# Adjustable Fuzzy Rough Reduction: A Nested Strategy

Authors: Ying Shi; Hui Qi; Xiaofang Mu; Mingxing Hou
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

As a crucial extension of Pawlak's rough set, a fuzzy rough set has been successfully applied in real-valued attribute reduction. Nevertheless, the traditional fuzzy rough set is not provided with adjustable ability due to the maximal and minimal operators. It follows that the associated measure for attribute evaluation is not always appropriate. To alleviate such problems, a novel adjustable fuzzy rough set model is presented and further introduced into the parameterized attribute reduction. Additionally, the inner relationship between the appointed parameter and the reduct result is discovered, and thereby a nested mechanism is adopted to accelerate the searching procedure of reduct. Experiments demonstrate that the proposed heuristic algorithm can offer us more stable reducts with higher computational efficiency as compared with the traditional approaches.


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## Body

## 1. Introduction

Rough set theory [1] is an effective mathematical tool to qualitatively and quantitatively describe the uncertainty information in data. Due to such characteristics, it has been frequently applied in attribute reduction [2–9], which aims to select a condition attribute subset that can retain the identifiable ability of the original data. It should be pointed out that among existing attribute reduction methods, fuzzy rough set-based ones [10–15] are widely concerned with handling indiscernibility and fuzziness in data with real-valued condition attributes.Up to now, attribute reduction with the fuzzy rough set has been studied extensively, leading to a large variety of algorithms for computing reducts [16–20]. Actually, the core of these algorithms lies in the fuzzy relation induced from the similarity between every two samples. Generally, a method will be regarded as favorable if it empirically performs well on the attribute evaluation used for identifying the discernibility power of candidate attributes. For example, Hu et al. [21] introduced Gaussian kernels to acquire T-fuzzy equivalence relation and gave a more effective approach for finding reducts; Hu et al. [22] also leveraged k-nearest neighbor rule to develop a robust attribute reduction method; Ye et al. [23] redefined two types of reflexive fuzzy-neighborhood operators to evaluate attributes appropriately. With a careful reviewing of these methods, unfortunately, these technologies are not provided with adjustable ability. Consequently, no matter how the analytical requirements change, the characterizations of fuzzy rough sets are always fixed. Such a case may result in inflexible attribute evaluation such as the commonly used measure, i.e., approximate quality, which can further cause the undesirable generalization performance of the final reduct.To fill such a gap, a novel adjustable parameterized fuzzy rough set model is presented firstly, and then the nested strategy is introduced into the process of searching reducts. Experimental results demonstrate that our proposal can offer us more stable reducts with higher computational efficiency as compared with the traditional approaches. The main contributions of this study are outlined as follows:(1)
An adjustable fuzzy rough set model is constructed. Two special fuzzy relations referred to as strong fuzzy relation and weak fuzzy relation are developed and discussed. Based on these two relations, a parameterized binary operator is reported and then introduced into the construction of the adjustable fuzzy relation, which can be adjusted by the setting of the parameter. Additionally, we have also suggested some properties of such fuzzy relation.(2)
The nested mechanism is adopted to perform the procedure of attribute reduction. Our algorithm consists of two aspects. One is attribute evaluation, on the basis of the adjustable fuzzy rough set model, the corresponding approximate quality is presented to measure whether the candidate attribute is qualified or not; the other is searching strategy, considering the appointed parameter in the designed model, the inner relationship between the parameters and the reducts is discovered, and three types of nested reduction are discussed to find the optimal or near optimal reduct, which is defined as forward nested reduction, reverse nested reduction, and weakly rested reduction, respectively.The rest of this paper is organized as follows. Section2 provides some background materials on the fuzzy rough set model and its attribute reduction. Section 3 introduces the proposed adjustable fuzzy rough set model, and then the nested strategy-based attribution reduction method is presented. Section 4 describes data sets, evaluation metrics, and experimental settings and then analyzes the results of comparative studies on 6 UCI data sets. Finally, Section 5 summarizes and sets up several issues for future work.


## 2. Related Work

### 2.1. Fuzzy Rough Set Model

Formally, a decision system DS can be considered as the 5-tuple <U, C ∪ D, V, f >, in which U = {x1, x2,. . ., xn} is a nonempty finite set of n samples called the universe of discourse; C = {a1, a2,. . ., am} is a nonempty finite set of condition attributes aimed to characterize the samples; D is the decision attributes; ∀a ∈ C ∪ D, Va is the domain of attribute a; f: U × A ⟶ V is an information function.LetU≠ø be a universe of discourse. F: U ⟶ [0, 1] is a fuzzy set [24] on U, F (x) ∈ [0, 1] is the membership function of F. F (U) is the set of all fuzzy sets on U, fuzzy subset R ∈ F (U × U) is a fuzzy relation, and (U, R) is referred to as a fuzzy approximation space. A fuzzy relation R may own the following properties: (1) R is linear if and only if ∀x ∈ U, ∃y ∈ U, R (x, y) = 1 holds; (2) R is reflexive if and only if ∀x ∈ U, R (x, x) = 1 holds; (3) R is symmetric if and only if ∀x, y ∈ U, R (x, y) = R (y, x) holds; (4) R is transitive if and only if ∀x, y, z ∈ U, ∧y (R (x, y), R (y, z)) ≤ R (x, z). The fuzzy relation R discussed in this paper is reflexive at least.Definition 1.
Let <U, C ∪ D, V, f > be a decision system, ∀at ∈ C, the fuzzy relation induced by attribute at is represented in the form of the following:(1)Rt=r11r12...r1nr21r22...r2nrn1rn2...rnn,where n is the number of samples in U, and ri j ∈ [0, 1] is the similarity between xi and xj.
By the fuzzy relation in Definition1, we can have the following:(1)
R1 = R2 ⇔ ∀xi, xj ∈ U, R1 (xi, xj) = R2 (xi, xj) holds.(2)
R = R1 ∪ R2 ⇔ ∀xi, xj ∈ U, R = max{R1 (xi, xj), R2 (xi, xj)} holds.(3)
R = R1 ∩ R2 ⇔ ∀xi, xj ∈ U, R = min{R1 (xi, xj), R2 (xi, xj)} holds.(4)
R1 ⊆ R2 ⇔∀xi, xj ∈ U, R1 (xi, xj) ≤ R2 (xi, xj) holds.Definition 2.
(see [24]). Let U≠φ be a universe of discourse, R is a fuzzy relation on U, ∀F ∈ F (U), the fuzzy lower and upper approximations of F in the fuzzy approximation space (U, R) are denoted by R¯F and R¯F, respectively,∀x ∈ U, the memberships that x belongs to, R¯F and R¯F, are defined as follows:(2)R¯Fx=infy∈Umax1−Rx,y,Fy,R¯Fx=supy∈UminRx,y,Fy,where R (x, y) is the similarity between x and y based on the fuzzy relation R.
The pair [R¯F, R¯F] is referred to as a fuzzy rough set of F.


### 2.2. Attribute Reduction in Fuzzy Rough Set

Attribute reduction is an effective way to eliminate irrelevant and redundant information in a decision system. It aims to achieve the minimal subset of condition attributes, which preserves a specified measure of discernibility power invariant. The common measures of discernibility power contain dependency function [25–27], information entropy [28, 29], and monotonic measure [30]. For simplicity, we take approximate quality as an example in this subsection.Definition 3.
Let <U, C ∪ D, V, f > be a decision system, ∀B ⊆ C, RB is the fuzzy relation generated by the set of attributes B, U|IND (D) = {d1, d2, ... , dp} is the partition induced by the decision D, and then the approximate quality of U|IND (D) with respect to B in the fuzzy rough set is defined as follows:(3)γB,D=Ui=1pRB¯diU=∑j=1Umax1≤i≤pRB¯dixjU,where U is the sample set, B is the attribute set, D is the decision set, RB is the fuzzy relation generated by the set of attributes B, xj is the jth sample combined with U, • is the cardinality of a set, and di is a decision class.
By Definition3, we can see that γ (B, D) reflects the approximation abilities of the granulated spaces generated by the set of attributes B to characterize the decision D. Literatures [21, 31] have proved that approximate quality is monotonic with the increasing or decreasing of condition attributes in a decision system, i.e, γ (B1, D) ≤ γ (B2, D) if B1 ⊆ B2 ⊆ C. In addition, it is obvious that 0 ≤ γ (B, D) ≤ 1 holds.Definition 4.
Let <U, C ∪ D, V, f > be a decision system, ∀B ⊆ C, B is referred to as a reduct of C if and only if(1)
γB,D≠γC,D(2)
∀B′⊂B,γB′,D≠γB,D
In a fuzzy rough set, the similarity of samples will decrease with the number of attributes increasing, and then the lower approximation of the decision system will increase. That is to say, the positive region will be enlarged. It is well known that the samples of the positive region are usually regarded as to be certain. Therefore, the certainty of the decision system will improve. It is consistent with our intuition that new attributes (features) will bring more information about granulation and classification.
Let <U, C ∪ D, V, f > be a decision system, ∀at ∈ B ⊆ C, and we define a coefficient as the significance of at relative to decision D.(4)Siginat,B,D=γB,D−γB−at,D,Siginat,B,D reflects the change of approximate quality if attribute at is eliminated from B. Similarly, we can also define the following:(5)Sigoutat,B,D=γB+at,D−γB,D,where at∈C − B, Sigoutat,B,D reflects the change of approximate quality if attribute at is introduced into B. From the above perspective, many researchers [9, 32–34] iteratively select the most significant attributes with forward greedy algorithm until no more deterministic rules generating with the increasing of attributes.
Formally, a forward attribute reduction algorithm can be designed as follows.
For Algorithm1, if approximate quality γ (·, ·) is replaced by some other measures of discernibility power, then it can also be used to compute reducts with different measurement indices.Algorithm 1: Heuristic approach to compute reduct (HACR).
(i)
Inputs: A decision system DS=<U,C∪D,V,f>(ii)
Outputs: A reduct B.(iii)
Step 1. B←∅, compute γC,D;Step 2.∀at∈C, compute Siginat,C,D=γC,D−γC−at,DStep 3., where Siginat′,C,D=maxSiginat,C,D:∀at∈CStep 4. Compute;γB,D(iv)
Step 5. Do(1)
compute∀at∈C−BSigoutat,B,D=γB+at,D−γB,D(2)
B←B∪at′ where Sigoutat′,B,D=maxSigoutat,B,D:∀at∈C−B(3)
ComputeγB,D;UntilγB,D=γC,DStep 6.∀at∈B, ifγB−at,D=γC,DthenB←B−at


## 2.1. Fuzzy Rough Set Model

Formally, a decision system DS can be considered as the 5-tuple <U, C ∪ D, V, f >, in which U = {x1, x2,. . ., xn} is a nonempty finite set of n samples called the universe of discourse; C = {a1, a2,. . ., am} is a nonempty finite set of condition attributes aimed to characterize the samples; D is the decision attributes; ∀a ∈ C ∪ D, Va is the domain of attribute a; f: U × A ⟶ V is an information function.LetU≠ø be a universe of discourse. F: U ⟶ [0, 1] is a fuzzy set [24] on U, F (x) ∈ [0, 1] is the membership function of F. F (U) is the set of all fuzzy sets on U, fuzzy subset R ∈ F (U × U) is a fuzzy relation, and (U, R) is referred to as a fuzzy approximation space. A fuzzy relation R may own the following properties: (1) R is linear if and only if ∀x ∈ U, ∃y ∈ U, R (x, y) = 1 holds; (2) R is reflexive if and only if ∀x ∈ U, R (x, x) = 1 holds; (3) R is symmetric if and only if ∀x, y ∈ U, R (x, y) = R (y, x) holds; (4) R is transitive if and only if ∀x, y, z ∈ U, ∧y (R (x, y), R (y, z)) ≤ R (x, z). The fuzzy relation R discussed in this paper is reflexive at least.Definition 1.
Let <U, C ∪ D, V, f > be a decision system, ∀at ∈ C, the fuzzy relation induced by attribute at is represented in the form of the following:(1)Rt=r11r12...r1nr21r22...r2nrn1rn2...rnn,where n is the number of samples in U, and ri j ∈ [0, 1] is the similarity between xi and xj.
By the fuzzy relation in Definition1, we can have the following:(1)
R1 = R2 ⇔ ∀xi, xj ∈ U, R1 (xi, xj) = R2 (xi, xj) holds.(2)
R = R1 ∪ R2 ⇔ ∀xi, xj ∈ U, R = max{R1 (xi, xj), R2 (xi, xj)} holds.(3)
R = R1 ∩ R2 ⇔ ∀xi, xj ∈ U, R = min{R1 (xi, xj), R2 (xi, xj)} holds.(4)
R1 ⊆ R2 ⇔∀xi, xj ∈ U, R1 (xi, xj) ≤ R2 (xi, xj) holds.Definition 2.
(see [24]). Let U≠φ be a universe of discourse, R is a fuzzy relation on U, ∀F ∈ F (U), the fuzzy lower and upper approximations of F in the fuzzy approximation space (U, R) are denoted by R¯F and R¯F, respectively,∀x ∈ U, the memberships that x belongs to, R¯F and R¯F, are defined as follows:(2)R¯Fx=infy∈Umax1−Rx,y,Fy,R¯Fx=supy∈UminRx,y,Fy,where R (x, y) is the similarity between x and y based on the fuzzy relation R.
The pair [R¯F, R¯F] is referred to as a fuzzy rough set of F.


## 2.2. Attribute Reduction in Fuzzy Rough Set

Attribute reduction is an effective way to eliminate irrelevant and redundant information in a decision system. It aims to achieve the minimal subset of condition attributes, which preserves a specified measure of discernibility power invariant. The common measures of discernibility power contain dependency function [25–27], information entropy [28, 29], and monotonic measure [30]. For simplicity, we take approximate quality as an example in this subsection.Definition 3.
Let <U, C ∪ D, V, f > be a decision system, ∀B ⊆ C, RB is the fuzzy relation generated by the set of attributes B, U|IND (D) = {d1, d2, ... , dp} is the partition induced by the decision D, and then the approximate quality of U|IND (D) with respect to B in the fuzzy rough set is defined as follows:(3)γB,D=Ui=1pRB¯diU=∑j=1Umax1≤i≤pRB¯dixjU,where U is the sample set, B is the attribute set, D is the decision set, RB is the fuzzy relation generated by the set of attributes B, xj is the jth sample combined with U, • is the cardinality of a set, and di is a decision class.
By Definition3, we can see that γ (B, D) reflects the approximation abilities of the granulated spaces generated by the set of attributes B to characterize the decision D. Literatures [21, 31] have proved that approximate quality is monotonic with the increasing or decreasing of condition attributes in a decision system, i.e, γ (B1, D) ≤ γ (B2, D) if B1 ⊆ B2 ⊆ C. In addition, it is obvious that 0 ≤ γ (B, D) ≤ 1 holds.Definition 4.
Let <U, C ∪ D, V, f > be a decision system, ∀B ⊆ C, B is referred to as a reduct of C if and only if(1)
γB,D≠γC,D(2)
∀B′⊂B,γB′,D≠γB,D
In a fuzzy rough set, the similarity of samples will decrease with the number of attributes increasing, and then the lower approximation of the decision system will increase. That is to say, the positive region will be enlarged. It is well known that the samples of the positive region are usually regarded as to be certain. Therefore, the certainty of the decision system will improve. It is consistent with our intuition that new attributes (features) will bring more information about granulation and classification.
Let <U, C ∪ D, V, f > be a decision system, ∀at ∈ B ⊆ C, and we define a coefficient as the significance of at relative to decision D.(4)Siginat,B,D=γB,D−γB−at,D,Siginat,B,D reflects the change of approximate quality if attribute at is eliminated from B. Similarly, we can also define the following:(5)Sigoutat,B,D=γB+at,D−γB,D,where at∈C − B, Sigoutat,B,D reflects the change of approximate quality if attribute at is introduced into B. From the above perspective, many researchers [9, 32–34] iteratively select the most significant attributes with forward greedy algorithm until no more deterministic rules generating with the increasing of attributes.
Formally, a forward attribute reduction algorithm can be designed as follows.
For Algorithm1, if approximate quality γ (·, ·) is replaced by some other measures of discernibility power, then it can also be used to compute reducts with different measurement indices.Algorithm 1: Heuristic approach to compute reduct (HACR).
(i)
Inputs: A decision system DS=<U,C∪D,V,f>(ii)
Outputs: A reduct B.(iii)
Step 1. B←∅, compute γC,D;Step 2.∀at∈C, compute Siginat,C,D=γC,D−γC−at,DStep 3., where Siginat′,C,D=maxSiginat,C,D:∀at∈CStep 4. Compute;γB,D(iv)
Step 5. Do(1)
compute∀at∈C−BSigoutat,B,D=γB+at,D−γB,D(2)
B←B∪at′ where Sigoutat′,B,D=maxSigoutat,B,D:∀at∈C−B(3)
ComputeγB,D;UntilγB,D=γC,DStep 6.∀at∈B, ifγB−at,D=γC,DthenB←B−at


## 3. Adjustable Fuzzy Rough Set

### 3.1. Strong and Weak Fuzzy Rough Set Model

In <U, C ∪ D, V, f >, ∀at ∈ C (1 ≤ t ≤ m), a series of fuzzy relations Rt can be induced by each attribute at. For all the m condition attributes, we can consider the following two special fuzzy relations.(1)
Strong fuzzy relation:RS=R1∩R2∩⋯∩Rm, RScorresponds to the strong fuzzy relation matrix MRS. In MRS, ∀xi, xj ∈ U, the similarity between xi and xj, denoted by rijs, is minimum.(2)
Weak fuzzy relation:Rw=R1∪R2∪⋯∪Rm, Rw corresponds to the weak fuzzy relation matrix MRw. In MRw, ∀xi, xj ∈ U, the similarity between xi and xj, denoted by rijw, is maximum.Definition 5.
LetU≠ø be a universe of discourse, RS is a strong fuzzy relation on U, ∀F ∈ F (U), the strong fuzzy lower and upper approximations of F in the fuzzy approximation space U,RS are denoted by RS¯F andRS¯F, respectively, ∀x ∈ U, the membership that x belongs to RS¯F and RS¯F are defined as follows:(6)Rs¯Fx=infy∈Umax1−Rsx,y,Fy,Rs¯Fx=supy∈UminRsx,y,Fy,where RSx,y is the similarity between x and y based on the strong fuzzy relation RS.
The pairRs¯F,Rs¯F is referred to as a strong fuzzy rough set of F.Definition 6.
LetU≠ø be a universe of discourse, Rw is a weak fuzzy relation on U, ∀F ∈ F (U), and the weak fuzzy lower and upper approximations of F in the fuzzy approximation space U,Rw are denoted by Rw¯F and Rw¯F, respectively, ∀x ∈ U, the memberships that x belongs to, Rw¯F and Rw¯F, are defined as follows:(7)Rw¯Fx=infy∈Umax1−Rwx,y,Fy,Rw¯Fx=supy∈UminRwx,y,Fy,where Rwx,y is the similarity between x and y based on the weak fuzzy relation Rw.
The pairRw¯F,Rw¯F is referred to as a weak fuzzy rough set of F.


### 3.2. Adjustable Fuzzy Rough Set Model

In a general way, suppose thatM is an index set, ∀t ∈ M, ∀kt ∈ [0, 1], then a parameterized binary operator [17] can be defined as follows:(8)∨t∈Mλkt=maxt∈Mkt:kt:t∈M⊆0,λ,mint∈Mkt:kt:t∈M⊆λ,1,λ:others,where λ ∈ [0, 1]. Especially, let index set M = 2, ∀k1, k2 ∈ [0, 1], then(9)k1∨λk2=maxk1,k2:k1,k2∈0,λ,mink1,k2:k1,k2∈λ,1,λ:others,where λ ∈  0, 1]. Obviously, ∨λ is a generalization of maximal operator (max) and minimal operator (min), ∨t∈Mλkt can be adjusted by setting up the value of λ. Especially, if λ = 0, then ∨0  = min; if λ = 1, then ∨1  = max.Theorem 1.
LetM is an index set, ∀t ∈ M, ∀kt ∈ [0, 1], thenmint∈Mkt≤∨t∈Mλkt≤maxt∈Mkt.Proof.
By formula (11), we can discuss in following three cases:(1)
Ifkt:t∈M⊆0,λ, then mint∈Mkt≤∨t∈Mλkt=maxt∈Mkt holds.(2)
Ifkt:t∈M⊆λ,1, then mint∈Mkt=∨t∈Mλkt≤maxt∈Mkt holds.(3)
Otherwise, it is obvious thatmint∈Mkt≤∨t∈Mλkt=λ≤maxt∈Mkt holds.
In <U, C ∪ D, V, f >, ∀at ∈ C (1 ≤ t ≤ m), a series of fuzzy relations Rt can be induced by each attribute at ∀xi, xj ∈ U, the similarity between xi and xj based on all the m fuzzy relations, denoted by rijλ, can be adjusted by the above binary operator. The fuzzy relation constructed by rijλ is referred to as an adjustable fuzzy relation, and it is denoted by Rλ. Rλ corresponds to the adjustable fuzzy relation matrix M (Rλ).Theorem 2.
Let <U, C ∪ D, V, f > be a decision system. The adjustable fuzzy relation Rλ can be obtained by adjusting the value of λ between strong fuzzy relation Rs and weak fuzzy relation Rw, namely, ∀xi, xj ∈ U, then rijλ=rijs∨λrijw.Proof.
Sincerijs≤rijw, we can discuss in the following three cases:(1)
∀xi, xj ∈ U, If rijs,rijw⊆0,λ, by formula (8), then rijλ=∨1≤t≤mλrij=max1≤t≤mrij=rijw−b±b2−4ac/2a, by formula (9), then rijs∨λrijw=maxrijs,rijw=rijw, accordingly, rijλ=rijs∨λrijw holds.(2)
∀xi, xj ∈ U, If rijs,rijw⊆λ,1, by formula (8), then rijλ=∨1≤t≤mλrij=min1≤t≤mrij=rijs, by formula (9), then rijs∨λrijw=minrijs,rijw=rijs, accordingly, rijλ=rijs∨λrijw holds.(3)
Otherwise, by formula (8), rijλ=∨1≤t≤mλrij=λ, by formula (9), rijs∨λrijw=λ, accordingly, rijλ=rijs∨λrijw holds.Definition 7.
LetU≠ø be a universe of discourse, Rλ is an adjustable fuzzy relation on U, ∀F ∈ F (U), the adjustable fuzzy lower and upper approximations of F in the fuzzy approximation space (U, Rλ) are denoted by Rλ¯F and Rλ¯F, respectively, ∀x ∈ U, the membership that x belongs to Rλ¯F and Rλ¯F are defined as follows:(10)Rλ¯Fx=infy∈Umax1−Rλx,y,Fy,(11)Rλ¯Fx=supy∈UminRλx,y,Fy,where Rλx,y is the similarity between x and y based on the adjustable fuzzy relation Rλ.
The pairRλ¯F,Rλ¯F is referred to as an adjustable fuzzy rough set of F.Theorem 3.
Let <U, C ∪ D, V, f > be a decision system, ∀F ∈ F (U), then(1)
R0¯F=Rs¯F;(2)
R0¯F=Rs¯F;(3)
R1¯F=Rw¯F;(4)
R1¯F=Rw¯F.Proof.
By formula (8), we have∨0=min and ∨1=max.(1)
By formula (10),∀x ∈ U,R0¯Fx=infy∈Umax1−R0x,y,Fy=infy∈Umax1−Rsx,y,Fy=Rs¯Fx, accordingly, R0¯F=Rs¯F holds.(2)
By formula (11), ∀x ∈ U, R0¯Fx=supy∈UminR0x,y,Fy=supy∈UminRsx,y,Fy=Rs¯Fx, accordingly, R0¯F=Rs¯F holds.(3)
By formula (10), ∀x ∈ U, R1¯Fx=infy∈Umax1−R1x,y,Fy=infy∈Umax1−Rwx,y,Fy=Rw¯Fx, accordingly, R1¯F=Rw¯F holds.(4)
By formula (11), ∀x ∈ U, R1¯Fx=supy∈UminR1x,y,Fy=supy∈UminRwx,y,Fy=Rw¯Fx, accordingly, R1¯F=Rw¯F holds.
Theorem3 tells us that both strong and weak fuzzy rough sets are special cases of an adjustable fuzzy rough set, and an adjustable fuzzy rough set is a generalization of strong and weak fuzzy rough sets.Theorem 4.
Let <U, C ∪ D, V, f > be a decision system, ∀F ∈ F (U), then(1)
Rw¯F⊆Rλ¯F⊆Rs¯F;(2)
Rs¯F⊆Rλ¯F⊆Rw¯F.Proof.
By Theorem1, we have mint∈Mkt≤∨t∈Mλkt≤maxt∈Mkt, then RSx,y≤Rλx,y≤Rwx,y.(1)
∀x ∈Rw¯F, Rw¯Fx=infy∈Umax1−Rwx,y,Fy≤infy∈Umax1−Rλx,y,Fy=Rλ¯Fx, accordingly,Rw¯F⊆Rλ¯F holds. Similarly, it is not difficult to prove Rλ¯F⊆Rs¯F holds.(2)
∀x ∈Rs¯F, Rs¯Fx=supy∈UminRsx,y,Fy≤supy∈UminRλx,y,Fy=Rλ¯Fx, accordingly, Rs¯F⊆Rλ¯F holds. Similarly, it is not difficult to prove Rλ¯F⊆Rw¯F holds.
Theorem4 tells us that adjustable fuzzy lower approximation is between weak and strong fuzzy lower approximations, and adjustable fuzzy upper approximation is between strong and weak fuzzy upper approximations.


### 3.3. Approximate Reduction in Adjustable Fuzzy Rough Set

As discussed in Subsection2.2, approximate quality reflects the relevance between condition attributes and decision, and it can be used to measure the significance of a candidate attribute. Similarly, we will give the definition of approximate quality in strong, weak, and adjustable fuzzy rough sets firstly.Definition 8.
Let <U, C ∪ D, V, f > be a decision system, ∀B ⊆ C, RBs,RBw and RBλ are the strong, weak, and adjustable fuzzy relations generated by the set of attributes B, respectively, U|IND (D) = {d1, d2,. . ., dp} is the partition induced by the decision D, then the approximate quality of U|IND (D) with respect to B in the strong, weak, and adjustable fuzzy rough sets are defined as follows, respectively:(12)γsB,D=Ui=1pRBsdiU=∑j=1Umax1≤i≤pRBsdixjU,γwB,D=Ui=1pRBwdiU=∑j=1Umax1≤i≤pRBw¯dixjU,γλB,D=Ui=1pRBλ¯diU=∑j=1Umax1≤i≤pRBλ¯dixjU,where RBs,RBw, and RBλ are the strong, weak, and adjustable fuzzy relations generated by the set of attributes B, respectively, U is the sample set, B is the attribute set, D is the decision set, | · | is the cardinality of a set, and di is a decision class.Theorem 5.
Let <U, C∪D, V, f > be a decision system, ∀B⊆C, then γw (B, D) ≤ γλ (B, D) ≤ γs (B, D).Proof.
By Theorem4, we have RBw¯di⊆RBλ¯di⊆RBs¯di, it is obvious that γwB,D≤γλB,D≤γsB,D holds.
Theorem5 tells us that approximation quality in the adjustable fuzzy rough set is between that in weak and strong fuzzy rough sets.
Algorithm1 can also be applicable for attribute reduction in the adjustable fuzzy rough set, which is similar to the classic fuzzy rough set. Nevertheless, in practice, the criteria of reduction are too harsh in Algorithm 1. To expand the application scope of attribute reduction (dimension reduction, feature selection), some researchers [9, 21, 27, 28, 31, 35] have introduced the threshold ε to control the change of discernibility power for relaxing the criteria of reduction, and they also consider B as a reduct of C satisfying the following conditions: (1) γ (C, D))−γ (B, D) ≤ ε; (2) ∀B′⊂B, γ (C, D))−γ (B′, D) > ε. The above conditions show that ε is aimed at eliminating redundant attributes as much as possible while maintaining the change of approximate quality in a smaller range. In general, ε is recommended to be 0%–10% of the original approximate quality in a decision system.
Formally, a tolerant forward approximate reduction algorithm in the adjustable fuzzy rough set can be designed as follows.


### 3.4. Nested Strategy-Based Reduction in Adjustable Fuzzy Rough Set

Algorithm2 is an enhancement version of Algorithm 1, which introduced a tolerant parameter to improve the efficiency of attribute reduction. However, in real-world applications, we always encounter data with too many objects or attributes. In this circumstance, starting with an empty set for the attribute reduction process may be time consuming and result in algorithm performance degradation. In order to deal with this issue, a nested strategy-based attribute reduction approach is introduced into the adjustable fuzzy rough sets model. In this subsection, some notions of the nested strategy-based reduction, i.e., Forward Nested Reduction, Reverse Nested Reduction, and Weakly Nested Reduction, are firstly presented.Algorithm 2: Tolerant heuristic approach to compute reduct (THACR).
Inputs: A decision systemDS=<U,C∪D,V,f>, the adjustable parameter λ, the threshold ε for controlling the change of approximate quality;Outputs: A reduct B.Step 1. compute γλC,D;Step 2. computeSiginat,C,D=γλC,D−γλC−at,DStep 3. where Siginat′,C,D=maxSiginat,C,D:∀at∈CStep 4. Compute γλB,DStep 5. Do(1)
compute;∀at∈C−BSigoutat,B,D=γλB+at,D−γλB,D(2)
where;B←B∪at′, Where Sigoutat′,B,D=maxSigγλat,B,D:∀at∈C−B(3)
Compute;γλB,DUntilγλC,D−γλB,D≤εStep 6.∀at∈B, ifγλC,D−γλB−at,D≤εthenB←BatDefinition 9.
(Forward Nested Reduction [36]). In a parameterized model, if the reducts on different approximations satisfy ∀Reduct (β), ∃Reduct (α), we have Reduct (α)⊆Reduct (β) which holds; then this type of nested structure is “Forward Nested Reduction.”Definition 10.
(Reverse Nested Reduction [36]). In a parameterized model, if the reducts on different approximations satisfy ∀Reduct (β), ∃Reduct (α), we have Reduct (α)⊇Reduct (β) which holds; then this type of nested structure is “Reverse Nested Reduction.”Definition 11.
(Weakly Nested Reduction [36]). In a parameterized model, if the reducts on different approximations satisfy ∀Reduct (β), ∃P⊇Reduct (β), we have P⊇Reduct (α) which holds; then this type of nested structure is “Forward Weakly Nested Reduction.”
By Definition9–Definition 11, we know that the forward nested structure satisfies that there exists a smaller reduct included in the known larger reduct (the former is a subset of the latter), and the reverse nested structure satisfies that there exists a larger reduct containing the known smaller reduct (the latter is a subset of the former). However, the weakly nested structure is a much weaker relation than the inclusion relation of forward and reverse nested structure, that is, the intersection relation of two reducts.Theorem 6.
Let <U, C∪D, V, f > be a decision system, ∀B⊆C, approximate quality γλB,D is monotonic with B in the adjustable fuzzy rough set, then with the parameter λ increasing, the weakly nested reduction holds.Theorem6 presents a necessary condition for constructing nested attribute reduction. In light of the nested strategy of reduction, an algorithm is designed as follows to search for a different reduct quickly with an existing reduct. The detailed processes are presented in Algorithm 3.The most striking difference between Algorithm3 and Algorithm 2 is that the prereduct is considered for searching new reduct in Algorithm 3. When the parameters are changed, the reduction under the new parameters can be obtained by changing only a few attributes sets of reduction with the original parameters. Since the proposed algorithm NS-THACR does not need to add attributes from an empty attribute set, the operation steps of approximation and approximation quality under repeated calculation are greatly reduced, and the efficiency of reduction execution is improved.Algorithm 3: Nested strategy-based tolerant heuristic approach to compute reduct (NS-THACR).
Inputs: A decision system. DS=<U,Reductλβ∪D,V,f>the existing parameter λβand a new parameter λα, the threshold ε for controlling the change of approximate quality;Outputs: A new reduct B of λα.Step 1.B←ReductλβStep 2. Compute the approximate qualities of γλαB,Dand γλαC,D, respectively;Step 3. IfγλαC,D−γλαB,D>εgo toStep 4Else IfγλαC,D−γλαB,D≤εgo toStep 5EndEndStep 4. Do(1)
∀at∈C−B, compute;Sigoutat,B,D=γλaB+at,D−γλaB,D(2)
where.B←B∪at′;Sigoutat′,B,D=maxSigoutat,B,D:∀at∈C−B;(3)
ComputeγλαB,D;UntilγλC,D−γλB,D≤εStep 5.∀at∈B, ifγλαC,D−γλαB−at,D≤εthenB←B∪at


## 3.1. Strong and Weak Fuzzy Rough Set Model

In <U, C ∪ D, V, f >, ∀at ∈ C (1 ≤ t ≤ m), a series of fuzzy relations Rt can be induced by each attribute at. For all the m condition attributes, we can consider the following two special fuzzy relations.(1)
Strong fuzzy relation:RS=R1∩R2∩⋯∩Rm, RScorresponds to the strong fuzzy relation matrix MRS. In MRS, ∀xi, xj ∈ U, the similarity between xi and xj, denoted by rijs, is minimum.(2)
Weak fuzzy relation:Rw=R1∪R2∪⋯∪Rm, Rw corresponds to the weak fuzzy relation matrix MRw. In MRw, ∀xi, xj ∈ U, the similarity between xi and xj, denoted by rijw, is maximum.Definition 5.
LetU≠ø be a universe of discourse, RS is a strong fuzzy relation on U, ∀F ∈ F (U), the strong fuzzy lower and upper approximations of F in the fuzzy approximation space U,RS are denoted by RS¯F andRS¯F, respectively, ∀x ∈ U, the membership that x belongs to RS¯F and RS¯F are defined as follows:(6)Rs¯Fx=infy∈Umax1−Rsx,y,Fy,Rs¯Fx=supy∈UminRsx,y,Fy,where RSx,y is the similarity between x and y based on the strong fuzzy relation RS.
The pairRs¯F,Rs¯F is referred to as a strong fuzzy rough set of F.Definition 6.
LetU≠ø be a universe of discourse, Rw is a weak fuzzy relation on U, ∀F ∈ F (U), and the weak fuzzy lower and upper approximations of F in the fuzzy approximation space U,Rw are denoted by Rw¯F and Rw¯F, respectively, ∀x ∈ U, the memberships that x belongs to, Rw¯F and Rw¯F, are defined as follows:(7)Rw¯Fx=infy∈Umax1−Rwx,y,Fy,Rw¯Fx=supy∈UminRwx,y,Fy,where Rwx,y is the similarity between x and y based on the weak fuzzy relation Rw.
The pairRw¯F,Rw¯F is referred to as a weak fuzzy rough set of F.


## 3.2. Adjustable Fuzzy Rough Set Model

In a general way, suppose thatM is an index set, ∀t ∈ M, ∀kt ∈ [0, 1], then a parameterized binary operator [17] can be defined as follows:(8)∨t∈Mλkt=maxt∈Mkt:kt:t∈M⊆0,λ,mint∈Mkt:kt:t∈M⊆λ,1,λ:others,where λ ∈ [0, 1]. Especially, let index set M = 2, ∀k1, k2 ∈ [0, 1], then(9)k1∨λk2=maxk1,k2:k1,k2∈0,λ,mink1,k2:k1,k2∈λ,1,λ:others,where λ ∈  0, 1]. Obviously, ∨λ is a generalization of maximal operator (max) and minimal operator (min), ∨t∈Mλkt can be adjusted by setting up the value of λ. Especially, if λ = 0, then ∨0  = min; if λ = 1, then ∨1  = max.Theorem 1.
LetM is an index set, ∀t ∈ M, ∀kt ∈ [0, 1], thenmint∈Mkt≤∨t∈Mλkt≤maxt∈Mkt.Proof.
By formula (11), we can discuss in following three cases:(1)
Ifkt:t∈M⊆0,λ, then mint∈Mkt≤∨t∈Mλkt=maxt∈Mkt holds.(2)
Ifkt:t∈M⊆λ,1, then mint∈Mkt=∨t∈Mλkt≤maxt∈Mkt holds.(3)
Otherwise, it is obvious thatmint∈Mkt≤∨t∈Mλkt=λ≤maxt∈Mkt holds.
In <U, C ∪ D, V, f >, ∀at ∈ C (1 ≤ t ≤ m), a series of fuzzy relations Rt can be induced by each attribute at ∀xi, xj ∈ U, the similarity between xi and xj based on all the m fuzzy relations, denoted by rijλ, can be adjusted by the above binary operator. The fuzzy relation constructed by rijλ is referred to as an adjustable fuzzy relation, and it is denoted by Rλ. Rλ corresponds to the adjustable fuzzy relation matrix M (Rλ).Theorem 2.
Let <U, C ∪ D, V, f > be a decision system. The adjustable fuzzy relation Rλ can be obtained by adjusting the value of λ between strong fuzzy relation Rs and weak fuzzy relation Rw, namely, ∀xi, xj ∈ U, then rijλ=rijs∨λrijw.Proof.
Sincerijs≤rijw, we can discuss in the following three cases:(1)
∀xi, xj ∈ U, If rijs,rijw⊆0,λ, by formula (8), then rijλ=∨1≤t≤mλrij=max1≤t≤mrij=rijw−b±b2−4ac/2a, by formula (9), then rijs∨λrijw=maxrijs,rijw=rijw, accordingly, rijλ=rijs∨λrijw holds.(2)
∀xi, xj ∈ U, If rijs,rijw⊆λ,1, by formula (8), then rijλ=∨1≤t≤mλrij=min1≤t≤mrij=rijs, by formula (9), then rijs∨λrijw=minrijs,rijw=rijs, accordingly, rijλ=rijs∨λrijw holds.(3)
Otherwise, by formula (8), rijλ=∨1≤t≤mλrij=λ, by formula (9), rijs∨λrijw=λ, accordingly, rijλ=rijs∨λrijw holds.Definition 7.
LetU≠ø be a universe of discourse, Rλ is an adjustable fuzzy relation on U, ∀F ∈ F (U), the adjustable fuzzy lower and upper approximations of F in the fuzzy approximation space (U, Rλ) are denoted by Rλ¯F and Rλ¯F, respectively, ∀x ∈ U, the membership that x belongs to Rλ¯F and Rλ¯F are defined as follows:(10)Rλ¯Fx=infy∈Umax1−Rλx,y,Fy,(11)Rλ¯Fx=supy∈UminRλx,y,Fy,where Rλx,y is the similarity between x and y based on the adjustable fuzzy relation Rλ.
The pairRλ¯F,Rλ¯F is referred to as an adjustable fuzzy rough set of F.Theorem 3.
Let <U, C ∪ D, V, f > be a decision system, ∀F ∈ F (U), then(1)
R0¯F=Rs¯F;(2)
R0¯F=Rs¯F;(3)
R1¯F=Rw¯F;(4)
R1¯F=Rw¯F.Proof.
By formula (8), we have∨0=min and ∨1=max.(1)
By formula (10),∀x ∈ U,R0¯Fx=infy∈Umax1−R0x,y,Fy=infy∈Umax1−Rsx,y,Fy=Rs¯Fx, accordingly, R0¯F=Rs¯F holds.(2)
By formula (11), ∀x ∈ U, R0¯Fx=supy∈UminR0x,y,Fy=supy∈UminRsx,y,Fy=Rs¯Fx, accordingly, R0¯F=Rs¯F holds.(3)
By formula (10), ∀x ∈ U, R1¯Fx=infy∈Umax1−R1x,y,Fy=infy∈Umax1−Rwx,y,Fy=Rw¯Fx, accordingly, R1¯F=Rw¯F holds.(4)
By formula (11), ∀x ∈ U, R1¯Fx=supy∈UminR1x,y,Fy=supy∈UminRwx,y,Fy=Rw¯Fx, accordingly, R1¯F=Rw¯F holds.
Theorem3 tells us that both strong and weak fuzzy rough sets are special cases of an adjustable fuzzy rough set, and an adjustable fuzzy rough set is a generalization of strong and weak fuzzy rough sets.Theorem 4.
Let <U, C ∪ D, V, f > be a decision system, ∀F ∈ F (U), then(1)
Rw¯F⊆Rλ¯F⊆Rs¯F;(2)
Rs¯F⊆Rλ¯F⊆Rw¯F.Proof.
By Theorem1, we have mint∈Mkt≤∨t∈Mλkt≤maxt∈Mkt, then RSx,y≤Rλx,y≤Rwx,y.(1)
∀x ∈Rw¯F, Rw¯Fx=infy∈Umax1−Rwx,y,Fy≤infy∈Umax1−Rλx,y,Fy=Rλ¯Fx, accordingly,Rw¯F⊆Rλ¯F holds. Similarly, it is not difficult to prove Rλ¯F⊆Rs¯F holds.(2)
∀x ∈Rs¯F, Rs¯Fx=supy∈UminRsx,y,Fy≤supy∈UminRλx,y,Fy=Rλ¯Fx, accordingly, Rs¯F⊆Rλ¯F holds. Similarly, it is not difficult to prove Rλ¯F⊆Rw¯F holds.
Theorem4 tells us that adjustable fuzzy lower approximation is between weak and strong fuzzy lower approximations, and adjustable fuzzy upper approximation is between strong and weak fuzzy upper approximations.


## 3.3. Approximate Reduction in Adjustable Fuzzy Rough Set

As discussed in Subsection2.2, approximate quality reflects the relevance between condition attributes and decision, and it can be used to measure the significance of a candidate attribute. Similarly, we will give the definition of approximate quality in strong, weak, and adjustable fuzzy rough sets firstly.Definition 8.
Let <U, C ∪ D, V, f > be a decision system, ∀B ⊆ C, RBs,RBw and RBλ are the strong, weak, and adjustable fuzzy relations generated by the set of attributes B, respectively, U|IND (D) = {d1, d2,. . ., dp} is the partition induced by the decision D, then the approximate quality of U|IND (D) with respect to B in the strong, weak, and adjustable fuzzy rough sets are defined as follows, respectively:(12)γsB,D=Ui=1pRBsdiU=∑j=1Umax1≤i≤pRBsdixjU,γwB,D=Ui=1pRBwdiU=∑j=1Umax1≤i≤pRBw¯dixjU,γλB,D=Ui=1pRBλ¯diU=∑j=1Umax1≤i≤pRBλ¯dixjU,where RBs,RBw, and RBλ are the strong, weak, and adjustable fuzzy relations generated by the set of attributes B, respectively, U is the sample set, B is the attribute set, D is the decision set, | · | is the cardinality of a set, and di is a decision class.Theorem 5.
Let <U, C∪D, V, f > be a decision system, ∀B⊆C, then γw (B, D) ≤ γλ (B, D) ≤ γs (B, D).Proof.
By Theorem4, we have RBw¯di⊆RBλ¯di⊆RBs¯di, it is obvious that γwB,D≤γλB,D≤γsB,D holds.
Theorem5 tells us that approximation quality in the adjustable fuzzy rough set is between that in weak and strong fuzzy rough sets.
Algorithm1 can also be applicable for attribute reduction in the adjustable fuzzy rough set, which is similar to the classic fuzzy rough set. Nevertheless, in practice, the criteria of reduction are too harsh in Algorithm 1. To expand the application scope of attribute reduction (dimension reduction, feature selection), some researchers [9, 21, 27, 28, 31, 35] have introduced the threshold ε to control the change of discernibility power for relaxing the criteria of reduction, and they also consider B as a reduct of C satisfying the following conditions: (1) γ (C, D))−γ (B, D) ≤ ε; (2) ∀B′⊂B, γ (C, D))−γ (B′, D) > ε. The above conditions show that ε is aimed at eliminating redundant attributes as much as possible while maintaining the change of approximate quality in a smaller range. In general, ε is recommended to be 0%–10% of the original approximate quality in a decision system.
Formally, a tolerant forward approximate reduction algorithm in the adjustable fuzzy rough set can be designed as follows.


## 3.4. Nested Strategy-Based Reduction in Adjustable Fuzzy Rough Set

Algorithm2 is an enhancement version of Algorithm 1, which introduced a tolerant parameter to improve the efficiency of attribute reduction. However, in real-world applications, we always encounter data with too many objects or attributes. In this circumstance, starting with an empty set for the attribute reduction process may be time consuming and result in algorithm performance degradation. In order to deal with this issue, a nested strategy-based attribute reduction approach is introduced into the adjustable fuzzy rough sets model. In this subsection, some notions of the nested strategy-based reduction, i.e., Forward Nested Reduction, Reverse Nested Reduction, and Weakly Nested Reduction, are firstly presented.Algorithm 2: Tolerant heuristic approach to compute reduct (THACR).
Inputs: A decision systemDS=<U,C∪D,V,f>, the adjustable parameter λ, the threshold ε for controlling the change of approximate quality;Outputs: A reduct B.Step 1. compute γλC,D;Step 2. computeSiginat,C,D=γλC,D−γλC−at,DStep 3. where Siginat′,C,D=maxSiginat,C,D:∀at∈CStep 4. Compute γλB,DStep 5. Do(1)
compute;∀at∈C−BSigoutat,B,D=γλB+at,D−γλB,D(2)
where;B←B∪at′, Where Sigoutat′,B,D=maxSigγλat,B,D:∀at∈C−B(3)
Compute;γλB,DUntilγλC,D−γλB,D≤εStep 6.∀at∈B, ifγλC,D−γλB−at,D≤εthenB←BatDefinition 9.
(Forward Nested Reduction [36]). In a parameterized model, if the reducts on different approximations satisfy ∀Reduct (β), ∃Reduct (α), we have Reduct (α)⊆Reduct (β) which holds; then this type of nested structure is “Forward Nested Reduction.”Definition 10.
(Reverse Nested Reduction [36]). In a parameterized model, if the reducts on different approximations satisfy ∀Reduct (β), ∃Reduct (α), we have Reduct (α)⊇Reduct (β) which holds; then this type of nested structure is “Reverse Nested Reduction.”Definition 11.
(Weakly Nested Reduction [36]). In a parameterized model, if the reducts on different approximations satisfy ∀Reduct (β), ∃P⊇Reduct (β), we have P⊇Reduct (α) which holds; then this type of nested structure is “Forward Weakly Nested Reduction.”
By Definition9–Definition 11, we know that the forward nested structure satisfies that there exists a smaller reduct included in the known larger reduct (the former is a subset of the latter), and the reverse nested structure satisfies that there exists a larger reduct containing the known smaller reduct (the latter is a subset of the former). However, the weakly nested structure is a much weaker relation than the inclusion relation of forward and reverse nested structure, that is, the intersection relation of two reducts.Theorem 6.
Let <U, C∪D, V, f > be a decision system, ∀B⊆C, approximate quality γλB,D is monotonic with B in the adjustable fuzzy rough set, then with the parameter λ increasing, the weakly nested reduction holds.Theorem6 presents a necessary condition for constructing nested attribute reduction. In light of the nested strategy of reduction, an algorithm is designed as follows to search for a different reduct quickly with an existing reduct. The detailed processes are presented in Algorithm 3.The most striking difference between Algorithm3 and Algorithm 2 is that the prereduct is considered for searching new reduct in Algorithm 3. When the parameters are changed, the reduction under the new parameters can be obtained by changing only a few attributes sets of reduction with the original parameters. Since the proposed algorithm NS-THACR does not need to add attributes from an empty attribute set, the operation steps of approximation and approximation quality under repeated calculation are greatly reduced, and the efficiency of reduction execution is improved.Algorithm 3: Nested strategy-based tolerant heuristic approach to compute reduct (NS-THACR).
Inputs: A decision system. DS=<U,Reductλβ∪D,V,f>the existing parameter λβand a new parameter λα, the threshold ε for controlling the change of approximate quality;Outputs: A new reduct B of λα.Step 1.B←ReductλβStep 2. Compute the approximate qualities of γλαB,Dand γλαC,D, respectively;Step 3. IfγλαC,D−γλαB,D>εgo toStep 4Else IfγλαC,D−γλαB,D≤εgo toStep 5EndEndStep 4. Do(1)
∀at∈C−B, compute;Sigoutat,B,D=γλaB+at,D−γλaB,D(2)
where.B←B∪at′;Sigoutat′,B,D=maxSigoutat,B,D:∀at∈C−B;(3)
ComputeγλαB,D;UntilγλC,D−γλB,D≤εStep 5.∀at∈B, ifγλαC,D−γλαB−at,D≤εthenB←B∪at


## 4. Experimental Analysis

To evaluate the performance of our nested strategy-based heuristic attribute reduction approach, 6 real-world UCI data sets have been employed in this paper. Table1 summarizes some details of these experimental data sets. All the experiments are run on a workstation equipped with a 3.10 GHz processor and a 8.00 G memory. The programming language is Matlab R2014b.Table 1
The characteristics of experimental data sets from UCI.
IDData setsAbbr.SamplesAttributesAbbr.Classes1DiabetesDiab7688Real value22E. coliE. coli3367Real value83GlassGlass21410Real value64IrisIris1504Real value35SeedsSeeds2107Real value36WineWine17813Real value3In the context of this paper, 10-fold cross-validation (10-CV) is used for evaluating the effectiveness of the proposed algorithm in our experiments. In the following experiments, all samples are broken into ten groups of the same size, the nine groups compose the training set, and the one group composes the testing set. The attribute reduction process is repeated 10 times in turn and the mean value and standard deviation of 10 experimental results are recorded.In this section, we focus on the elapsed time comparison for computing reduct by using the nested reduction strategy or not. The comparison in this part is set up as follows: (1) Ten values of the parameterλ are selected on the interval [0, 1] with the step length 0.1. (2) Seven values of ε are chosen to control the changes of approximate quality, i.e., 0%, 2%, 4%, 6%, 8%, 10%. In fact, ε = 0% means that the approximate quality of obtained reduct is equal to the original decision system and similar explanations can be achieved for another six values. (3) In order to show the effectiveness of the nested reduction strategy, THACR discussed in Algorithm 2 is presented as a benchmark algorithm. The detailed results are shown in the following Tables 2 –7. In these six tables, the first row is the elapsed time of computing THACR and the second row is the elapsed time of computing NS-THACR and the minimum elapsed time is bold. With a careful study of these tables, we can obtain the following remarks.(1)
From the results in Tables2–7, it can be seen that for all experiments, the elapsed time of computing NS-THACR is better than the elapsed time of computing THACR, and the calculation time is reduced by more than 80 percent. In addition, the difference between ε and λ also has a certain impact on the experimental results, and the time consumed by the two methods is also different. The choice of these two parameters also requires different attempts in the experiment to search for an appropriate subset of attributes.(2)
The elapsed time of computing reduct by THACR is higher than that of computing reduct by NS-THACR. For example, when we search for a reduct for datasetE. coli (Table 3), if we set λ = 0.3 and ε = 8%, then we can note from Table 3 that the average elapsed time of THACR is 2.5724, whereas the average elapsed time of NS-THACR is 0.2282, the computing time is decreased by 91.12 percent. For dataset Seeds (Table 6), in the attribute reduction process, if we choose λ = 0.7 and ε = 2%, then we can obtain the average elapsed time of THACR, which is 0.7392 and the average elapsed time of NS-THACR, which is 0.1376, and the computing time, which is decreased by 81.38 percent.(3)
In general, the elapsed time ofε = 0 is higher than others in these two algorithms. For instance, in dataset Glass (Table 4), if we set λ = 0.8, we can note that the average elapsed time of THACR is 2.0171 when ε = 0%, whereas the average elapsed time of THACR is 1.7782 when ε = 6%. As far as NS-THACR is concerned, the average elapsed time is 0.2985 when ε = 0%, whereas the average elapsed time is 0.2032 when ε = 6%. It is because that ε = 0% is too strict; that is, the approximate quality of reduct must be equal to the original dataset. Therefore, more time is needed to search for an appropriate attribute subset.Table 2
Computational time of attribute reduction with using nested reduction or not (mean ± std. deviation) on diabetes.
Diabetesλ=0λ=0.1λ=0.2λ=0.3λ=0.4λ=0.5λ=0.6λ=0.7λ=0.8λ=0.9λ=1.0ε=10%4.5692 ± 0.17034.7531 ± 0.16214.3546 ± 0.14784.5263 ± 0.18614.4215 ± 0.18784.1463 ± 0.15275.1201 ± 0.16385.1879 ± 0.10244.9872 ± 0.16324.7832 ± 0.18434.6547 ± 0.16451.2313 ± 0.07031.0216 ± 0.08120.9841 ± 0.10310.8791 ± 0.11310.9941 ± 0.08610.9833 ± 0.08780.9167 ± 0.05271.1023 ± 0.09781.0578 ± 0.10210.9221 ± 0.06320.9489 ± 0.0843ε=8%4.0167 ± 0.15274.2361 ± 0.12265.0167 ± 0.09895.1668 ± 0.10325.0333 ± 0.07034.9354 ± 0.07034.7656 ± 0.08614.8972 ± 0.10984.9951 ± 0.09755.1021 ± 0.06324.9941 ± 0.10781.0167 ± 0.04371.1021 ± 0.05470.9167 ± 0.08420.9915 ± 0.03210.9386 ± 0.07030.9386 ± 0.07030.9667 ± 0.08610.9988 ± 0.07650.9745 ± 0.08230.9288 ± 0.06320.9476 ± 0.0843ε=6%4.0310 ± 0.06554.1278 ± 0.07424.6167 ± 0.05275.0132 ± 0.07145.0021 ± 0.12045.0167 ± 0.06984.9120 ± 0.06325.0667 ± 0.08614.9756 ± 0.06715.0101 ± 0.07524.9982 ± 0.00121.0310 ± 0.06551.1573 ± 0.04620.9168 ± 0.05270.9962 ± 0.03450.9987 ± 0.00121.0167 ± 0.05830.9998 ± 0.06411.0667 ± 0.08611.1271 ± 0.08750.9942 ± 0.05461.0400 ± 0.0343ε=4%4.9286 ± 0.06024.8352 ± 0.08755.0286 ± 0.06025.1021 ± 0.06904.9875 ± 0.08794.8562 ± 0.02145.0167 ± 0.05275.0214 ± 0.06425.0512 ± 0.08055.0126 ± 0.64784.9975 ± 0.54231.0286 ± 0.06021.0542 ± 0.03460.9286 ± 0.04020.9578 ± 0.03210.9745 ± 0.05781.1024 ± 0.02451.0578 ± 0.04320.9167 ± 0.04780.9576 ± 0.04050.9432 ± 0.04890.9863 ± 0.0543ε=2%5.0143 ± 0.08525.0278 ± 0.04524.9877 ± 0.06524.9963 ± 0.07425.2342 ± 0.08794.8956 ± 0.06025.3475 ± 0.89455.1478 ± 0.09785.1245 ± 0.10245.3214 ± 0.13455.0245 ± 0.11781.0143 ± 0.04520.9945 ± 0.03731.0345 ± 0.08781.1135 ± 0.06421.2012 ± 0.06121.0286 ± 0.06021.0478 ± 0.05411.0781 ± 0.03450.9977 ± 0.05360.9876 ± 0.05470.9496 ± 0.0843ε=0%5.5679 ± 0.13525.3214 ± 0.11475.7836 ± 0.14525.3245 ± 0.06985.4312 ± 0.16785.3369 ± 0.12695.7548 ± 0.16895.8963 ± 0.13525.6312 ± 0.09735.3475 ± 0.10425.4875 ± 0.16821.1204 ± 0.04521.2354 ± 0.09871.2245 ± 0.08751.3012 ± 0.10201.1546 ± 0.08341.1136 ± 0.85421.1345 ± 0.63211.9989 ± 0.10401.1032 ± 0.08791.1346 ± 0.46781.2320 ± 0.0843Table 3
Computational time of attribute reduction with using nested reduction or not (mean ± std. deviation) onE. coli.
E. coliλ=0λ=0.1λ=0.2λ=0.3λ=0.4λ=0.5λ=0.6λ=0.7λ=0.8λ=0.9λ=1.0ε=10%2.3705 ± 0.08482.4221 ± 0.11132.4765 ± 0.11182.4578 ± 0.13682.4517 ± 0.14382.3782 ± 0.09132.432 ± 0.05983.0164 ± 0.48283.0452 ± 0.22722.7472 ± 0.22781.2564 ± 0.15850.1637 ± 0.01590.1713 ± 0.01040.169 ± 0.00430.1731 ± 0.00460.1724 ± 0.00480.1782 ± 0.00820.1762 ± 0.00510.1829 ± 0.00580.1814 ± 0.00350.1886 ± 0.00950.1597 ± 0.0060ε=8%2.443 ± 0.08962.5058 ± 0.09882.543 ± 0.08582.5724 ± 0.09072.5684 ± 0.11732.593 ± 0.12202.498 ± 0.13892.4141 ± 0.01432.4395 ± 0.01332.4392 ± 0.13480.9964 ± 0.00680.211 ± 0.08600.2167 ± 0.09750.1927 ± 0.08410.228 ± 0.09840.2276 ± 0.11360.299 ± 0.15890.296 ± 0.16060.1948 ± 0.01400.1876 ± 0.01030.1849 ± 0.00520.1591 ± 0.0087ε=6%2.431 ± 0.09962.6975 ± 0.26722.7796 ± 0.29263.0122 ± 0.11572.9942 ± 0.10713.028 ± 0.14033.082 ± 0.12363.0489 ± 0.15612.8670 ± 0.02872.9031 ± 0.15081.1553 ± 0.01270.1845 ± 0.05870.2175 ± 0.06470.1659 ± 0.00280.199 ± 0.00630.1971 ± 0.09200.2577 ± 0.11740.3921 ± 0.16590.4266 ± 0.20730.2089 ± 0.00460.2312 ± 0.12430.1552 ± 0.0065ε=4%3.022 ± 0.15993.1649 ± 0.10873.200 ± 0.09393.190 ± 0.11813.0848 ± 0.03733.104 ± 0.03022.6788 ± 0.15223.1241 ± 0.10873.0336 ± 0.18522.8982 ± 0.07211.1558 ± 0.01350.3286 ± 0.08750.3915 ± 0.14910.330 ± 0.06530.3973 ± 0.14980.1956 ± 0.08410.233 ± 0.09850.2022 ± 0.00600.3091 ± 0.16590.4717 ± 0.19230.1871 ± 0.00410.1558 ± 0.0071ε=2%3.1095 ± 0.17663.1861 ± 0.09433.2045 ± 0.07333.249 ± 0.05633.2665 ± 0.08613.291 ± 0.10912.9392 ± 0.23892.7398 ± 0.13513.1648 ± 0.17923.2586 ± 0.04961.1197 ± 0.09110.2062 ± 0.06040.2202 ± 0.06160.189 ± 0.05280.240 ± 0.08310.3248 ± 0.05560.3705 ± 0.08710.235 ± 0.10220.2411 ± 0.10570.3140 ± 0.16790.4970 ± 0.01040.1592 ± 0.0094ε=0%3.9261 ± 0.26883.2094 ± 0.07443.381 ± 0.08383.3222 ± 0.08793.3427 ± 0.09213.356 ± 0.08293.303 ± 0.09183.3516 ± 0.09403.3662 ± 0.07753.3492 ± 0.07731.1840 ± 0.01370.3877 ± 0.00520.4030 ± 0.00520.3799 ± 0.00750.401 ± 0.00580.4742 ± 0.00400.4304 ± 0.06730.395 ± 0.06880.5046 ± 0.06930.5085 ± 0.07040.5968 ± 0.01870.1773 ± 0.0088Table 4
Computational time of attribute reduction with using nested reduction or not (mean ± std. deviation) on Glass.
Glassλ=0λ=0.1λ=0.2λ=0.3λ=0.4λ=0.5λ=0.6λ=0.7λ=0.8λ=0.9λ=1.0ε=10%1.5656 ± 0.07431.6236 ± 0.08341.6282 ± 0.09051.6455 ± 0.09231.6063 ± 0.08161.653 ± 0.09081.606 ± 0.05801.6047 ± 0.01281.6440 ± 0.02831.5314 ± 0.08241.2579 ± 0.16930.0657 ± 0.00650.0674 ± 0.00730.0703 ± 0.00810.0749 ± 0.00680.0720 ± 0.00800.0749 ± 0.00650.0766 ± 0.00480.0781 ± 0.00030.0812 ± 0.00670.0796 ± 0.00510.1938 ± 0.0974ε=8%1.601 ± 0.03851.6610 ± 0.05761.6766 ± 0.05791.6907 ± 0.05801.6845 ± 0.07391.712 ± 0.04901.7376 ± 0.05931.6360 ± 0.07691.6472 ± 0.07781.6390 ± 0.09221.2595 ± 0.16780.1125 ± 0.07350.1219 ± 0.07810.118 ± 0.08000.1250 ± 0.07540.1156 ± 0.07500.151 ± 0.09090.125 ± 0.08320.0954 ± 0.05430.1313 ± 0.08950.1719 ± 0.09900.0719 ± 0.0079ε=6%1.613 ± 0.00661.6813 ± 0.01681.6923 ± 0.01281.7032 ± 0.01291.7189 ± 0.01051.7423 ± 0.01141.7501 ± 0.01071.7751 ± 0.01841.7782 ± 0.04541.7112 ± 0.04211.2484 ± 0.16530.081 ± 0.04880.0922 ± 0.04540.0860 ± 0.05250.084 ± 0.04800.0906 ± 0.05080.1109 ± 0.06920.1095 ± 0.07010.2142 ± 0.07170.2032 ± 0.08310.1610 ± 0.09770.0703 ± 0.0081ε=4%1.601 ± 0.00821.6704 ± 0.01151.684 ± 0.01221.7110 ± 0.03721.7125 ± 0.00821.734 ± 0.01031.7486 ± 0.01681.7657 ± 0.01461.7689 ± 0.04291.7188 ± 0.06551.2454 ± 0.16380.064 ± 0.00490.0704 ± 0.00800.071 ± 0.01120.0720 ± 0.00800.0844 ± 0.03680.0782 ± 0.00040.0782 ± 0.00040.0766 ± 0.00510.0830 ± 0.00760.0860 ± 0.00800.0719 ± 0.0081ε=2%1.6767 ± 0.05901.7048 ± 0.04971.7032 ± 0.03531.736 ± 0.04551.7438 ± 0.03221.7408 ± 0.01971.7610 ± 0.01801.7783 ± 0.01611.7954 ± 0.02581.7235 ± 0.07141.2469 ± 0.16470.128 ± 0.05650.0904 ± 0.05160.084 ± 0.03690.084 ± 0.03690.0845 ± 0.03700.0782 ± 0.00040.0765 ± 0.00470.0780 ± 0.00000.0812 ± 0.00670.0845 ± 0.00800.0686 ± 0.0083ε=0%1.797 ± 0.01461.8814 ± 0.01861.886 ± 0.01491.9142 ± 0.02001.9281 ± 0.01701.946 ± 0.01531.9672 ± 0.01721.9892 ± 0.01292.0171 ± 0.02812.1236 ± 0.03261.3546 ± 0.16540.1780 ± 0.05210.2483 ± 0.06000.253 ± 0.03410.2578 ± 0.03600.2469 ± 0.04790.2797 ± 0.00520.281 ± 0.00040.2874 ± 0.00830.2985 ± 0.01140.3328 ± 0.06160.2718 ± 0.0082Table 5
Computational time of attribute reduction with using nested reduction or not (mean ± std. deviation) on Iris.
Irisλ=0λ=0.1λ=0.2λ=0.3λ=0.4λ=0.5λ=0.6λ=0.7λ=0.8λ=0.9λ=1.0ε=10%0.1266 ± 0.00510.1282 ± 0.01450.100 ± 0.00800.095 ± 0.01860.0516 ± 0.00760.112 ± 0.04220.1423 ± 0.00920.1297 ± 0.01970.1016 ± 0.00820.1219 ± 0.02200.0969 ± 0.00640.021 ± 0.00840.0265 ± 0.00770.0234 ± 0.00820.0250 ± 0.00820.0328 ± 0.02160.022 ± 0.04310.051 ± 0.04830.0266 ± 0.00760.0250 ± 0.00800.0268 ± 0.01780.0375 ± 0.0198ε=8%0.1266 ± 0.00510.1344 ± 0.00810.0968 ± 0.00670.092 ± 0.01710.0532 ± 0.00820.139 ± 0.00500.134 ± 0.00810.1359 ± 0.01470.1063 ± 0.01630.1250 ± 0.02190.0969 ± 0.00670.023 ± 0.00830.0266 ± 0.00760.0250 ± 0.00820.023 ± 0.00800.0391 ± 0.02110.049 ± 0.03800.0251 ± 0.00780.0266 ± 0.00730.0251 ± 0.00810.0281 ± 0.01450.0250 ± 0.0082ε=6%0.1250 ± 0.00000.1360 ± 0.00760.123 ± 0.02140.101 ± 0.01490.0579 ± 0.02450.139 ± 0.00500.139 ± 0.00460.1360 ± 0.00760.1156 ± 0.02100.1328 ± 0.01540.0938 ± 0.00040.018 ± 0.00640.0250 ± 0.00820.023 ± 0.00810.0235 ± 0.00790.0438 ± 0.02930.0220 ± 0.00820.0266 ± 0.00760.0266 ± 0.00760.0266 ± 0.00760.0360 ± 0.01990.0234 ± 0.0084ε=4%0.1407 ± 0.00730.1563 ± 0.01280.148 ± 0.03240.1078 ± 0.02000.1406 ± 0.04360.139 ± 0.00490.1375 ± 0.00660.1375 ± 0.00660.1360 ± 0.02090.1391 ± 0.00880.0938 ± 0.00040.0275 ± 0.01100.0500 ± 0.01000.0278 ± 0.01930.0250 ± 0.00770.0531 ± 0.02450.0297 ± 0.00880.0265 ± 0.00750.0281 ± 0.00670.0281 ± 0.00990.0328 ± 0.02270.0234 ± 0.0087ε=2%0.1454 ± 0.01050.1595 ± 0.00660.1626 ± 0.00790.1173 ± 0.03470.1422 ± 0.04670.1562 ± 0.01470.1376 ± 0.01440.1375 ± 0.01440.1562 ± 0.01960.1390 ± 0.00880.0954 ± 0.00510.0266 ± 0.01070.0297 ± 0.01390.0391 ± 0.03070.036 ± 0.03130.0312 ± 0.00730.0483 ± 0.01370.0266 ± 0.01050.0281 ± 0.01450.0316 ± 0.01960.0250 ± 0.00800.0234 ± 0.0080ε=0%0.1469 ± 0.01070.1625 ± 0.00800.1725 ± 0.01380.1594 ± 0.00670.1641 ± 0.00810.164 ± 0.00830.1610 ± 0.00760.1610 ± 0.00710.1626 ± 0.00810.1476 ± 0.00990.0989 ± 0.00640.035 ± 0.00770.0381 ± 0.00690.0351 ± 0.00780.0766 ± 0.02890.0581 ± 0.00670.051 ± 0.01030.056 ± 0.01330.0531 ± 0.01530.0543 ± 0.02320.0350 ± 0.00820.0465 ± 0.0079Table 6
Computational time of attribute reduction with using nested reduction or not (mean ± std. deviation) on Seeds.
Seedsλ=0λ=0.1λ=0.2λ=0.3λ=0.4λ=0.5λ=0.6λ=0.7λ=0.8λ=0.9λ=1.0ε=10%0.4375 ± 0.01030.5250 ± 0.03970.554 ± 0.03250.534 ± 0.04340.5547 ± 0.04040.5375 ± 0.01080.5594 ± 0.02170.5720 ± 0.02570.5766 ± 0.06690.4609 ± 0.02210.6030 ± 0.02900.039 ± 0.00810.0469 ± 0.00030.046 ± 0.00040.0470 ± 0.00000.0485 ± 0.00470.048 ± 0.00480.0517 ± 0.00760.0500 ± 0.00660.0562 ± 0.00790.0530 ± 0.00800.0475 ± 0.0249ε=8%0.4328 ± 0.00750.4938 ± 0.00800.5032 ± 0.00670.510 ± 0.01050.5297 ± 0.00480.532 ± 0.00500.5422 ± 0.00730.5531 ± 0.00810.5594 ± 0.00980.4266 ± 0.00760.5938 ± 0.03040.039 ± 0.00840.0468 ± 0.00040.046 ± 0.00040.0452 ± 0.00500.0484 ± 0.00510.0516 ± 0.00740.0499 ± 0.00670.0531 ± 0.00790.0531 ± 0.00810.0547 ± 0.00810.0470 ± 0.0231ε=6%0.5063 ± 0.02490.5782 ± 0.02820.5923 ± 0.03330.6047 ± 0.03760.6171 ± 0.03640.623 ± 0.03310.653 ± 0.04800.5470 ± 0.00000.5547 ± 0.00810.4234 ± 0.00480.5922 ± 0.02900.040 ± 0.03000.0565 ± 0.03420.035 ± 0.03240.0377 ± 0.03280.0380 ± 0.03930.043 ± 0.03440.0516 ± 0.03750.0515 ± 0.00770.0532 ± 0.00800.0500 ± 0.00660.0422 ± 0.0077ε=4%0.5188 ± 0.02300.5969 ± 0.02740.6187 ± 0.02880.6204 ± 0.00740.6297 ± 0.00760.6437 ± 0.00650.6469 ± 0.00780.6579 ± 0.00500.5517 ± 0.01050.5001 ± 0.06410.5906 ± 0.03210.0515 ± 0.02770.0657 ± 0.04040.0734 ± 0.03980.0579 ± 0.03480.0719 ± 0.04500.060 ± 0.03400.059 ± 0.03430.1656 ± 0.00760.0515 ± 0.00750.0485 ± 0.00470.0421 ± 0.0077ε=2%0.5813 ± 0.02730.6843 ± 0.05290.7015 ± 0.03790.6859 ± 0.02780.7078 ± 0.00770.7219 ± 0.00670.735 ± 0.00500.7392 ± 0.03010.6672 ± 0.00770.5516 ± 0.00760.6172 ± 0.02110.089 ± 0.02320.1093 ± 0.03450.1047 ± 0.03700.1203 ± 0.02650.1249 ± 0.02840.134 ± 0.00810.142 ± 0.00490.1376 ± 0.02760.1703 ± 0.00890.0765 ± 0.05080.0640 ± 0.0228ε=0%0.606 ± 0.00990.7142 ± 0.01470.7375 ± 0.00660.753 ± 0.00650.7672 ± 0.00520.7859 ± 0.00770.7985 ± 0.00470.8127 ± 0.01050.8189 ± 0.00780.8219 ± 0.00830.6423 ± 0.00480.0892 ± 0.02100.1062 ± 0.02420.1094 ± 0.02870.1287 ± 0.02450.1341 ± 0.02640.149 ± 0.00050.158 ± 0.00800.1450 ± 0.02850.2094 ± 0.00830.3186 ± 0.00830.0709 ± 0.0260Table 7
Computational time of attribute reduction with using nested reduction or not (mean ± std. deviation) on Wine.
Irisλ=0λ=0.1λ=0.2λ=0.3λ=0.4λ=0.5λ=0.6λ=0.7λ=0.8λ=0.9λ=1.0ε=10%1.4704 ± 0.06771.5689 ± 0.06981.6157 ± 0.07161.6173 ± 0.06491.6406 ± 0.02081.5126 ± 0.09871.383 ± 0.11151.3283 ± 0.09661.1422 ± 0.10120.8843 ± 0.12651.0656 ± 0.08150.0376 ± 0.00770.0422 ± 0.00750.0407 ± 0.00790.045 ± 0.00500.0470 ± 0.00000.050 ± 0.00660.048 ± 0.00480.0501 ± 0.00680.0515 ± 0.00770.0484 ± 0.00510.0515 ± 0.1160ε=8%1.484 ± 0.01631.6109 ± 0.02011.6783 ± 0.02571.6907 ± 0.05791.6439 ± 0.07081.604 ± 0.08891.459 ± 0.09961.3548 ± 0.01291.1547 ± 0.07160.9766 ± 0.01081.0579 ± 0.08500.0876 ± 0.06620.0984 ± 0.07730.1094 ± 0.07760.0781 ± 0.06220.0781 ± 0.06580.0485 ± 0.00470.048 ± 0.00480.0501 ± 0.00650.0485 ± 0.00510.0515 ± 0.00750.0422 ± 0.0073ε=6%1.5720 ± 0.03221.6969 ± 0.04961.775 ± 0.05391.779 ± 0.06511.7766 ± 0.06001.653 ± 0.08961.512 ± 0.10451.3751 ± 0.07841.1860 ± 0.02120.9750 ± 0.01511.1048 ± 0.08990.123 ± 0.04520.1390 ± 0.05340.1407 ± 0.05390.1437 ± 0.06750.1547 ± 0.07450.0657 ± 0.05410.050 ± 0.00690.0500 ± 0.00660.0500 ± 0.00660.0515 ± 0.00720.0437 ± 0.0067ε=4%1.717 ± 0.12011.7799 ± 0.04971.8641 ± 0.05371.8907 ± 0.06031.9204 ± 0.02631.8549 ± 0.09701.841 ± 0.09431.7579 ± 0.32401.4310 ± 0.11121.1108 ± 0.06291.2240 ± 0.10550.112 ± 0.03530.1265 ± 0.04310.1170 ± 0.05320.1328 ± 0.06070.1547 ± 0.05720.209 ± 0.00830.2391 ± 0.00750.1141 ± 0.10470.0750 ± 0.07830.0516 ± 0.00790.0437 ± 0.0067ε=2%1.9824 ± 0.07832.1187 ± 0.08962.132 ± 0.05352.1876 ± 0.08532.2345 ± 0.09432.232 ± 0.13001.9678 ± 0.17001.8470 ± 0.24991.6112 ± 0.28111.1905 ± 0.10811.1880 ± 0.15640.132 ± 0.04690.1406 ± 0.03910.165 ± 0.04960.168 ± 0.04290.1783 ± 0.03900.2078 ± 0.04730.184 ± 0.06930.2126 ± 0.08750.0937 ± 0.09490.0969 ± 0.10550.0437 ± 0.0067ε=0%2.116 ± 0.05412.9588 ± 0.04302.232 ± 0.01792.294 ± 0.02222.2506 ± 0.02062.3263 ± 0.02712.293 ± 0.04562.2867 ± 0.04931.7365 ± 0.15522.4604 ± 0.11581.4571 ± 0.07840.1875 ± 0.01980.2078 ± 0.02990.185 ± 0.03540.189 ± 0.04210.1783 ± 0.04820.2750 ± 0.05390.503 ± 0.08510.6843 ± 0.10120.3359 ± 0.10290.2656 ± 0.07720.0821 ± 0.0077In addition, the proposed method is compared with two other popular attribute reduction methods, one is the classical Nested Strategy-based Heuristic Approach to Computing Reduct (NS-HACR), and another is DMF-FN [36, 37]. MF-FN is the nested reduction approaches designed by the discernibility matrix. The detailed results of the above six data sets are shown in Figure 1, in which xx axis denotes different data sets and yy is the average elapsed time of three different algorithms. It is difficult to note from Figure 1 that the proposed method NS-THACR obtained the minimum elapsed time. With the above discussions, we can conclude that the proposed NS-THACR method is an effective and efficient attribute reduction method.Figure 1
Elapsed time of three different algorithms.


## 5. Conclusions

In this paper, we have discussed some weaknesses in attribute reduction based on the traditional fuzzy rough set, which is often constructed by the operator without adjustability, and thereby a new adjustable fuzzy rough set has been presented. In our approach, a parameterized operator has been applied to develop strong and weak fuzzy relations, and such two special relations can offer fuzzy rough set model adjustability. Furthermore, inspired by the inner relationship between parameter and reduct, we have also employed a nested mechanism to accelerate the searching process of parameterized reduct.The following topics deserve our further investigations: (1) some evolutionary algorithms [2] such as particle swarm optimization algorithm and ant colony algorithm can be applied to find the optimal parameter used in our proposed model; (2) some accelerated searching strategy [12] such as sampling technique and attribute group technique can be used to further reduce the elapsed time over one single parameter.



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*Source: 5513722-2021-07-23.xml*