute reduction algorithm for numerical information system is designed. Experimental analysis shows the effectiveness and superiority of the proposed method.


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## Body

## 1. Introduction

Rough set theory was first proposed by Pawlak, a Polish scholar. At present, it has become an important mathematical analysis tool in the fields of machine learning, pattern recognition, and knowledge discovery [1, 2]. Attribute reduction is an important application of rough set theory. Its main purpose is to eliminate redundant and inconsistent attributes in the original data set and improve the performance of data classification and knowledge discovery. Based on rough set theory, many kinds of attribute reduction algorithms have been proposed [3–6].Equivalence relation is used to partition data in classical rough set model, so attribute reduction can only be carried out for discrete information system, but there are a large number of numerical information systems in the actual environment. In order to solve this problem, scholars have established neighborhood rough set model under numerical information system and designed a variety of attribute reduction algorithms [7–10]. Hu et al. [11] proposed the earliest attribute reduction algorithm of numerical information system through neighborhood rough set. On the basis of Hu, several scholars improved neighborhood rough set, and Fan et al. [12] proposed attribute reduction algorithm based on maximum decision neighborhood rough set, Wang et al. [13] proposed an attribute reduction algorithm based on k-nearest neighbor rough set, Hu et al. [14] proposed an attribute reduction algorithm based on weighted neighborhood rough set. On the other hand, based on the neighborhood granulation of neighborhood rough set, many attribute evaluation methods of numerical information system are defined and used to construct attribute reduction algorithms. For example, He et al. [15] combined neighborhood granularity with neighborhood roughness and proposed an attribute reduction method of neighborhood combination measurement. Yao et al. [16] and Zhao and Yang [17] respectively defined the concepts of neighborhood rough mutual information entropy and neighborhood conditional entropy and designed the corresponding attribute reduction method. Wang et al. [18, 19] proposed neighborhood discrimination and neighborhood self-information degree on neighborhood rough set for attribute reduction of numerical information system. In short, attribute reduction based on neighborhood rough set has attracted more and more attention.However, in the neighborhood rough set model, the similarity relation between data objects is measured by distance [7–19]. Therefore, attribute reduction based on neighborhood rough set is realized by distance measurement similarity. In the field of data analysis, the kernel method [20] is a general tool for processing, analyzing, and comparing various types of data. In the kernel method, the kernel function is used to map the data to a higher dimensional feature space, in which the data can become easier to separate or better structured. Because kernel function provides a method to calculate the point product between vectors in high-dimensional feature space, it can be used as a kind of similarity measurement method in machine learning and also has good similarity measurement effect [21, 22].For numerical information system, in order to further improve the similarity measurement effect between data objects, a new rough set model is proposed by using kernel function to measure the similarity between objects, and the uncertainty measurement and attribute reduction of information system are studied. Firstly, the similarity between objects in numerical information system is calculated by kernel function, a concept of kernel similarity relation is proposed, and the granular structure model and rough set model based on kernel similarity relation are constructed. Then, based on the kernel similarity relation rough set model, an information system uncertainty measurement method of kernel approximation precision and kernel approximation roughness is proposed. Because the kernel approximation precision and kernel approximation roughness do not satisfy the strict monotonicity of information granulation, the concept of kernel knowledge granularity of granular structure model based on kernel similarity relation is defined. By combining kernel approximation precision and kernel approximation roughness with kernel knowledge granularity, an uncertainty measurement method of kernel similarity combination measurement is proposed. Finally, using the strict monotonicity of kernel similarity combination measurement, an attribute reduction algorithm for numerical information system is proposed. Experimental analysis shows the effectiveness and superiority of the proposed algorithm.


## 2. Basic Theory

In this section, we mainly introduce the concepts of neighborhood rough set and kernel function under numerical information system.In rough set theory, numerical information system is a numerical attribute value information system [11], which is usually expressed as S=U,AT,V. Here, U is called the universe of numerical information system, AT is the attribute set of numerical information system, and V is the attribute value domain of numerical information system.Definition 1 (see [11]).
Given the numerical information systemS=U,AT,V and the attribute subset B⊆AT, the neighborhood relation induced by the attribute subset B under the numerical information system is defined as(1)NBδ=x,y∈U×U|dBx,y≤δ,where dBx,y=∑∀a∈Bax−ayk1/k is the distance measurement between objects x and y and ax represents the attribute value of object x under attribute a. Generally, k takes 1, 2 or +∞, δ is called the neighborhood radius of neighborhood relation NBδ.Definition 2 (see [11]).
Given the numerical information systemS=U,AT,V, the neighborhood relation determined by the attribute subset B⊆AT is NBδ. The neighborhood class of ∀x∈U about NBδ under the universe U is defined as δBx=y∈U|x,y∈NBδ. According to the neighborhood class of each object in the universe U, the neighborhood granular structure of the universe U under the neighborhood relation NBδ can be further obtained. If U=x1,x2,…,xn, the neighborhood granular structure is expressed as(2)GSBδU=δBx1,δBx2,…,δBxn.
Definition2 shows that given the neighborhood relation, the universe of information system can be induced into a neighborhood granular structure of the universe, that is, the neighborhood granulation of the universe from the perspective of granular computing [11]. The neighborhood granulation can be used for various rough approximate calculations of information system [11–14].Definition 3 (see [11]).
Given the numerical information systemS=U,AT,V, the neighborhood relation determined by the attribute subset B⊆AT is NBδ, and the induced granular structure is GSBδU. Then, for the approximation object set X⊆U, the lower approximation set and the upper approximation set with respect to the neighborhood relation NBδ are defined as(3)N¯BδX=x∈U|δBx⊆X,N¯BδX=x∈U|δBx∩X≠∅,where N¯BδX,N¯BδX is called the neighborhood rough set of the approximation object set X under neighborhood relation NBδ.
Kernel function provides an effective method for the calculation of inner product in high-dimensional space. Usually, kernel function is interpreted as a similarity measurement between a pair of objects.Definition 4 (see [20]).
Given the numerical information systemS=U,AT,V, when the function mapping K:U×U⟶ℜ (ℜ is a set of real numbers) satisfies the following two relations at the same time, the function is called kernel function:(1)
∀x,y∈U,Kx,y=Ky,x(2)
∀x1,x2,…,xn∈U,∀r1,r2,…,rn∈ℜ,∑i=1n∑j=1nr1r2Kxi,xj≥0
Table1 provides three common kernel functions [20], where x−y represents the Euclidean distance between the object x and y, and the parameter σ is a variable value. The larger the parameter value is, the closer the value of these kernel functions will be to 1, and the smaller the parameter value is, the closer their value will be to 0. The selection of the optimal value of parameter σ is beyond the scope of this paper, which is not discussed in this paper. In the following, we assume that the parameter σ has given an appropriate value.Table 1
Common kernel functions.
NameKernel functionsGaussian kernelKG=exp−x−y2σExponential kernelKE=exp−x−yσRational square kernelKE=1−x−y2x−y2+σ


## 3. Rough Set Model and Uncertainty Measurement Based on Kernel Similarity Relation

Since kernel function provides a method to measure the similarity between objects, in this section, the binary relation in numerical information system will be defined through kernel function, which is called kernel similarity relation. Then, the kernel similarity relation is used to induce the information granulation of numerical information system, and the corresponding rough set model is proposed. At the same time, the uncertainty measurement of numerical information system is further studied.


### 3.1. Granular Structure and Rough Set Model Based on Kernel Similarity Relation

Definition 5.
Given the numerical information systemS=U,AT,V, attribute subset B⊆AT, and kernel similarity threshold θ∈0,1, the kernel similarity relation KRBθ is defined as(4)KRBθ=x,y∈U×U|Kx,y≥θ,where Kx,y is the kernel similarity measurement result between x and y. For ∀x∈U, the kernel similarity class kBθx under the kernel similarity relation KRBθ is defined as(5)kBθx=y∈U|x,y∈KRBθ.
Through the definition of kernel function in Definition4 and the definition of kernel similarity relation in Definition 5, it can be seen that the kernel similarity relation satisfies reflexivity and symmetry, but not necessarily transitivity.Property 1.
Given the numerical information systemS=U,AT,V, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1. For ∀x∈U, so satisfied(1)
IfA⊆B, then kBθ1x⊆kAθ1x(2)
Ifθ1≤θ2, then kBθ2x⊆kBθ1xProof.
(1)
For∀x,y∈U, according to the definition of Euclidean distance, if x−yA≤x−yB exists when A⊆B is met, KAx,y≥KBx,y is established, and kBθ1x⊆kAθ1x can be obtained based on Definition 5.(2)
For∀x,y∈U, there must be Kx,y≥θ1 when Kx,y≥θ2, but not Kx,y≥θ2 when Kx,y≥θ1. Therefore, based on Definition 5, if θ1≤θ2, there is kBθ2x⊆kBθ1x.
In Property1, (1) it shows that the kernel similarity class decreases monotonically with the increase of attributes and (2) it shows that the kernel similarity class decreases monotonically with the increase of kernel similarity threshold.
In Definition4, the kernel similarity relation of numerical information system is established by kernel function, and the kernel similarity class of each object can be calculated by kernel similarity relation. From the perspective of granular computing, each kernel similarity class is a subset of the universe, so it can be regarded as a unit of information granular computing. The collection of all object kernel similarity classes in the universe forms the coverage of a universe. This paper calls it kernel similarity granular structure. At the same time, by using different attribute subsets and kernel similarity thresholds, different kernel similarity granular structures can be constructed. Therefore, the granular structure framework model of numerical information system based on kernel similarity relation can be established.Definition 6.
Given the numerical information systemS=U,AT,V, U=x1,x2,…,xn, attribute subset B⊆AT, and kernel similarity threshold θ∈0,1, the kernel similarity relation determined by attribute subset B is KRBθ. Then, the kernel similarity granular structure induced by KRBθ in universe U is defined as(6)GSBθU=kBθx1,kBθx2,…,kBθxn.where kBθxi,1≤i≤n is called the kernel similarity granular of object xi under KRBθ.
In the numerical information system, each object can generate a kernel similarity granular. Given different attribute subsets and kernel similarity thresholds, different kernel similarity granular structures can be constructed, which requires us to compare the two kernel similarity granular structures. Therefore, next, we give the refinement and coarsening relationship of kernel similarity granular structure.Definition 7.
Given the numerical information systemS=U,AT,V, attribute subsets A,B⊆AT and kernel similarity threshold θ∈0,1:(1)
GSAθU=GSBθU⇔∀x∈U,kAθx=kBθx(2)
GSAθU≺¯GSBθU⇔∀x∈U,kAθx⊆kBθx
In Definition7, when ∀x∈U,kAθx=kBθx, we call the kernel similarity granular structure GSAθU equivalent to GSBθU, expressed as GSAθU=GSBθU. When ∀x∈U,kAθx⊆kBθx, we call the kernel similarity granular structure GSAθU refined in GSBθU and GSBθU coarsened in GSAθU, which is expressed as GSAθU≺¯GSBθU. In particular, when ∀x∈U,kAθx⊆kBθx and kAθx≠kBθx, we call the kernel similarity granular structure GSAθU strictly refined in GSBθU and GSBθU strictly coarsened in GSAθU, expressed as GSAθU≺GSBθU.
The refinement and coarsening relationships satisfy the transitivity of the relationship. For example, ifA,B,C⊆AT satisfies the refinement and coarsening relationships GSAθU≺¯GSBθU and GSBθU≺¯GSCθU, there is GSAθU≺¯GSCθU, and so is the strict refinement and strict coarsening relationship.Property 2.
Given the numerical information systemS=U,AT,V, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1. Then, the kernel similarity granular structure is satisfied(1)
IfA⊆B, then GSBθ1U≺¯GSAθ1U(2)
Ifθ1≤θ2, then GSBθ2U≺¯GSBθ1UProof.
According to Property1 and Definition 7, Property 2 can be obtained directly.
In Property2, (1) it shows the monotonic refinement of kernel similarity granular structure with the increase of attribute, and (2) it shows the monotonic refinement of kernel similarity granular structure with the increase of kernel similarity threshold.
Granular structure shows the spatial structure of information system based on granular and takes granular as the basic calculation unit, so it can carry out a variety of rough approximate calculations of information system. Next, a rough set model of numerical information system based on kernel similarity relation is proposed.Definition 8.
Given the numerical information systemS=U,AT,V, attribute subset B⊆AT, and kernel similarity threshold θ∈0,1, the kernel similarity relation determined by B is KRBθ. For the target approximation set X⊆U, the kernel lower approximation set and the kernel upper approximation set under KRBθ are defined as(7)KR¯BθX=x∈U|kBθx∈GSBθU,kBθx⊆X,KR¯BθX=x∈U|kBθx∈GSBθU,kBθx∩X≠∅,where KR¯BθX,KR¯BθX is called the kernel similarity rough set of the target approximation set X⊆U under the kernel similarity relation KRBθ.
At the same time, the kernel positive region set, kernel negative region set, and kernel boundary region set of the target approximation setX⊆U under KRBθ are defined as(8)POSBθX=KR¯BθX;NEGBθX=U−KR¯BθX;BUNBθX=KR¯BθX−KR¯BθX.
In particular, for the numerical decision information systemS=U,AT=C∪D,V, the decision class of universe U under the decision attribute set D is divided into U/D=D1,D2,…,Dm. Then, the kernel lower approximation set and kernel upper approximation set of decision attribute set D under KRBθ are defined as(9)KR¯BθD=∪Di∈U/DKR¯BθDi;KR¯BθD=∪Di∈U/DKR¯BθDi.
At the same time, the kernel positive region set, kernel negative region set, and kernel boundary region set of decision attribute setD under KRBθ are defined as(10)POSBθD=KR¯BθD;NEGBθD=U−KR¯BθD;BUNBθD=KR¯BθD−KR¯BθD.
In the neighborhood rough set model, the similarity relation between data objects is measured by distance. In the kernel similarity rough set model, the similarity relation between data objects is measured by kernel function, and the universe of information system is granulated by kernel function. The kernel similarity rough set model proposed in Definition8 provides a new solution for the rough approximation of numerical information system, which further expands the research results of rough set model of numerical information system.


### 3.2. Uncertainty Measurement Based on Kernel Similarity Relation

In rough set theory, approximation precision and approximation roughness are a common method to measure the uncertainty of information system from the perspective of rough approximation [22–24]. In this section, based on the kernel similarity rough set model, the kernel approximation precision and kernel approximation roughness of numerical information system are proposed.Definition 9.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, the kernel similarity relation determined by attribute subset B⊆AT is KRBθ, θ∈0,1. The kernel approximation precision αBθD and kernel approximation roughness ρBθD of decision attribute set D with respect to B are defined as(11)αBθD=∑i=1mKR¯BθD∑i=1mKR¯BθD;ρBθD=1−∑i=1mKR¯BθD∑i=1mKR¯BθD=1−αBθD,where ∘ represents the cardinality of the set. Obviously, 0≤αBθD,ρBθD≤1.
Through Definition9, it can be seen that the kernel approximation precision reflects the approximation degree of the kernel similarity relation to the rough approximation of the decision attribute set, and the kernel approximation roughness is the inverse measure of the kernel approximation precision, which reflects the uncertainty of the approximation.
The kernel approximation precision and kernel approximation roughness satisfy the following properties.Property 3.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfA⊆B, then αAθ1D≤αBθ1D, ρAθ1D≥ρBθ1D(2)
Ifθ1≤θ2, then αAθ1D≤αAθ2D, ρAθ1D≥ρAθ2DProof.
(1)
According to Property1, if A⊆B, ∀x∈U meets kBθ1x⊆kAθ1x. If x∈KR¯Aθ1D, ∃Di∈U/D meets kAθ1x⊆Di, that is, kBθ1x⊆kAθ1x⊆Di, then x∈KR¯Bθ1D, so KR¯Aθ1D⊆KR¯Bθ1D can be obtained. Similarly, if x∈KR¯Bθ1D, then ∃Di∈U/D meets kBθ1x∩Di≠∅, and because kBθ1x⊆kAθ1x, so kAθ1x∩Di≠∅, then x∈KR¯Aθ1D, KR¯Bθ1D⊆KR¯Aθ1D can be obtained. According to Definition 9, αAθ1D≤αBθ1D, ρAθ1D≥ρBθ1D can be obtained.(2)
According to Property1, similar to the proof of (1), we can get that (2) holds.Property 4.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfGSBθ1U≺¯GSAθ1U, then αAθ1D≤αBθ1D, ρAθ1D≥ρBθ1D(2)
IfGSAθ2U≺¯GSAθ1U, then αAθ1D≤αAθ2D, ρAθ1D≥ρAθ2DProof.
Property4 can be obtained by combining Property 2 and Property 3.
It can be seen from Properties2 and 3 that the kernel approximation precision and kernel approximation roughness can evaluate the uncertainty of information system. However, in some cases, these two measurement methods cannot provide enough information to evaluate the uncertainty caused by kernel similarity relation in numerical information systems, which can be illustrated by the following example.Example 1.
Given the numerical decision information systemS, universe U=x1,x2,x3,x4,x5,x6,x7,x8,x9,x10, which contains two classes D1=x1,x2,x3,x4,x5 and D2=x6,x7,x8,x9,x10. Let the two attribute subsets A and B of the information system satisfy A⊆B, and the kernel similarity granular structure is(12)GSAU=kAx1,kAx2,…,kAx2.where
kAx1=x1,x2,x3,x4; kAx2=x1,x2,x4 ; kAx3=x1,x3,x6; kAx4=x1,x2,x4,x5,x8;
kAx5=x4,x5,x6,x7,x8; kAx6=x3,x5,x6,x9; kAx7=x5,x7,x10; kAx8=x4,x5,x8,x9,x10;
kAx9=x6,x8,x9,x10; and kAx10=x7,x8,x9,x10.(13)GSBU=kBx1,kBx2,…,kBx2.where
kBx1=x1,x3; kBx2=x1,x4 ; kBx3=x1,x3,x6; kBx4=x2,x4,x5,x8; kBx5=x4,x5,x6,x7; kBx6=x3,x5,x6 ; kBx7=x5,x7,x10; kBx8=x4,x8,x9; kBx9=x8,x9,x10; and kBx10=x7,x9,x10.
Obviously, the relationshipGSBU≺¯GSAU is satisfied. It can be obtained according to Definitions 8 and 9,(14)KR¯AD1=KR¯BD1=x1,x2;KR¯AD1=KR¯BD1=x1,x2,x3,x4,x5,x6x7,x8;KR¯AD2=KR¯BD2=x9,x10;KR¯AD2=KR¯BD2=x3,x4,x5,x6x7,x8,x9,x10;
So,(15)αAD=KR¯AD1+KR¯AD2KR¯AD1+KR¯AD2=416;ρAD=1−αAD=1216,αBD=KR¯BD1+KR¯BD2KR¯BD1+KR¯BD2=416;ρBD=1−αBD=1216,that is, αAD=αBD,ρAD=ρBD.Example 1.
shows that with the increase of attributes, the kernel similarity class of the object decreases monotonically, while the kernel approximation precision and kernel approximation roughness do not change, that is, they do not meet the strict monotonicity.
In order to overcome the fact that approximation precision and approximation roughness cannot measure the uncertainty of information system well, scholars introduce the measurement of knowledge granularity of information system and propose a combined measurement method by combining approximation precision (approximation roughness) with knowledge granularity, which can make up for the defect of uncertainty measurement of approximation precision (approximation roughness) [22–24]. Based on this method, a combined measurement method of kernel approximation precision (kernel approximation roughness) and kernel knowledge granularity is proposed in this paper.Definition 10.
Given the numerical information systemS=U,AT,V, U=x1,x2,…,xn, attribute subset B⊆AT and kernel similarity threshold θ∈0,1. The kernel knowledge granularity of numerical information system under kernel similarity granular structure GSBθU=kBθx1,kBθx2,…,kBθxn is defined as(16)KKGBθU=1n∑i=1nkBθxiU.
The kernel knowledge granularity satisfies1/n≤KKGBθU≤1. When ∀x∈U, kBθx=x, KKGBθU=1/n; when ∀x∈U, kBθx=U, KKGBθU=1.
The kernel knowledge granularity satisfies the following property.Property 5.
Given the numerical information systemS=U,AT,V, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfA⊆B, then KKGBθ1U≤KKGAθ1U(2)
Ifθ1≤θ2, then KKGAθ2U≤KKGAθ1UProof.
According to the definition of kernel knowledge granularity in Property1 and Definition 10, Property 5 can be proved.
In Property5, (1) shows that the kernel knowledge granularity decreases monotonically with the increase of attributes, (2) shows that the kernel knowledge granularity decreases monotonically with the increase of kernel similarity threshold, and the decreases of (1) and (2) are strict. When at least one kernel similarity class in the kernel similarity granular structure decreases monotonically, the kernel knowledge granularity decreases. The situation shown in Example 1 will not occur.
Based on kernel approximation precision (kernel approximation roughness) and kernel knowledge granularity, a new uncertainty measurement method for numerical information systems will be proposed, which is called kernel similarity combination measurement.Definition 11.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subset B⊆AT and kernel similarity threshold θ∈0,1. Then, the kernel similarity combination measurement KCMθB is defined as(17)KCMθB=KKGBθU⋅1−αBθD=KKGBθU⋅ρBθD.
The kernel similarity combination measurement satisfies0≤KCMθB≤1.
The kernel similarity combination measurement satisfies the following properties.Property 6.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfA⊆B, then KCMθ1B≤KCMθ1A(2)
Ifθ1≤θ2, then KCMθ2A≤KCMθ1AProof.
According to Properties1, 3, and 5 and Definition 11, Property 6 can be obtained directly.Property 7.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfGSBθ1U≺¯GSAθ1U, then KCMθ1B≤KCMθ1A(2)
IfGSAθ2U≺¯GSAθ1U, then KCMθ2A≤KCMθ1AProof.
According to Property6, Property 7 can be obtained directly.
The kernel similarity combination measurement reflects the uncertainty of the information system from two aspects: one is to evaluate the uncertainty through the approximation degree of the kernel lower approximation set and the kernel upper approximation set, and the other is to evaluate the uncertainty through the average granularity of the kernel similarity granular structure under the kernel similarity relation. As shown in Definition10, average the size of each kernel similarity class in universe to define the kernel knowledge granularity. When at least one kernel similarity class in the universe decreases, the kernel knowledge granularity will decrease, since the kernel similarity combination measurement is the product of the kernel approximation precision (the kernel approximation roughness) and the kernel knowledge granularity. This makes the kernel similarity combination measurement have more strict monotonicity with the increase of attributes and also has higher uncertainty measurement effect in numerical information system.Example 2.
For the results of kernel similarity granular structure of the information system in Example1, according to Definition 10, the following can be obtained(18)KKGAU=110∑i=110kAxi10=40100;KKGBU=110∑i=110kBxi10=30100.
Therefore, it can be obtained according to Definition11,(19)KCMA=KKGAU⋅1−αAD=40100×1216=310;KCMB=KKGBU⋅1−αBD=30100×1216=940.
So,KCMB<KCMA.


## 3.1. Granular Structure and Rough Set Model Based on Kernel Similarity Relation

Definition 5.
Given the numerical information systemS=U,AT,V, attribute subset B⊆AT, and kernel similarity threshold θ∈0,1, the kernel similarity relation KRBθ is defined as(4)KRBθ=x,y∈U×U|Kx,y≥θ,where Kx,y is the kernel similarity measurement result between x and y. For ∀x∈U, the kernel similarity class kBθx under the kernel similarity relation KRBθ is defined as(5)kBθx=y∈U|x,y∈KRBθ.
Through the definition of kernel function in Definition4 and the definition of kernel similarity relation in Definition 5, it can be seen that the kernel similarity relation satisfies reflexivity and symmetry, but not necessarily transitivity.Property 1.
Given the numerical information systemS=U,AT,V, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1. For ∀x∈U, so satisfied(1)
IfA⊆B, then kBθ1x⊆kAθ1x(2)
Ifθ1≤θ2, then kBθ2x⊆kBθ1xProof.
(1)
For∀x,y∈U, according to the definition of Euclidean distance, if x−yA≤x−yB exists when A⊆B is met, KAx,y≥KBx,y is established, and kBθ1x⊆kAθ1x can be obtained based on Definition 5.(2)
For∀x,y∈U, there must be Kx,y≥θ1 when Kx,y≥θ2, but not Kx,y≥θ2 when Kx,y≥θ1. Therefore, based on Definition 5, if θ1≤θ2, there is kBθ2x⊆kBθ1x.
In Property1, (1) it shows that the kernel similarity class decreases monotonically with the increase of attributes and (2) it shows that the kernel similarity class decreases monotonically with the increase of kernel similarity threshold.
In Definition4, the kernel similarity relation of numerical information system is established by kernel function, and the kernel similarity class of each object can be calculated by kernel similarity relation. From the perspective of granular computing, each kernel similarity class is a subset of the universe, so it can be regarded as a unit of information granular computing. The collection of all object kernel similarity classes in the universe forms the coverage of a universe. This paper calls it kernel similarity granular structure. At the same time, by using different attribute subsets and kernel similarity thresholds, different kernel similarity granular structures can be constructed. Therefore, the granular structure framework model of numerical information system based on kernel similarity relation can be established.Definition 6.
Given the numerical information systemS=U,AT,V, U=x1,x2,…,xn, attribute subset B⊆AT, and kernel similarity threshold θ∈0,1, the kernel similarity relation determined by attribute subset B is KRBθ. Then, the kernel similarity granular structure induced by KRBθ in universe U is defined as(6)GSBθU=kBθx1,kBθx2,…,kBθxn.where kBθxi,1≤i≤n is called the kernel similarity granular of object xi under KRBθ.
In the numerical information system, each object can generate a kernel similarity granular. Given different attribute subsets and kernel similarity thresholds, different kernel similarity granular structures can be constructed, which requires us to compare the two kernel similarity granular structures. Therefore, next, we give the refinement and coarsening relationship of kernel similarity granular structure.Definition 7.
Given the numerical information systemS=U,AT,V, attribute subsets A,B⊆AT and kernel similarity threshold θ∈0,1:(1)
GSAθU=GSBθU⇔∀x∈U,kAθx=kBθx(2)
GSAθU≺¯GSBθU⇔∀x∈U,kAθx⊆kBθx
In Definition7, when ∀x∈U,kAθx=kBθx, we call the kernel similarity granular structure GSAθU equivalent to GSBθU, expressed as GSAθU=GSBθU. When ∀x∈U,kAθx⊆kBθx, we call the kernel similarity granular structure GSAθU refined in GSBθU and GSBθU coarsened in GSAθU, which is expressed as GSAθU≺¯GSBθU. In particular, when ∀x∈U,kAθx⊆kBθx and kAθx≠kBθx, we call the kernel similarity granular structure GSAθU strictly refined in GSBθU and GSBθU strictly coarsened in GSAθU, expressed as GSAθU≺GSBθU.
The refinement and coarsening relationships satisfy the transitivity of the relationship. For example, ifA,B,C⊆AT satisfies the refinement and coarsening relationships GSAθU≺¯GSBθU and GSBθU≺¯GSCθU, there is GSAθU≺¯GSCθU, and so is the strict refinement and strict coarsening relationship.Property 2.
Given the numerical information systemS=U,AT,V, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1. Then, the kernel similarity granular structure is satisfied(1)
IfA⊆B, then GSBθ1U≺¯GSAθ1U(2)
Ifθ1≤θ2, then GSBθ2U≺¯GSBθ1UProof.
According to Property1 and Definition 7, Property 2 can be obtained directly.
In Property2, (1) it shows the monotonic refinement of kernel similarity granular structure with the increase of attribute, and (2) it shows the monotonic refinement of kernel similarity granular structure with the increase of kernel similarity threshold.
Granular structure shows the spatial structure of information system based on granular and takes granular as the basic calculation unit, so it can carry out a variety of rough approximate calculations of information system. Next, a rough set model of numerical information system based on kernel similarity relation is proposed.Definition 8.
Given the numerical information systemS=U,AT,V, attribute subset B⊆AT, and kernel similarity threshold θ∈0,1, the kernel similarity relation determined by B is KRBθ. For the target approximation set X⊆U, the kernel lower approximation set and the kernel upper approximation set under KRBθ are defined as(7)KR¯BθX=x∈U|kBθx∈GSBθU,kBθx⊆X,KR¯BθX=x∈U|kBθx∈GSBθU,kBθx∩X≠∅,where KR¯BθX,KR¯BθX is called the kernel similarity rough set of the target approximation set X⊆U under the kernel similarity relation KRBθ.
At the same time, the kernel positive region set, kernel negative region set, and kernel boundary region set of the target approximation setX⊆U under KRBθ are defined as(8)POSBθX=KR¯BθX;NEGBθX=U−KR¯BθX;BUNBθX=KR¯BθX−KR¯BθX.
In particular, for the numerical decision information systemS=U,AT=C∪D,V, the decision class of universe U under the decision attribute set D is divided into U/D=D1,D2,…,Dm. Then, the kernel lower approximation set and kernel upper approximation set of decision attribute set D under KRBθ are defined as(9)KR¯BθD=∪Di∈U/DKR¯BθDi;KR¯BθD=∪Di∈U/DKR¯BθDi.
At the same time, the kernel positive region set, kernel negative region set, and kernel boundary region set of decision attribute setD under KRBθ are defined as(10)POSBθD=KR¯BθD;NEGBθD=U−KR¯BθD;BUNBθD=KR¯BθD−KR¯BθD.
In the neighborhood rough set model, the similarity relation between data objects is measured by distance. In the kernel similarity rough set model, the similarity relation between data objects is measured by kernel function, and the universe of information system is granulated by kernel function. The kernel similarity rough set model proposed in Definition8 provides a new solution for the rough approximation of numerical information system, which further expands the research results of rough set model of numerical information system.


## 3.2. Uncertainty Measurement Based on Kernel Similarity Relation

In rough set theory, approximation precision and approximation roughness are a common method to measure the uncertainty of information system from the perspective of rough approximation [22–24]. In this section, based on the kernel similarity rough set model, the kernel approximation precision and kernel approximation roughness of numerical information system are proposed.Definition 9.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, the kernel similarity relation determined by attribute subset B⊆AT is KRBθ, θ∈0,1. The kernel approximation precision αBθD and kernel approximation roughness ρBθD of decision attribute set D with respect to B are defined as(11)αBθD=∑i=1mKR¯BθD∑i=1mKR¯BθD;ρBθD=1−∑i=1mKR¯BθD∑i=1mKR¯BθD=1−αBθD,where ∘ represents the cardinality of the set. Obviously, 0≤αBθD,ρBθD≤1.
Through Definition9, it can be seen that the kernel approximation precision reflects the approximation degree of the kernel similarity relation to the rough approximation of the decision attribute set, and the kernel approximation roughness is the inverse measure of the kernel approximation precision, which reflects the uncertainty of the approximation.
The kernel approximation precision and kernel approximation roughness satisfy the following properties.Property 3.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfA⊆B, then αAθ1D≤αBθ1D, ρAθ1D≥ρBθ1D(2)
Ifθ1≤θ2, then αAθ1D≤αAθ2D, ρAθ1D≥ρAθ2DProof.
(1)
According to Property1, if A⊆B, ∀x∈U meets kBθ1x⊆kAθ1x. If x∈KR¯Aθ1D, ∃Di∈U/D meets kAθ1x⊆Di, that is, kBθ1x⊆kAθ1x⊆Di, then x∈KR¯Bθ1D, so KR¯Aθ1D⊆KR¯Bθ1D can be obtained. Similarly, if x∈KR¯Bθ1D, then ∃Di∈U/D meets kBθ1x∩Di≠∅, and because kBθ1x⊆kAθ1x, so kAθ1x∩Di≠∅, then x∈KR¯Aθ1D, KR¯Bθ1D⊆KR¯Aθ1D can be obtained. According to Definition 9, αAθ1D≤αBθ1D, ρAθ1D≥ρBθ1D can be obtained.(2)
According to Property1, similar to the proof of (1), we can get that (2) holds.Property 4.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfGSBθ1U≺¯GSAθ1U, then αAθ1D≤αBθ1D, ρAθ1D≥ρBθ1D(2)
IfGSAθ2U≺¯GSAθ1U, then αAθ1D≤αAθ2D, ρAθ1D≥ρAθ2DProof.
Property4 can be obtained by combining Property 2 and Property 3.
It can be seen from Properties2 and 3 that the kernel approximation precision and kernel approximation roughness can evaluate the uncertainty of information system. However, in some cases, these two measurement methods cannot provide enough information to evaluate the uncertainty caused by kernel similarity relation in numerical information systems, which can be illustrated by the following example.Example 1.
Given the numerical decision information systemS, universe U=x1,x2,x3,x4,x5,x6,x7,x8,x9,x10, which contains two classes D1=x1,x2,x3,x4,x5 and D2=x6,x7,x8,x9,x10. Let the two attribute subsets A and B of the information system satisfy A⊆B, and the kernel similarity granular structure is(12)GSAU=kAx1,kAx2,…,kAx2.where
kAx1=x1,x2,x3,x4; kAx2=x1,x2,x4 ; kAx3=x1,x3,x6; kAx4=x1,x2,x4,x5,x8;
kAx5=x4,x5,x6,x7,x8; kAx6=x3,x5,x6,x9; kAx7=x5,x7,x10; kAx8=x4,x5,x8,x9,x10;
kAx9=x6,x8,x9,x10; and kAx10=x7,x8,x9,x10.(13)GSBU=kBx1,kBx2,…,kBx2.where
kBx1=x1,x3; kBx2=x1,x4 ; kBx3=x1,x3,x6; kBx4=x2,x4,x5,x8; kBx5=x4,x5,x6,x7; kBx6=x3,x5,x6 ; kBx7=x5,x7,x10; kBx8=x4,x8,x9; kBx9=x8,x9,x10; and kBx10=x7,x9,x10.
Obviously, the relationshipGSBU≺¯GSAU is satisfied. It can be obtained according to Definitions 8 and 9,(14)KR¯AD1=KR¯BD1=x1,x2;KR¯AD1=KR¯BD1=x1,x2,x3,x4,x5,x6x7,x8;KR¯AD2=KR¯BD2=x9,x10;KR¯AD2=KR¯BD2=x3,x4,x5,x6x7,x8,x9,x10;
So,(15)αAD=KR¯AD1+KR¯AD2KR¯AD1+KR¯AD2=416;ρAD=1−αAD=1216,αBD=KR¯BD1+KR¯BD2KR¯BD1+KR¯BD2=416;ρBD=1−αBD=1216,that is, αAD=αBD,ρAD=ρBD.Example 1.
shows that with the increase of attributes, the kernel similarity class of the object decreases monotonically, while the kernel approximation precision and kernel approximation roughness do not change, that is, they do not meet the strict monotonicity.
In order to overcome the fact that approximation precision and approximation roughness cannot measure the uncertainty of information system well, scholars introduce the measurement of knowledge granularity of information system and propose a combined measurement method by combining approximation precision (approximation roughness) with knowledge granularity, which can make up for the defect of uncertainty measurement of approximation precision (approximation roughness) [22–24]. Based on this method, a combined measurement method of kernel approximation precision (kernel approximation roughness) and kernel knowledge granularity is proposed in this paper.Definition 10.
Given the numerical information systemS=U,AT,V, U=x1,x2,…,xn, attribute subset B⊆AT and kernel similarity threshold θ∈0,1. The kernel knowledge granularity of numerical information system under kernel similarity granular structure GSBθU=kBθx1,kBθx2,…,kBθxn is defined as(16)KKGBθU=1n∑i=1nkBθxiU.
The kernel knowledge granularity satisfies1/n≤KKGBθU≤1. When ∀x∈U, kBθx=x, KKGBθU=1/n; when ∀x∈U, kBθx=U, KKGBθU=1.
The kernel knowledge granularity satisfies the following property.Property 5.
Given the numerical information systemS=U,AT,V, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfA⊆B, then KKGBθ1U≤KKGAθ1U(2)
Ifθ1≤θ2, then KKGAθ2U≤KKGAθ1UProof.
According to the definition of kernel knowledge granularity in Property1 and Definition 10, Property 5 can be proved.
In Property5, (1) shows that the kernel knowledge granularity decreases monotonically with the increase of attributes, (2) shows that the kernel knowledge granularity decreases monotonically with the increase of kernel similarity threshold, and the decreases of (1) and (2) are strict. When at least one kernel similarity class in the kernel similarity granular structure decreases monotonically, the kernel knowledge granularity decreases. The situation shown in Example 1 will not occur.
Based on kernel approximation precision (kernel approximation roughness) and kernel knowledge granularity, a new uncertainty measurement method for numerical information systems will be proposed, which is called kernel similarity combination measurement.Definition 11.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subset B⊆AT and kernel similarity threshold θ∈0,1. Then, the kernel similarity combination measurement KCMθB is defined as(17)KCMθB=KKGBθU⋅1−αBθD=KKGBθU⋅ρBθD.
The kernel similarity combination measurement satisfies0≤KCMθB≤1.
The kernel similarity combination measurement satisfies the following properties.Property 6.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfA⊆B, then KCMθ1B≤KCMθ1A(2)
Ifθ1≤θ2, then KCMθ2A≤KCMθ1AProof.
According to Properties1, 3, and 5 and Definition 11, Property 6 can be obtained directly.Property 7.
Given the numerical decision information systemS=U,AT=C∪D,V, U/D=D1,D2,…,Dm, attribute subsets A,B⊆AT and kernel similarity thresholds θ1,θ2∈0,1, so satisfied,(1)
IfGSBθ1U≺¯GSAθ1U, then KCMθ1B≤KCMθ1A(2)
IfGSAθ2U≺¯GSAθ1U, then KCMθ2A≤KCMθ1AProof.
According to Property6, Property 7 can be obtained directly.
The kernel similarity combination measurement reflects the uncertainty of the information system from two aspects: one is to evaluate the uncertainty through the approximation degree of the kernel lower approximation set and the kernel upper approximation set, and the other is to evaluate the uncertainty through the average granularity of the kernel similarity granular structure under the kernel similarity relation. As shown in Definition10, average the size of each kernel similarity class in universe to define the kernel knowledge granularity. When at least one kernel similarity class in the universe decreases, the kernel knowledge granularity will decrease, since the kernel similarity combination measurement is the product of the kernel approximation precision (the kernel approximation roughness) and the kernel knowledge granularity. This makes the kernel similarity combination measurement have more strict monotonicity with the increase of attributes and also has higher uncertainty measurement effect in numerical information system.Example 2.
For the results of kernel similarity granular structure of the information system in Example1, according to Definition 10, the following can be obtained(18)KKGAU=110∑i=110kAxi10=40100;KKGBU=110∑i=110kBxi10=30100.
Therefore, it can be obtained according to Definition11,(19)KCMA=KKGAU⋅1−αAD=40100×1216=310;KCMB=KKGBU⋅1−αBD=30100×1216=940.
So,KCMB<KCMA.


## 4. Attribute Reduction Algorithm Based on Kernel Similarity Combination Measurement

In the previous section, the uncertainty measurement method of numerical information system based on kernel similarity combination measurement is proposed by combining kernel approximation precision (kernel approximation roughness) and kernel knowledge granularity. Due to the strict monotonicity of the kernel similarity combination measurement with the increase of attributes, this measurement is used to design the attribute reduction of information system in this section, and a numerical information system attribute reduction algorithm based on the kernel similarity combination measurement is proposed.Definition 12.
Given the numerical decision information systemS=U,AT=C∪D,V and the kernel similarity threshold θ∈0,1. If the attribute subset red is the kernel similarity combination measurement attribute reduction set of information system S, then if and only if(1)
KCMθred=KCMθC(2)
∀a∈red,KCMθred−a>KCMθred
In Definition12, condition (1) requires that the kernel similarity combination measurement of the attribute reduction set is equal to that of the complete attribute set, and condition (2) requires that the attribute reduction set is minimal, that is, it does not contain redundant attributes.
Generally, heuristic method is an important way to search for attribute reduction of information system [4]. The main strategy of this method is to select the optimal attribute in each iteration. The strict monotonicity of kernel similarity combination measurement is an important property to evaluate attribute significance. We define the following methods to evaluate the attribute importance of numerical information systems based on kernel similarity combination measurement.Definition 13.
Given the numerical decision information systemS=U,AT=C∪D,V, attribute subset B⊆AT and kernel similarity threshold θ∈0,1. The external importance of attribute ∀a∈C−B with respect to attribute set B is defined as(20)Sigoutθa,B=KCMθB−KCMθB∪a.
The internal importance of attribute∀b∈B with respect to attribute set B is defined as(21)Siginθb,B=KCMθB−b−KCMθB.
In Definition13, the external attribute importance is to evaluate the attribute importance of attributes other than candidate attributes, and the internal attribute importance is to evaluate the attribute importance of attributes in candidate attributes. Using the attribute importance shown in Definition 13 to evaluate the significance of attributes in information system, an attribute reduction algorithm for numerical information system is proposed, as shown in Algorithm 1.
In Algorithm1, step 2 searches for attributes from the complete attribute set through the external attribute importance and selects the attribute with the maximal importance as the candidate attribute. Because every time a new attribute is selected, the kernel similarity granular structure of the candidate attribute reduction result is changed, and step 3 needs to continue the heuristic search for attributes in the remaining attributes until the external attribute importance of the remaining attributes is 0. In order to prevent redundant attributes in the selected attribute reduction results, step 4 performs reverse redundancy elimination through the internal attribute importance, that is, the attribute with internal attribute importance equal to 0 will be deleted, and the final attribute reduction result will be obtained after completing steps 2 to 4. In Algorithm 1, the time complexity of calculating attribute importance is OC⋅U2, so the time complexity of the whole Algorithm 1 is OC2⋅U2.Algorithm1: Attribute reduction algorithm of numerical information system based on kernel similarity combination measurement.
Input: Numerical decision information systemS=U,AT=C∪D,V, specific kernel function K, kernel similarity threshold θ∈0,1.Output: Attribute reduction resultred.Step 1. Initialize attribute reduction resultred=∅;Step 2. For attribute∀a∈C−red, the external attribute importance Sigoutθa,red of each attribute a is calculated. In particular, Sigoutθa,∅=0.Step 3. Find the attributeamax corresponding to the maximum external attribute importance of ∀a∈C−red in step 2. If Sigoutθamax,red>0, then red←red∪amax and return to step 2; otherwise, jump to step 4.Step 4. For∀a∈red, calculate the internal attribute importance Siginθa,red of attribute a. If Siginθa,red=0 exists, then red←red−a.Step 5. Return attribute reduction resultred.


## 5. Experimental Analysis

In this section, we will test the uncertainty measurement method and attribute reduction algorithm proposed in this paper through UCI machine learning classification data set, so as to verify the effectiveness and superiority of this method. Table2 shows the details of UCI data sets. The conditional attribute values of these data sets are numerical, so they can be directly applied to the algorithm proposed in this paper. In order to prevent the influence of the dimension of attribute values of data sets on the experiment, the conditional attribute values of these data sets are normalized to the [0,1] interval by range normalization method. At the same time, the kernel similarity measurement of all experiments adopts Gaussian kernel function, and the parameters σ are uniformly set to 0.8. All experiments are coded by MATLAB 2015b and run in a personal PC environment with Intel(R) Core(TM) i7-4790 CPU @3.60 GHz and 16 GB memory.Table 2
Experimental data sets.
No.Data setsObjectsAttributesClasses1wine1781332sonar2086023iono3513424wdbc5693125biodeg10554126messidor11511927winequality-red15991168winequality-white48981179magic19020102


### 5.1. Experiment on Uncertainty Measurement

In the kernel similarity combination measurement proposed in this paper, the kernel similarity thresholdθ∈0,1 plays a key role in information granulation, so it is particularly important to select an appropriate kernel similarity threshold. In this experiment, the kernel similarity threshold θ is taken in the interval 0.57,0.95 through the interval of 0.02, and then the uncertainty measurement is calculated for the data sets in the order of increasing attribute serial number. The experimental results of all data sets are shown in Figure 1. The abscissa of each graph is the increasing number of attributes, the ordinate is the kernel similarity threshold θ, and the vertical coordinate is the uncertainty measurement result. By observing the subgraphs in Figure 1, it can be found that when the kernel similarity threshold θ is small, such as 0.57 to 0.71, with the increase of attributes, the kernel similarity combination measurement of most data sets decreases slowly, and the difference between the uncertainty measurement of the first attribute and the last attribute is not particularly large. When the kernel similarity threshold θ is large, for example, 0.87 to 0.95, with the increase of attributes, the kernel similarity combination measurement of most data sets decreases rapidly, and for the last several attributes, the uncertainty measurement result is equal to or close to 0. This shows that too large or too small θ value makes the kernel similarity class too refined or too coarse, which cannot measure the uncertainty of numerical information system well. Based on the experimental results of each subgraph in Figure 1, the kernel similarity threshold θ=0.75 is selected for the experiment, and this value is also used in subsequent experiments.Figure 1
Uncertainty measurement results of each data set under differentθ. (a) wine. (b) sonar. (c) iono. (d) wdbc. (e) biodeg. (f) messidor. (g) winequality-red. (h) winequality-white. (i) magic.
(a)(b)(c)(d)(e)(f)(g)(h)(i)In order to verify the effectiveness of the kernel similarity combination measurement proposed in this paper, this section compares the uncertainty measurement results of each data set with the kernel approximation roughness, kernel similarity knowledge granularity, and kernel similarity combination measurement, respectively. The experimental results of all data sets are shown in Figure2. The abscissa of each graph is the increasing number of attributes, and the ordinate is the uncertainty measurement result. In Figure 2, the three uncertainty measurement results gradually decrease with the increase of attributes. This is mainly because more knowledge is acquired with the increase of attributes, so the uncertainty of information system gradually decreases. Among these experimental results, for the kernel similarity roughness of the data set messidor, when the number of information system attributes increases from 1 to 8, the uncertainty measurement results basically remain unchanged. Similarly, there are data sets winequality-red, winequality-white, and magic. At this time, the kernel similarity knowledge granularity decreases, indicating that the granulation of information system is gradually refined, so the kernel similarity roughness does not meet the strict monotonicity of data granulation. For kernel similarity knowledge granularity and kernel similarity combination measurement, the uncertainty measurement results decrease strictly monotonically with the increase of attributes. At the same time, the measurement results of kernel similarity combination measurement integrate the results of kernel similarity roughness and kernel similarity knowledge granularity, so kernel similarity combination measurement has better uncertainty measurement effect of information system.Figure 2
Comparison of three uncertainty measurement results. (a) wine. (b) sonar. (c) iono. (d) wdbc. (e) biodeg. (f) messidor. (g) winequality-red. (h) winequality-white. (i) magic.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


### 5.2. Experimental Comparison of Attribute Reduction

This section carries out the comparative experiment of attribute reduction. In order to make comparison, this experiment selects three comparison algorithms among the same type of attribute reduction algorithms proposed in recent years, namely, (1) a novel hybrid feature selection method considering feature interaction in neighborhood rough set [25] (it is recorded as the comparison Algorithm 1); a novel approach to attribute reduction based on weighted neighborhood rough sets [14] (it is recorded as the comparison Algorithm 2); and attribute reduction with fuzzy rough self-information measures [5] (it is recorded as the comparison Algorithm 3). In this experiment, SVM classifier and kNN (k = 3) classifier will be used to evaluate the attribute reduction performance of all algorithms through 10 cross validations. In the training stage, all algorithms are used to reduce the attributes of the training set, and the corresponding attribute reduction results of each algorithm are obtained. In the verification stage, the obtained attribute reduction set is used to extract subdata from the verification data set to calculate the classification accuracy of SVM and kNN. Repeat the above experiment for 10 times, and take the average value of 10 times as the final experimental result.The classification accuracy calculated by SVM and kNN on the original attribute sets and the reduction sets of the four algorithms are shown in Tables3 and 4, respectively, in which the best experimental results are highlighted in bold. It can be seen from Tables 3 and 4 that the average classification accuracy of all reduction results is better than the original data. Compared with the average classification accuracy of original data, the average classification accuracy of comparison Algorithms 1–3, and the algorithm proposed in this paper under SVM are improved by 4.3%, 4.5%, 2.1%, and 5.3%, respectively, and the average classification accuracy under kNN is improved by 3.5%, 4.6%, 2.0%, and 5.8%, respectively. For these four algorithms, the proposed algorithm has higher SVM and KNN classification accuracy in most data sets. Therefore, the attribute reduction classification accuracy performance of the proposed algorithm is obviously better than the three comparison algorithms.Table 3
Comparison of SVM classification accuracy of attribute reduction results
Data setsOriginal dataThe comparison Algorithm 1The comparison Algorithm 2The comparison Algorithm 3The algorithm proposed in this paperwine0.9250 ± 0.00270.9493 ± 0.00640.9526 ± 0.00570.9372 ± 0.00270.9657 ± 0.0019sonar0.8066 ± 0.00900.8550 ± 0.01720.8425 ± 0.01450.8237 ± 0.01260.8564 ± 0.0132iono0.8109 ± 0.00430.8432 ± 0.00430.8521 ± 0.00480.8329 ± 0.00620.8570 ± 0.0057wdbc0.9251 ± 0.00220.9572 ± 0.00290.9735 ± 0.00450.9356 ± 0.00180.9697 ± 0.0023biodeg0.8250 ± 0.00270.8383 ± 0.00490.8551 ± 0.00330.8290 ± 0.00270.8478 ± 0.0036messidor0.8458 ± 0.00610.8557 ± 0.00670.8608 ± 0.00580.8593 ± 0.00690.8713 ± 0.0062winequality-red0.7671 ± 0.00740.8129 ± 0.00810.7864 ± 0.00620.7781 ± 0.00580.7932 ± 0.0051winequality-white0.7484 ± 0.00170.8364 ± 0.00770.8242 ± 0.00420.7992 ± 0.00650.8425 ± 0.0047magic0.8266 ± 0.00110.8596 ± 0.00230.8691 ± 0.00310.8431 ± 0.00260.8793 ± 0.0018average0.8311 ± 0.00410.8675 ± 0.00670.8684 ± 0.00570.8486 ± 0.00530.8758 ± 0.0049The best experimental results are highlighted in bold.Table 4
Comparison of kNN classification accuracy of attribute reduction results.
Data setsOriginal dataThe comparison Algorithm 1The comparison Algorithm 2The comparison Algorithm 3The algorithm proposed in this paperwine0.9011 ± 0.01700.9321 ± 0.00870.9493 ± 0.00470.9254 ± 0.00680.9551 ± 0.0053sonar0.8164 ± 0.01910.8437 ± 0.01870.8608 ± 0.01750.8237 ± 0.01630.8498 ± 0.0139iono0.8493 ± 0.00620.8988 ± 0.00680.9183 ± 0.00490.8841 ± 0.00590.9268 ± 0.0048wdbc0.9378 ± 0.00460.9468 ± 0.00620.9560 ± 0.00570.9424 ± 0.00390.9698 ± 0.0031biodeg0.8250 ± 0.00580.8328 ± 0.00460.8273 ± 0.00510.8257 ± 0.00380.8378 ± 0.0040messidor0.8273 ± 0.00440.8562 ± 0.00420.8476 ± 0.00490.8455 ± 0.00570.8693 ± 0.0050winequality-red0.7563 ± 0.00820.8015 ± 0.00740.7962 ± 0.00690.7760 ± 0.00560.7927 ± 0.0053winequality-white0.7598 ± 0.00390.8019 ± 0.00330.8250 ± 0.00450.7828 ± 0.00480.8056 ± 0.0042magic0.8019 ± 0.00220.8278 ± 0.00170.8352 ± 0.00360.8159 ± 0.00330.8434 ± 0.0029average0.8305 ± 0.00790.8601 ± 0.00670.8684 ± 0.00640.8468 ± 0.00620.8790 ± 0.0053The best experimental results are highlighted in bold.Running time is an important index to evaluate the feasibility of attribute reduction algorithm. The average running time of the four algorithms for attribute reduction is shown in Figure3. As can be seen from Figure 3, the running time of the algorithm proposed in this paper is slightly worse than that of the comparison Algorithm 3. The main reason is that the comparison Algorithm 3 uses the fuzzy rough self-information for heuristic search attribute reduction, only needs to calculate the fuzzy similarity class of each object, and does not need to calculate the upper and lower approximation sets of the decision class, thus reducing a part of the amount of calculation. The running time of the algorithm proposed in this paper is obviously better than that of the comparison Algorithm 1 and the comparison, Algorithm 2. The main reason is that the comparison Algorithm 1 needs to perform the operation of attribute interaction in attribute reduction, and the comparison Algorithm 2 needs to perform the operation of attribute weight in attribute reduction, so more calculation will be generated, and the comparison Algorithm 2 needs to perform a large number of matrix operations when calculating attribute weight, so the calculation time of the comparison Algorithm 2 is the largest. Obviously, under the condition of comprehensive calculation efficiency and classification accuracy, the attribute reduction algorithm in this paper is the best choice.Figure 3
Attribute reduction runtime comparison. (a) Partial data sets runtime comparison. (b) Partial data sets runtime comparison.
(a)(b)The length of attribute reduction results is another important index to evaluate the performance of attribute reduction. Table5 shows the average length results of attribute reduction results of all algorithms. Comparing these four algorithms, it can be found that the length of the attribute subset selected by the algorithm proposed in this paper is the smallest in most data sets, and there are slightly larger attribute subsets in data sets iono, messidor, and magic. The synthesis shows that the algorithm proposed in this paper has good attribute reduction effect and obvious advantages in dimension reduction.Table 5
Length comparison of attribute reduction results.
Data setsOriginal dataThe comparison Algorithm 1The comparison Algorithm 2The comparison Algorithm 3The algorithm proposed in this paperwine1310.28.79.38.2sonar6012.713.510.29.4iono349.310.57.98.3wdbc3112.29.510.39.4biodeg4113.012.512.211.2messidor1911.210.89.810.2winequality-red1111.010.611.010.4winequality-white1111.011.010.510.5magic106.78.07.87.2average25.610.810.59.89.4The best experimental results are highlighted in bold.Combining all the experimental results, it can be proved that the attribute reduction algorithm based on kernel similarity combination measurement proposed in this paper can quickly select smaller attribute subsets and obtain higher classification performance.


## 5.1. Experiment on Uncertainty Measurement

In the kernel similarity combination measurement proposed in this paper, the kernel similarity thresholdθ∈0,1 plays a key role in information granulation, so it is particularly important to select an appropriate kernel similarity threshold. In this experiment, the kernel similarity threshold θ is taken in the interval 0.57,0.95 through the interval of 0.02, and then the uncertainty measurement is calculated for the data sets in the order of increasing attribute serial number. The experimental results of all data sets are shown in Figure 1. The abscissa of each graph is the increasing number of attributes, the ordinate is the kernel similarity threshold θ, and the vertical coordinate is the uncertainty measurement result. By observing the subgraphs in Figure 1, it can be found that when the kernel similarity threshold θ is small, such as 0.57 to 0.71, with the increase of attributes, the kernel similarity combination measurement of most data sets decreases slowly, and the difference between the uncertainty measurement of the first attribute and the last attribute is not particularly large. When the kernel similarity threshold θ is large, for example, 0.87 to 0.95, with the increase of attributes, the kernel similarity combination measurement of most data sets decreases rapidly, and for the last several attributes, the uncertainty measurement result is equal to or close to 0. This shows that too large or too small θ value makes the kernel similarity class too refined or too coarse, which cannot measure the uncertainty of numerical information system well. Based on the experimental results of each subgraph in Figure 1, the kernel similarity threshold θ=0.75 is selected for the experiment, and this value is also used in subsequent experiments.Figure 1
Uncertainty measurement results of each data set under differentθ. (a) wine. (b) sonar. (c) iono. (d) wdbc. (e) biodeg. (f) messidor. (g) winequality-red. (h) winequality-white. (i) magic.
(a)(b)(c)(d)(e)(f)(g)(h)(i)In order to verify the effectiveness of the kernel similarity combination measurement proposed in this paper, this section compares the uncertainty measurement results of each data set with the kernel approximation roughness, kernel similarity knowledge granularity, and kernel similarity combination measurement, respectively. The experimental results of all data sets are shown in Figure2. The abscissa of each graph is the increasing number of attributes, and the ordinate is the uncertainty measurement result. In Figure 2, the three uncertainty measurement results gradually decrease with the increase of attributes. This is mainly because more knowledge is acquired with the increase of attributes, so the uncertainty of information system gradually decreases. Among these experimental results, for the kernel similarity roughness of the data set messidor, when the number of information system attributes increases from 1 to 8, the uncertainty measurement results basically remain unchanged. Similarly, there are data sets winequality-red, winequality-white, and magic. At this time, the kernel similarity knowledge granularity decreases, indicating that the granulation of information system is gradually refined, so the kernel similarity roughness does not meet the strict monotonicity of data granulation. For kernel similarity knowledge granularity and kernel similarity combination measurement, the uncertainty measurement results decrease strictly monotonically with the increase of attributes. At the same time, the measurement results of kernel similarity combination measurement integrate the results of kernel similarity roughness and kernel similarity knowledge granularity, so kernel similarity combination measurement has better uncertainty measurement effect of information system.Figure 2
Comparison of three uncertainty measurement results. (a) wine. (b) sonar. (c) iono. (d) wdbc. (e) biodeg. (f) messidor. (g) winequality-red. (h) winequality-white. (i) magic.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 5.2. Experimental Comparison of Attribute Reduction

This section carries out the comparative experiment of attribute reduction. In order to make comparison, this experiment selects three comparison algorithms among the same type of attribute reduction algorithms proposed in recent years, namely, (1) a novel hybrid feature selection method considering feature interaction in neighborhood rough set [25] (it is recorded as the comparison Algorithm 1); a novel approach to attribute reduction based on weighted neighborhood rough sets [14] (it is recorded as the comparison Algorithm 2); and attribute reduction with fuzzy rough self-information measures [5] (it is recorded as the comparison Algorithm 3). In this experiment, SVM classifier and kNN (k = 3) classifier will be used to evaluate the attribute reduction performance of all algorithms through 10 cross validations. In the training stage, all algorithms are used to reduce the attributes of the training set, and the corresponding attribute reduction results of each algorithm are obtained. In the verification stage, the obtained attribute reduction set is used to extract subdata from the verification data set to calculate the classification accuracy of SVM and kNN. Repeat the above experiment for 10 times, and take the average value of 10 times as the final experimental result.The classification accuracy calculated by SVM and kNN on the original attribute sets and the reduction sets of the four algorithms are shown in Tables3 and 4, respectively, in which the best experimental results are highlighted in bold. It can be seen from Tables 3 and 4 that the average classification accuracy of all reduction results is better than the original data. Compared with the average classification accuracy of original data, the average classification accuracy of comparison Algorithms 1–3, and the algorithm proposed in this paper under SVM are improved by 4.3%, 4.5%, 2.1%, and 5.3%, respectively, and the average classification accuracy under kNN is improved by 3.5%, 4.6%, 2.0%, and 5.8%, respectively. For these four algorithms, the proposed algorithm has higher SVM and KNN classification accuracy in most data sets. Therefore, the attribute reduction classification accuracy performance of the proposed algorithm is obviously better than the three comparison algorithms.Table 3
Comparison of SVM classification accuracy of attribute reduction results
Data setsOriginal dataThe comparison Algorithm 1The comparison Algorithm 2The comparison Algorithm 3The algorithm proposed in this paperwine0.9250 ± 0.00270.9493 ± 0.00640.9526 ± 0.00570.9372 ± 0.00270.9657 ± 0.0019sonar0.8066 ± 0.00900.8550 ± 0.01720.8425 ± 0.01450.8237 ± 0.01260.8564 ± 0.0132iono0.8109 ± 0.00430.8432 ± 0.00430.8521 ± 0.00480.8329 ± 0.00620.8570 ± 0.0057wdbc0.9251 ± 0.00220.9572 ± 0.00290.9735 ± 0.00450.9356 ± 0.00180.9697 ± 0.0023biodeg0.8250 ± 0.00270.8383 ± 0.00490.8551 ± 0.00330.8290 ± 0.00270.8478 ± 0.0036messidor0.8458 ± 0.00610.8557 ± 0.00670.8608 ± 0.00580.8593 ± 0.00690.8713 ± 0.0062winequality-red0.7671 ± 0.00740.8129 ± 0.00810.7864 ± 0.00620.7781 ± 0.00580.7932 ± 0.0051winequality-white0.7484 ± 0.00170.8364 ± 0.00770.8242 ± 0.00420.7992 ± 0.00650.8425 ± 0.0047magic0.8266 ± 0.00110.8596 ± 0.00230.8691 ± 0.00310.8431 ± 0.00260.8793 ± 0.0018average0.8311 ± 0.00410.8675 ± 0.00670.8684 ± 0.00570.8486 ± 0.00530.8758 ± 0.0049The best experimental results are highlighted in bold.Table 4
Comparison of kNN classification accuracy of attribute reduction results.
Data setsOriginal dataThe comparison Algorithm 1The comparison Algorithm 2The comparison Algorithm 3The algorithm proposed in this paperwine0.9011 ± 0.01700.9321 ± 0.00870.9493 ± 0.00470.9254 ± 0.00680.9551 ± 0.0053sonar0.8164 ± 0.01910.8437 ± 0.01870.8608 ± 0.01750.8237 ± 0.01630.8498 ± 0.0139iono0.8493 ± 0.00620.8988 ± 0.00680.9183 ± 0.00490.8841 ± 0.00590.9268 ± 0.0048wdbc0.9378 ± 0.00460.9468 ± 0.00620.9560 ± 0.00570.9424 ± 0.00390.9698 ± 0.0031biodeg0.8250 ± 0.00580.8328 ± 0.00460.8273 ± 0.00510.8257 ± 0.00380.8378 ± 0.0040messidor0.8273 ± 0.00440.8562 ± 0.00420.8476 ± 0.00490.8455 ± 0.00570.8693 ± 0.0050winequality-red0.7563 ± 0.00820.8015 ± 0.00740.7962 ± 0.00690.7760 ± 0.00560.7927 ± 0.0053winequality-white0.7598 ± 0.00390.8019 ± 0.00330.8250 ± 0.00450.7828 ± 0.00480.8056 ± 0.0042magic0.8019 ± 0.00220.8278 ± 0.00170.8352 ± 0.00360.8159 ± 0.00330.8434 ± 0.0029average0.8305 ± 0.00790.8601 ± 0.00670.8684 ± 0.00640.8468 ± 0.00620.8790 ± 0.0053The best experimental results are highlighted in bold.Running time is an important index to evaluate the feasibility of attribute reduction algorithm. The average running time of the four algorithms for attribute reduction is shown in Figure3. As can be seen from Figure 3, the running time of the algorithm proposed in this paper is slightly worse than that of the comparison Algorithm 3. The main reason is that the comparison Algorithm 3 uses the fuzzy rough self-information for heuristic search attribute reduction, only needs to calculate the fuzzy similarity class of each object, and does not need to calculate the upper and lower approximation sets of the decision class, thus reducing a part of the amount of calculation. The running time of the algorithm proposed in this paper is obviously better than that of the comparison Algorithm 1 and the comparison, Algorithm 2. The main reason is that the comparison Algorithm 1 needs to perform the operation of attribute interaction in attribute reduction, and the comparison Algorithm 2 needs to perform the operation of attribute weight in attribute reduction, so more calculation will be generated, and the comparison Algorithm 2 needs to perform a large number of matrix operations when calculating attribute weight, so the calculation time of the comparison Algorithm 2 is the largest. Obviously, under the condition of comprehensive calculation efficiency and classification accuracy, the attribute reduction algorithm in this paper is the best choice.Figure 3
Attribute reduction runtime comparison. (a) Partial data sets runtime comparison. (b) Partial data sets runtime comparison.
(a)(b)The length of attribute reduction results is another important index to evaluate the performance of attribute reduction. Table5 shows the average length results of attribute reduction results of all algorithms. Comparing these four algorithms, it can be found that the length of the attribute subset selected by the algorithm proposed in this paper is the smallest in most data sets, and there are slightly larger attribute subsets in data sets iono, messidor, and magic. The synthesis shows that the algorithm proposed in this paper has good attribute reduction effect and obvious advantages in dimension reduction.Table 5
Length comparison of attribute reduction results.
Data setsOriginal dataThe comparison Algorithm 1The comparison Algorithm 2The comparison Algorithm 3The algorithm proposed in this paperwine1310.28.79.38.2sonar6012.713.510.29.4iono349.310.57.98.3wdbc3112.29.510.39.4biodeg4113.012.512.211.2messidor1911.210.89.810.2winequality-red1111.010.611.010.4winequality-white1111.011.010.510.5magic106.78.07.87.2average25.610.810.59.89.4The best experimental results are highlighted in bold.Combining all the experimental results, it can be proved that the attribute reduction algorithm based on kernel similarity combination measurement proposed in this paper can quickly select smaller attribute subsets and obtain higher classification performance.


## 6. Conclusion

Deleting low correlation and redundant attributes in the data sets before classification and regression can improve performance and computational efficiency. For the numerical information systems, distance is used to evaluate the similarity between objects in neighborhood rough sets. In order to further improve the effect of similarity measurement between data objects, a rough set model of kernel similarity relation is proposed in this paper. Then, based on the combination of kernel approximation precision (kernel approximation roughness) and kernel knowledge granularity, an uncertainty measurement method of kernel similarity combination measurement is proposed. Lastly, using the strict monotonicity of kernel similarity combination measurement, an attribute reduction algorithm for numerical information system is proposed. Experiments show its effectiveness and superiority. Due to the dynamic situation of data in the actual environment, we will further study the incremental attribute reduction of kernel similarity combination measurement method.



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*Source: 5675200-2022-02-22.xml*