Attribute reduction is considered as an important processing step for pattern recognition, machine learning and data mining. In this paper, we combine soft set and rough set to use them in applications. We generalize rough set model and introduce a soft metric rough set model to deal with the problem of heterogeneous numerical feature subset selection. We construct a soft metric on the family of knowledge structures based on the soft distance between attributes. The proposed model will degrade to the classical one if we specify a zero soft real number. We also provide a systematic study of attribute reduction of rough sets based on soft metric. Based on the constructed metric, we define co-information systems and consistent co-decision systems, and we provide a new method of attribute reductions of each system. Furthermore, we present a judgement theorem and discernibility matrix associated with attribute of each type of system. As an application, we present a case study from Zoo data set to verify our theoretical results.
Attribute reduction, also known as feature selection is the process whereby superfluous attributes are removed from a given data base of knowledge while preserving the consistency of classifications. It plays an important role in machine learning, artifical intelligence, pattern recognition and other fields. Pawlak [1, 2] was the first to propose the concept of attribute reduction for decision tables. In order to obtain minimal subset of attributes that induces the same in discernibility as the whole set of attributes in as information system, Skowron and Rauszer [3] were the first to propose the concept of discernibility matrix, which transforms the discernibility function from its conjunctive normal form into disjunctive normal form. The minimal reduction subset of attributes can then be obtained. Pawlak and Skowron [4, 5], Skowron and Rauszer [3] proposed an algorithm for attribute reduction based on discernibility matrix with equivalence relations. This equivalence relation depends on equality which is unnatural in life applications.
However, Pawlak rough sets can only deal with complete and symbolic datasets[6]. In order to deal with different types of dataset, many extened rough set models have been proposed.There are now many different types of attribute reductions [7–15] with respect to different criteria. Jia et al. [16] gave a brief description of twenty-two kinds of existing reduction approaches, e.g., variable precision reductions [17, 18], mutual information reductions [19], distribution reductions [20], postive region reductions [5], covering reductions [21] and cost senstive reductions [22].
As equivalence is too restrictive for many applications; therefore, several authors [23, 24] have recently studied certain types of reduction using dominance relation-based attribute reduction. For a given decision table, Wei et al. [25] derived a compacted decision table that can preserve all information contained in the original decision table. Yamany et al. [26] proposed an intelligent optimization method called the “flower search algorithm” which adaptively searches for optimal attributes for the fitness function used in rough sets-based classification. Liu et al. [27] proposed a general attribute reduction algorithm for relation decision system. Zhang and Xu [28, 29] extended the concepts of the lower and upper approximation reduction to ordered information systems. C. Wang [30] constructed a fuzzy rough set model based on distance measure with a fixed parameter. Then, the fixed distance parameter was replaced by a variable one to better characterize attribute reduction with fuzzy rough sets. C. Wang [31] proposed a new neighborhood rough set model called k-nearest neighborhood rough sets. Furthermore, an attribute reduction algorithm based on this model was designed. In [32] C. Wang proposed a new fuzzy-rough-set model for categorical data by introducing a variable parameter to control the similarity of samples. In [33] a neighborhood rough set model with nominal metric emb eddingIn [34]. Gonga, Hongyun Zhang develop two quick general reduction algorithms. In [35] an acceleration strategy of attribute reduction based on attribute group was proposed was proposed. In [36] incremental attribute reduction algorithms for incomplete decision systems when adding multiple objects and when deleting multiple objects were proposed. In [37] a multi-objective attribute reduction (MOAR)was modeled.
In 1999, Molodtsov [38] introduced the concept of soft sets which can be seen as a new mathematical tool for dealing with uncertainties. This so-called soft set theory is free from the difficulties affecting existing methods. With the establishment of soft set theory, its application has boomed in recent years [39–45]. In [46], Maji et al. introduced the notion of reduct-soft set, and described the application of soft set theory to a decision-making problem using rough sets. Chen et al. [47] presented a new definition of soft set parameterization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. Kong et al. [48] introduced the notion of normal parameter reduction of soft sets and constructed a reduction algorithm based on the importance degree of parameters. In [49, 50], Das and Samanta introduced a notion of soft real sets, soft real numbers, soft complex sets, soft complex numbers and some of their basic properties have been investigated. Some applications of soft real sets and soft real numbers have been presented in real life problem. In [51] Tantawy introduced the operations on soft real numbers and defined countable and uncountabl- soft real sets. In [52] Das and Samanta introduced a notion of soft metric space which is based on ‘soft point’ of soft sets. Tantawy et al. [53] introduced a new definition of the soft metric function using the concept of soft elements. By this definition, each soft metric in view of Das and Samanta [50] is also a soft metric in this concept but the converse is not true.
In this paper, we introduce the soft metric model for attribute reductions. Pawlak’s rough set model employs equivalence relations to partition the universe and generate mutually exclusive equivalence classes as elemental concepts. This is just applicable to data with nominal attributes. To deal with heterogeneous numerical attributes, we construct a soft metric on the family of knowledge structures based on the soft distance between attributes. Based on the constructed metric, we define a relation between objects which can be used to generate a family of soft open balls from the universe characterized with numerical features, and then we can use these balls to approximate decision classes. Based on this observation, a soft metric rough set model is constructed. The proposed soft metric is an effective tool for attribute reduction especially for numerical data. Most of the algorithms for computing the reduction sets need the notion of discernability matrix, so as a result this paper proposes an improved discernability matrix and introduce an attribute reduction algorithms for decision tables based on the soft metric. We can use this soft metric to discover the important attributes and solve attribute reduction. Based on the proposed model, the dependency between heterogeneous numerical features and decision is defined for constructing measures of attribute significance for heterogeneous numerical data. The main contributions of the work are three-fold. First, we extend the rough set model to deal with data with heterogeneous numerical features and discuss two classes of monotonicity in terms of soft open ball radii and attributes; second, an efficient algorithm is designed for searching an effective feature subset; third, the radius of soft open ball which represents by a soft real number can take a distinct values for different attributes.
The remainder of this paper is arranged as follows: Section 2 presents some basic notions of soft set theory and soft metric. The purpose of Section 3 is to introduce the notion of co-information system. Therefore, a new soft metric is proposed based on soft distance between attributes. Also, we defined the discernibility matrix for each co information system.as a generalization of discernibility matrix defined by Pawlak. The main properties of current branch are studied Also, we establish an attribute reduction algorithms associated with every arbitrary co information system. The purpose of Section 4 is to present the concept of co-decision system. Many concepts on co decision system are introduced such as consistency, dependency and attribute reduction. Furthermore we present a judgement theorem and discernibility matric associated with attribute of such system. As an application, in Section 5, we present an algorithm for simplification of co decision system by using soft metric. We present a case study from Zoo data set to verify our theoritical results. Finally, in the last section, we generalize Pawlak’s decision logic by using the soft metric. We also have proposed a decision rule that enables us to extract characteristics based on comparison between values of different attributes.
Preliminaries
In this section, the basic definitions and results of soft set theory which will be needed in the sequel are presented. Throughout this study, X refers to an initial universe, P (X) is the power set of X, E is a set of parameters and A ⊆ E.
Definition 2.1. [38] A soft set FA on the universe X is defined as a set of ordered pairs
where FA : E → P (X), such that FA (e) = φ(empty set) ∀e ∉ A, and A is called the support of FA.
The collection of all soft sets with support A is denoted by .
Definition 2.2. [38] Let FA = (F, A) and GB = (G, B) be two soft sets over X
(1)(F, A) is called a soft subset of (G, B) denoted if F (e) ⊆ G (e) ∀ , e ∈ A, A ⊆ B ⊆ E, and they are equals if and .
(2) The union denoted by is the soft set HC = (H, C) where C = A ∪ B and ∀ c ∈ C
(3) The intersection denoted by is the soft set HC = (H, C), where C = A ∩ B and H (e) = F (e) ∩ G (e) ∀ e ∈ C.
(4) The difference (H, A) denoted by is defined by H (e) = F (e) - G (e) ∀ e ∈ A.
(5) The complement is denoted by where (FA) c : E → P (X) is a function given by Fc (e) = X - F (e) ∀ e ∈ E, clearly ((FA) c) c = FA.
Definition 2.3. [38] A soft set FA over X is called a null soft set with support A, denoted by if F (e) = φ ∀ e ∈ A and is called absolute soft set with support A denoted By if F (e) = X ∀ e ∈ A, A ⊆ E clearly and .
Definition 2.4. [53] A soft element with support A is a soft set such that F (e) = {xe} is a singleton set ∀ e ∈ A. The collection of all soft elements is denoted by where [X] = {{x} : x ∈ X} ∪ {φ}.
For simplicity, we will use the notations to denote the soft elements, for example
Definition 2.5. [53] A soft element , is said to belongs to a soft set FA, denoted by , if {x} ⊂ F (e) ∀ e ∈ A ⊆ E.
Lemma 2.6. [53] For any two soft sets FA and GA, iff for any soft element and hence FA = GA iff , where denote that belongs to FA as a soft subset.
Proposition 2.7. [53] (i) Every soft subset FAA ⊆ E can be considered as a union of soft elements all with the same support A.
(ii) Every soft element is a union of a collection of one point support soft elements.
We are going to define and give a concept of soft real numbers and soft metric
Definition 2.8. [51] By a soft real number we mean a soft element of real numbers with support A which is a parametrized collection of real numbers denoted by i.e. and is called a soft +ve real number if where [R+] = {{r} : r ∈ R+}.
i.e. a soft real number with support A is given by
For instance, and are the soft real numbers where and ∀e ∈ A.
An ordered relation on the set of all soft elements of real numbers is given in the following definition.
Proposition 2.9. [51] For any two soft real numbers where is the collection of all soft real numbers with support equal or contained in E,
i) (resp. ) if re ≤ le (resp. le ≤ re) ∀e ∈ E,
ii) Define the operation ⊕, ⊖ , ⊙, respectively on by
le, le and le ∀e ∈ E
Definition 2.10. [53] Let be the collection of all soft element of X with support equal or contained in E, A mapping is said to be soft metric or distance on the soft set if ds satisfies the following conditions
(d1) ∀
(d2) iff
(d3) ∀
(d4) ∀
Definition 2.11. [51] A soft metric ds on is called equal support soft metric or (E-soft metric) denoted by dES if the conditions (d1), (d2), (d3),(d4) in Definition 2.10 are holds for all soft elements with equal supports and ()(e) =0 ∀ e∉support
To have a natural relation between the soft metric structures and the metric structures we reconsider a special case of soft metrics in which the soft distance is defined between soft elements with the same support only.
Soft metric based rough set model
Co information system
In this section, we introduce the notion of co-information system Therefore, a new soft metric is proposed based on soft distance between attributes. Also, we propose a new discernibility matrix and establish an attribute reduction algorithm for an arbitrary co information system. The main result of this section shows that this reduction is a generalization of Pawlak,s reduction.
Definition 3.1. A co information system (CIS for short) can be written as a tuple, ((U, A) for short) where U = {x1, x2, . . . . . . xn} is the non empty set of objects, A is the nonempty set of variables (also called as features, inputs, attributes) and Va is the value domain of attribute a. Each object xi defines a soft element , where called the co information function for xi .
Example 3.2. Let (U, A) be a co information system given in Table 1
U
a
b
c
x1
1.9
0
1
x2
2.1
0.5
0
x3
3.7
0.8
2
x4
2
0.3
1
U = {x1, . , x4} , A = {a, b, c} , Va ={1.9, 2.1, 3.7, 2},Vb = {0, 0.5, 0.8, 0.3} , Vc = {0, 1, 2} . So such that For example
Each soft element represent the values of the attributes for the object xi for each i .
In the following definition, we are going to define a soft metric on the set of objects.
Definition 3.3. Let (U, A) be a co information system. For any two objects xi, xj ∈ U, the soft metric is defined as Sd (xi, xj) = ds (αi, αj) =
Soft metric rough set
Definition 3.4. Let (U, A) be a co information system. For each object xi and each soft real number with support B ⊂ A, the soft open ball with center xi and radius , denoted by is the set of objects {xj ∈ U : Sd (xi, xj)
Remark 3.5. 1-
∀ a ∈ B} .
i) For each object xi and each soft real number
ii)
iii)
Definition 3.6. Let (U, A) be a co information system. For every set of attributes φ ≠ B ⊆ A, we define indiscernibility relation for any soft real number as follows:
Two objects xi and xj are indiscernible with respect to B i.eInd (B)) xj if Sd (xi, xj) i.e ∀ a ∈ B
Remark 3.7. i) It is easy to show that the relation is reflexive and symmetric.
ii) is an equivalence class and is an equivalence relation (Pawlak case) if this case is applicable to discrete data. So Pawlak,s approximations is a special case of our approximations.
iii) can take a distinct values for different attributes.
Based on relation, every subset of objects X ⊆ U corresponds -lower and -upper approximations defined as follows:
Definition 3.8. Let (U, A) be a co information system, X ⊆ U and φ ≠ B ⊆ A The -lower and -upper approximations of X in terms indiscernibility relation are defined as
The boundary region of X in the co information system (U, A) is formulated as
The size of the boundary region reflects the degree of roughness of the set X in the co information system (U, A). Assuming that X is the sample subset with a decision label, usually we hope that the boundary region of the decision is as little as possible for decreasing uncertainty in decision. The sizes of the boundary regions depend on X, attributes B to describe U, and the soft real number
Remark 3.9. By Definition 3.8 and Remark 3.5 . So The -lower and -upper approximations of X can be written as
Example 3.10. Let (U, A) be a co information system given in Table 1. Let , and So ds (x1, x2) = {0.2, 0.5, 1} , ds (x1, x3) = {1.8, 0.8, 1} and ds (x1, x4) = {0.1, 0.3, 0} . Hence . Simillarly, and
Theorem 3.11. Given a co information system (U, A) and two soft real numbers and if we have
i) ∀ xi ∈ U:,
ii) ∀ X ⊆ U :
Proof. If
obviously, we have Assuming we have Therefore, we must have if However, xi is not sure in if we have . Hence Similarly, we can get
Theorem 3.11 shows that a finer indiscernibility relation is produced with a smaller soft real number; accordingly, the lower approximation is larger than that with a great soft real number.
The soft metric attribute reduction of co information systems
In this subsection, we provide approachs to attribute reduction in co information system
Let (U, A) be co information system and be a soft real number. For any given xi ∈ U, We consider a reduction type which keep the soft open ball unchanged. We now provide its definition.
Definition 3.12. Let (U, A) be co information system and be a soft real number The attribute a ∈ A is called -dispensable (or superflous) in A if ∀ xi ∈ U. Otherwise a is -indispensable in A.
The set of attributes B ⊆ A is -independent if every a ∈ B is - indispensable in B, otherwise B is - dependent.
Definition 3.13. Let (U, A) be a co information system and be a soft real number. The set of attributes φ ≠ B ⊆ A is - reduct of A if B is -independent and ∀ xi ∈ U. The set of all - indispensable attributes a ∈ A is called the core of A.
Next, we define the -discernibility matrix for each co-information system (U, A). This help to define the reduction by using the -discernability matrix.
Definition 3.14. Let (U, A) be a co information system, where U = {x1, x2, . . . . . xn} . For every soft real number , the -discernibility matrix of (U, A) is an n × n matrix defined as follows:
(cij) = {a ∈ A : for i, j=1,2,.....n, cij is the set of all attributes which discern objects xi and xj.w.r.t soft real number
The -core of A w.r.t soft real number is the set of all single elements entires of the discernibility matrix , i.e.
Next, we provide the properties of the -discernibility matrix for each co-information system (U, A).
Proposition 3.15. Let (U, A) be a co information system and φ ≠ B ⊆ A . If then cij ≠ φ
Proof. Obvious
Theorem 3.16. Let (U, A) be co information system and φ ≠ B ⊆ A .Then the following conditions are equivalent;
i) ∀ xi ∈ U.
ii) If cij ≠ φ, then B ∩ cij ≠ φ.
Proof.i ⇒ ii If cij ≠ φ and B ∩ cij = φ. By the definition of cij, we assume that B ∩ cij = φ implies ; using condition 1, . This is a contradition.
ii ⇒ i Since B ⊆ A, then xj ∈ U. We show that If , then cij ≠ φ. So by condition 2, B ∩ cij ≠ φ. That is for some b ∈ B. This means that . This proves
Corollary 3.17. Let (U, A) be a co information system and φ ≠ B ⊆ A. Then B is a -reduct of A iff it is a minimal subset satisfying B ∩ cij ≠ φ, for any cij ≠ φ.
Proof. IfB is -reduct of A, then by Definition 3.12, ∀ xi ∈ U. By using Theorem 3.16, B satisfies B ∩ cij ≠ φ for any cij ≠ φ. Using Definition 3.12 since B is -independent, B is a minimal subset of A satisfying B ∩ cij ≠ φ for any cij ≠ φ.
Conversely, if B is a minimal subset of A satisfying B ∩ cij ≠ φ, for any cij ≠ φ, by Theorem 3.16, ∀ xi ∈ U. Moreover, the minimal property of B implies that B is -independent.
Example 3.18. Let (U, A) be a co information system given in Table 2
U
a
b
c
d
x1
0
1
2
0
x2
1
2
0
2
x3
1
0
1
0
x4
2
1
0
1
x5
1
1
0
2
Let , and So c12 = {a ∈ A : } = {a, b, c, d}.
The -discernibility matrix of (U, A) is givin in Table 3:
U
1
2
3
4
5
1
2
a,b,c,d
3
a,b,c
b,c,d
4
a,c
a,b
a,b,c
5
a,c,d
b
b,c,d
a
The -core of A is {a, b} which also -reduct.
The soft metric attribute reduction of co decision system
In this section we present the concept of co-decision system. Many concepts in co decision systems are introduced; such as consistency, dependency and attribute reduction. Furthermore we present a judgement theorem and discernibility matrix associated with attribute of such system.
Definition 4.1. A co-decision system (CDS for short) can be seen as a system S = (U, A = C ∪ D, {Va : a ∈ A} , {αxi : xi ∈ U}) where U = {x1, x2, . . . . . . xn} is a non empty set of objects, A is a non empty set of attributes, C is a set of condition attributes, D is a set of decision attributes,Va is a non empty set of values and αxi is a co information function for each object xi .
Definition 4.2. Let (U, A = C ∪ D) be co a decision system. If ∀xi ∈ U, then (U, C ∪ D) is called consistent, otherwise (U, C ∪ D) is called inconsistent.
Definition 4.3. Let (U, A = C ∪ D) be a co decision system. An postive region of C relative to D denoted by , where xi ∈ U} .
Definition 4.4. Let (U, A = C ∪ D) be a co decision system and B ⊆ C. The lower and upper approximations of the decision with respect to attributes B are defined as
The decision boundary region of D with respect to attributes B is defined as
Next, we define the dependency in a co decision system (U, C ∪ D) as follows:
Definition 4.5. Let (U, A = C ∪ D) be a co decision system and B ⊆ C . The dependency degree of D toB is defined as
Where | • | is the cardinality of a set, reflects the ability of B to approximate D. Obviously, We say that D completely depends on B with respect to if , denoted by ; otherwise we say that D depends on B with respect to in the degree of γ.
Proposition 4.6 Let (U, A = C ∪ D) be a co decision system. For B1 ⊆ B2, ⊆ C, we have
Proof. i) If (U, C ∪ D) is -consistent, then ∀xi ∈ U. In general, -posC (D) ⊆ U . On the other hand, if xi ∈ U . Then clearly Then s.t , i.e. Hence .
Theorem 4.7. (Type-1 monotonicity) Let (U, A = C ∪ D) be a co decision system. For B1 ⊆ B2, ⊆ C, we have
i)
ii) ∀ X ⊆ U :
iii)
Proof. i and ii are obvious.
iii) ∀ xi ∈ U, we have if B1 ⊆ B2 . Assume that where X ∈ U/D, then we have At the same time, there may be xj, and . Therefore, . Accordingly, we have
Theorem 4.7 shows that dependece monotonically increases with attributes, which means that adding a new attribute in the attribute subset at least does not decrease the dependence. This property is very important for constructing attribute reduction algorithms. Generally speaking, we hope to find a minimal subset which has the same characterizing power as the whole samples.
Theorem 4.8. (Type-2 monotonicity) Let (U, A =C ∪ D) be a co decision system. For two soft real numbers and if we have
i) ∀ X ⊆ U :
ii) .
Proof. Refer to the proof of Theorem 3.11.
Definition 4.9. Let (U, A = C ∪ D) be a co decision system and φ ≠ B ⊆ C .If , then we call the attribute b is necessary for the attribute set B relative to decision attribte D, otherwise not necessary. If the attribute set B satisfies the following condition,
i)
ii) (D),
then the attribute set B is called a reduction relative to the attribute set A.
The goal of attribute reduction is to remove the redundant attributes which are not necessary for classifying objects to its true label. The first condition guarantees that . The second condition shows that each attribute in a reduction B is necessary. Therefore, a reduct is the minimal subset of attributes which has the same approximating power as the whole attribute set. This definition presents a feasible direct to find optimal feature subsets.
Discernibility matrix associated with consistent co decision system
Definition 4.10. Let (U, A = C ∪ D) .be an -consistent co decision system, where U = {x1, x2, . . . . . xn} . For soft real numbers , , the -decision discernibility matrix of (U, C ∪ D) is an n × n matrix defined as follows:
where
The computational complexity of the -decision discernibility matrix is O (n × n) .
Lemma 4.11. Let U = {x1, x2,............,xn} and (U, A = C ∪ D) . be an - consistent co decision information system. Then Dij ≠ φ for 1 ≤ i, j ≤ n .
Proof. Suppose that Dij = φ for xi, xj ∈ U. By the definition of Dij, if , then because Dij = φ, this contradicts to
Theorem 4.12. Let (U, A = C ∪ D) be an -consistent co decision system, B ⊆ C and B ≠ φ. Then (U, B ∪ D) is -consistent iff B ∩ Dij ≠ φ.
Proof. Suppose that (U, B ∪ D) is -consistent. If , then B ∩ Dij = B ∩ C ≠ Φ. If Since (U, B ∪ D) is -consistent, then . So . Thus ∃ a ∈ B s.t . That is, a ∈ B ∩ Dij .
Conversely, we show that ∀xj ∈ U. Suppose that Let a ∈ B ∩ Dij . Then and i.e. Consequently (U, B ∪ D) is -consistent.
Corollary 4.13. Let (U, A = C ∪ D) .be an -consistent co decision system, and B ⊆ C .Then B is -reduction of C iff it is a minimal subset satisfying B ∩ Dij ≠ φ, 1 ≤ i, j ≤ n .
Proof. Obvious.
According to Corollary 4.13, we now give a reduction algorithm for an -consistent co decision system (U, C ∪ D) :
Algorithm 4.14.
Find the discernibility matrix D = (Dij) n×n for suitable soft real numbers
Computing discernibility function Dij), where ∨ Dij is the disjunction of all attributes in Dij,
, Bi ⊆ C is obtained by Boolean operators.
Red (C) = {B1, B2, . . . . . . . . Bn} and .
End the algorithm
Now in the following, we present a case study from Zoo data set to verify our theoritical results.
Example 4.15. The Zoo data set (http://archive.ics.uci.edu/ml/datasets/Zoo) has 10 instances, 3 condition attributes, and one decision attribute. The animal name attribute is superfluous in our example so we remove it. Let xi (i = 1, 2, 3, . . .10) represents 10 instances, a1, a2, a3 denote the condition attributes, and d denote the decision attribute, as shown in the Table 4 below. As a result we obain an consistent co decision system (U, A = C ∪ D) , where U = {xi : i = 1, 2, . . . . .10} , C = {a1, a2, a3} and D = {d}
U
a1
a2
a3
d
x1
2.5
0.7
3.5
2.2
x2
3.4
1.9
0.9
2.8
x3
1.8
0.8
3
1.9
x4
2
1.9
2.8
2.8
x5
0.1
0.6
4
0.8
x6
0.1
0.9
4.1
0.8
x7
4.1
2.3
1
2.9
x8
0.1
1
3.9
0.8
x9
2.3
0.5
3.3
2
x10
3.8
2.4
0.7
3.1
Let , and
So, , =SB (x10, , and ,
So, xi ∈ U} = {SB (xi, xi ∈ U} = {{x1, x3, x9} , {x2, x4, x7, x10} , {x5, x6, x8}} and hence , x5, x6, x7, x8, x9, x10} =U . Therefore Thus (U, C ∪ D) .be an consistent co decision system.
Now, we compute the reduction of C by using Definition 4.9. If B1 = (a2, a3} = C - {a1}, then . So and a1 is indispensable attribute. Simillarly if B2 = (a1, a3} = C - {a2}, then . So a2 is indispensable attribute. If B3 = (a1, a2} = C - {a3}, then posB3 (D) = {x1, x2, x3, x4, x5, x6, x7, x8, x9, . So a3 is dispensable attribute. Hence the set of attributes B3 = (a1, a2} is dependent. Also, and Consequently B3 = (a1, a2} is reduct and core of C.
On the other hand we can compute the reduction of C by using the decision discernability matrix of (U, C ∪ D) according to Corollary 4.13. , x6, x8, x9} and Thus D11 = C, D11 = C, D21 = C, D31 = C, D41 = {a2} , D51 = {a1} , D61 = {a1} , D71 = C, D81 = {a1} , D91= C and D10 1 = C . Simillarly, we can compute the other element of the decision discernibility matrix of (U, C ∪ D) as in the following table (Table 5)
U
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x1
C
x2
C
C
x3
C
C
C
x4
a2
C
a2
C
x5
a1
C
a1
a1, a2
C
x6
a1
a1, a2
a1
a1, a2
C
C
x7
C
C
C
C
C
C
C
x8
a1
C
a1
a1, a2
C
C
C
C
x9
C
C
C
a2
a1
a1
C
C
C
x10
C
C
C
C
C
C
C
C
C
C
It can be easily seen from Table 5 that B3 = (a1, a2} is reduct and core of C
A soft metric based attribute reduction algorithm
The motivation of rough set based attribute reduction is to select a minimal attribute subset, which has the same characterizing power as the whole attribute set, and without any redundant attribute. In other words, the dependency of the selected attributes is the same as that of the original attributes. And the dependency will decrease if any selected attribute is deleted. There are two key problems in constructing a feature selection algorithm. One is how to evaluate the selected features; the other is how to search for a good feature subset. We will discuss them in the following. Here dependence function can be introduced to evaluate the goodness of selected features. In this section, we construct a measure for attribute evaluation, and then present a feature selection algorithm.
Definition 4.6. Let (U, A = C ∪ D) be a co decision system and φ ≠ B ⊆ C, a ∈ B, the significance of an attribute is defined as
Note that the significance of an attribute is related with three variables: a, B and D. An attribute a may be of great significance in B1 but of little significance in B2. What’s more, the attribute’s significance is different for each decision attribute if they are multiple decision attributes in a decision table. The above definition is applicable to backward feature selection, where redundant features are eliminated from the original set of features one by one. Similarly, a measure applicable to forward selection can be written as
Definition 4.16. Let (U, A = C ∪ D) be a co decision system and φ ≠ B ⊆ C, a ∉ B, the significance of an attribute is defined as
As and ∀a ∈ B : we have
Although there usually are multiple reducts for a given co decision table, in the most of applications, it is enough to find one of them. With the proposed distane, an algorithm for attribute reduction can be formulated as follows
Algorithm 4.17
Input: (U, A = C ∪ D)
, // Control the size of the soft open ball
Output: reduct green.
Step 1: φ⟶green;// green is the pool to contain the selected attributes
Step 2: For each ai∈ C-green
Compute
Compute //
Here we define : end
Step 3: Select the attribute ak satisfying SIG (ak, green, D)=
Step 4: If SIG (ak, green, D) ≻0,
green∪ ak ⟶ green: go to step2
else
return green
Step 5: end
Algorithm 4.17. adds an attribute with the great increment of dependence into the reduct in each soft open ball until the dependence does not increase, namely, adding any new attribute will not increase the dependence in this case. The time complexity of the algorithm is O (N × N) , whereN is the number of candidate attributes.
Example 4.17. Let U = {u1, u2, u3, u4, u5}be five amino acids (for short, AAs). The (AAs) are described in terms of seven attributes: a1 = PIE, a2=PIF (two measures of the side chain lipophilicity), a3 = DGR = G of transfer from the protein interior to water, a4 =SAC=surface area, [54, 55]. As a result we obain an consistent co decision system (U, A = C ∪ D), shown in Table 6, where C = {a1, a2, a3, a4} and D = {d}
U
a1
a2
a3
a4
d
u1
–0.11
–0.22
0.29
335
6.3
u2
–0.51
–0.64
0.76
311.6
8.8
u3
0
0
0
244.9
7.1
u4
0.15
0.13
–0.25
337.2
10.1
u5
1.2
1.8
–2.1
322.6
16.8
Let , , and
By applying Algorithm 4.17, we find that B = {a1, a2, a4} is reduct of C
Simplification of the co decision systems
In traditional rough set theory, the simplification of a decision table was investigated and minimal algorithms were introduced to express its result. In this article, we generalize Pawlak’s rough approach for simplifying the decision table in an information system.
The approach to table simplification consist of the following steps
Computation of reducts of condition sttributes which is equivalent to elimination of some column from the decision table.
Elimination of duplicate rows.
Elimination of superfluous values of attributes.
Definition 5.1. [2] Let F ={X1, X2, Xn}, be a family of sets suh that Xi ⊆ U and a subset Y ⊆ U, such that ∩F ⊆ Y .
We say that Xi is Y- dispensable in ∩F, if ∩ (F - {Xi}) ⊆ Y; otherwise the set Xi is Y-indispensable in ∩F .
The family F is Y-independent in ∩F if all of its component are Y-indispensable in ∩F ; otherwiswe F is Y-dependent in ∩F .
The family H ⊆ F is a Y-reduct of ∩F, if H is Y-independent in ∩F and ∩H ⊆ U .
The family of all Y-indispensable sets in ∩F is called the Y-core of F, denoted COREY(F).
Now, we simplify the co-decision information system of the Zoo data given in Example 4.16 by using the soft metric.
Example 5.2. Let (U, A = C ∪ D) be a consistent co decision system of the Zoo data set given in Example 4.15. The only d- dispensable attribute is a3 ;consequently, we can remove column a3 ;in Table 4, which yields Table 7 shown below
U
a1
a2
d
x1
2.5
0.7
2.2
x2
3.4
1.9
2.8
x3
1.8
0.8
1.9
x4
2
1.9
2.8
x5
0.1
0.6
0.8
x6
0.1
0.9
0.8
x7
4.1
2.3
2.9
x8
0.1
1
0.8
x9
2.3
0.5
2
x10
3.8
2.4
3.1
Now we have to reduce superfluous values of condition attributes, in every decision rule. We have to compute core values of condition attributes in every decision rule.
Let us compute the core value of condition attributes for the second decision rule, i.e. the core of the family of sets.
{x1, x3, x5, x6, x8, x9}} . Moreover a1 (x1) =2, a2 (x1) =1. In order to find dispensable categories, we have to drop one category at a time and check whether the intersection of remaining categories is still included in the decision category , i.e.
and
This means that the core values are a1 (x1) =2 and a2 (x1) =1 . Similarly we can compute remaining core values of condition attributes in every decision rule and the final results are presented in Table 8
U
a1
a2
d
x1
2.5
0.7
2.2
x2
—
1.9
2.8
x3
1.8
0.8
1.9
x4
2
—
2.8
x5
—
0.6
0.8
x6
—
0.9
0.8
x7
—
—
2.9
x8
—
1
0.8
x9
2.3
0.5
2
x10
—
—
3.1
Having computed core values of condition attributes, we can proceed to compute value reducts. Let us compute value reducts for the second decision rule. Accordingly to Definition 5.1, in order to compute reducts of the family we have to find all subfamilies G ⊆ F such that ∩G ⊆ {x2, x4, x7, x10} = ∩ F . The two subfamilies and are included in ∩F . Hence we have two value reducts: a1 (x2) =3 or a2 (x2) =2 . In Table 9 we list value reducts for all decision rules in Table 8
U
a1
a2
d
x1
2.5
0.7
2.2
x2
3.4
×
2.8
×
1.9
2.8
x3
1.8
0.8
1.9
x4
×
1.9
2.8
x5
0.1
×
0.8
x6
0.1
×
0.8
x7
4.1
2.3
2.9
×
2.3
2.9
x8
0.1
×
0.8
x9
2.3
1
2
×
2.4
3
There are many (not necessarily different) solutions to this problem. One such solution is presented in the Table 10 below
U
a1
a2
d
x1
2.5
0.7
2.2
×
1.9
2.8
x3
1.8
0.8
1.9
x4
×
1.9
2.8
x5
0.1
×
0.8
x6
0.1
×
0.8
x7
×
2.3
2.9
x8
0.1
×
0.8
x9
2.3
0.5
2
x10
3.8
×
3.1
×
2.4
3.1
Because decision rules and x4 are identical, and so are x5, x6 and x8, we have Table 11.
U
a1
a2
d
x1
2.5
0.7
2.2
×
1.9
2.8
x3
1.8
0.8
1.9
x5,x6,x8
0.1
×
0.8
×
2.3
2.9
x9
2.3
0.5
2
×
2.4
3.1
Decision logic based on soft metric
Decision rules and decision algorithm based on soft metric
Decision rule aquistion is one of the important topics in rough set theory and drawing more and more attention. Rough set-based decision rule extraction is a useful tool for data mining, however, almost all the approaches of decision rule extraction are based on comparison between values of the same attribute. This may cause a limitation for the various reasoning.
In this section we generalize Pawlak,s decision logic by using the soft metric. We have studied a method of rule reduction and rule set minimization based on soft metric
The language of decision logic (DL-language) consists of atomic formulas, which attribute-value pairs, combined by means of sentenial connectives(and, or, not etc.) in a standard way, forming compound formulas
Definition 6.1. The alphabet of the language consists of:
A-the set of attribute which are constants.
V = ∪ Va- the set of attribute value constants a ∈ A .
Set {∽, ∨, ∧, ⟶, ≊} of propositional connectives, called negation, disjunstion, conjunction, implication and equivalence respectively.
Definition 6.2. The set of formulas in DL-language is the least set satisfying the following condition:
Expression of the form (a, v), or in short av, called elementary (atomic) formulas, are formulas of the DL-language for any a ∈ A and v ∈ Va .
If φ and ψ are formulas of the DL-language, then so are ∽φ, (φ ∨ ψ), (φ ∧ ψ), (φ ⟶ ψ), and (φ ≊ ψ).
Next, we show that the atomic formula (a, v) is interpreted as a description of all objects belong to the soft open ball of center a (x) and radius
Definition 6.3. Let (U, A = C ∪ D) be a co decision system. An object x ∈ U satisfies a formula φ = (a, v) in (U, A) with respect to the soft real number denoted if and only if the following conditions are satisfied:
1) iff
2) iff non
3) iff or
4) iff and
As a corollary from the obove conditions we get
5) iff
6) iff and
If φ is a formula then the set defined as follows
will be called the meaning of the formula φ in (U, A) .
iff ∀ a ∈ A
Definition 6.4. Let (U, A = C ∪ D) be a co decision system. A formula φ is said to be true in (U, A) iff
Definition 6.5. Let (U, A = C ∪ D) be a co decision system. Formula of the form
(a1, v1) ∧ (a1, v2) ∧ . . . . . . . ∧ (an, vn) , where vi ∈ Vai, P = {a1, a2, . . . . . . , an} , will be called a C-basic formula or in short C-formula.
Definition 6.6. Let (U, A = C ∪ D) be a co decision system. Any implication φ ⟶ ψ will be called a decision rule; φ and ψ are referred to as the predecessor and successor of φ ⟶ ψ respectively.
A decision rule φ ⟶ ψ is consistent in (U, A) if it is true in (U, A); otherwise, the decision rule is incosistent in (U, A)
If φ ⟶ ψ is a decision rule and φ and ψ are C-basic and D-basic formulas respectively, then the decision rule φ ⟶ ψ will be called a CD- basic decision rule, (in short CD-rule), or basic rule when CD is known.
Definition 6.7. Let (U, A = C ∪ D) be a co decision system. Any finite set of basic decision rules will be called a basic decision algorithm. If all decision rules in a basic decision algoriyhm are CD-decision rules, then the algorithm is said to be CD-decision algorithm, or in short CD-algorithm, and will be denoted by (C, D) .
Definition 6.8. Let (U, A = C ∪ D) be a co decision system. The CD-algorithm is consistent in (U, C ∪ D), if all its decision rules are consistent(true) in (U, C ∪ D); otherwise the algorithm is inconsistent.
Reduction of consistent algorithm by using soft metric
In this article, we will simplify a consistent CD-decision algorithms This corresponds exactly to that discussed in Section 4.
Definition 6.9. Let (U, A = C ∪ D) be a consistent algorithm, and a ∈ C . We will say that a is dispensable in (C, D) if the algorithm ((C - {a}) , D) is consistent; otherwise a is indispensable.
If all attributes a ∈ C are indispensable in the algorithm (C, D), then (C, D) will be called independent.
The subset B ⊆ C will be called an reduct of C in (C, D), if the algorithm (B, D) is independent and consistent.
If B is reduct of C in the algorithm (C, D), then the algorithm (BD) is said to be reduct of (C, D) .
The set of all indispensable attributes in the algorithm (C, D) will be called the core of (C, D) .
Example 6.10. Let (U, A = C ∪ D) .be an consistent co decision system, given in Table 12 below
U
a
b
c
d
e
x1
1
0
2
1
1
x2
2
1
0
1
0
x3
2
0.5
2
0
2
x4
0.5
2
3
1
1
x5
0.7
2
0
0
2
There are three condition attributes C = {a, b, c}, and two decision attributes D = {d, e}. Let , , and . The CD-algorithm of the system shown below
Let us first compute reducts of condition attributes in the algorithm. The only dispensable condition attribute is b. So there is one reducts of C, namely {a, c} .The CD-algorithm can be reduced as
In order to find the core set of attributes we have to drop condition atrributes, one by one and see whether the obtained co decision information system is consistent. Removing attribute a, we get the Table 13
U
b
c
d
e
x1
0
2
1
1
x2
1
0
1
0
x3
0.5
2
0
2
x4
2
3
1
1
x5
2
0
0
2
which is inconsistent. Since the decision rule 3 {b0.5c2 ⟶ d0e2} is inconsistent. Dropping attribute b we get consistent Table 14
U
a
c
d
e
x1
1
2
1
1
x2
2
0
1
0
x3
2
2
0
2
x4
0.5
3
1
1
x5
0.7
0
0
2
However when atribute c is dropped the result is an inconsistent Table 15
U
a
b
d
e
x1
1
0
1
1
x2
2
1
1
0
x3
2
0.5
0
2
x4
0.5
2
1
1
x5
0.7
2
0
2
Since the decision rule 2{a2b1⟶d1e0} is not consistent. So the core is {a, c}.
Reduction of decision rules by using soft metric
In this subsection, we show that how decision logic based on soft metric can be used to simplify decision algorithms by elimination of unnecassary conditions in each decision rule of a decision algorithm separately, in contrast to reduction performed on all decion rulrs simultaneously, as defined in Subsection 6.2.
If φ is C-basic formula and B ⊆ C, then by φ/B we mean the B- basic formula obtained from the formula φ by removing from φ all elementary formulas (a, va) such that a ∈ C - B .
Definition 6.11. Let φ ⟶ ψ be CD-rule, and let a ∈ C We will say that the attribute a is dispensable in the rule φ ⟶ ψ if and only if
Otherwise, attribute a is indispensable in φ ⟶ ψ.
If all attributes a ∈ C are indispensable in φ ⟶ ψ, then φ ⟶ ψ will be called independent.
The subset of attributes B ⊆ C will be called reduct of CD-rule φ ⟶ ψ, if φ/B ⟶ ψ is -independent and implies .
If B is reduct of the CD-rule φ ⟶ ψ, then φ/B ⟶ ψ is said to be reduced.
The set of all indispensable attributes in φ ⟶ ψ will be called the core of φ ⟶ ψ, and will be denoted by CORE(φ ⟶ ψ) .
Proposition 6.12.
where RED(C ⟶ D) is the set of all reducts of (C ⟶ D)
Example 6.13. Let (U, A = C ∪ D) be -consistent co decision system, given as in Table 10. We are now going to eliminate unnecessary conitions in each deciion rule of a decision algorithm separately, i.e. compute the cor and reducts of each decision rule in the algorithm
Let us start with the first decision rule.a1b0c2 ⟶ d1e1. The core of this rule is a, because a is indispensable while b and c are dispensable in the rule. There are two reducts of the rule, namely {a, c} and {a, b}. Hence the rule a1b0c2 ⟶ d1e1 can be replaced by either one of the rules a1c2 ⟶ d1e1 or a1b0 ⟶ d1e1 . Each of the remaining four rules has one core attribute c, and consequently two reducts {a, c} and{b, c} . Thus, for example, the second rule, a2b1c0 ⟶ d1e0 has two reduced forms, a2c0 ⟶ d1e0 and b1c0 ⟶ d1e0 .
The following Table 16 contains cores of each decision rule
Because each decision rule in the algorithm has two -reduced forms, hence in order to simplify the algorithm we have to choose one of them. As a result we get the algorithm with reduced decision rules as shown in Table 18
U
a
b
c
d
e
x1
1
—
2
1
1
—
1
0
1
0
x3
2
—
2
0
2
—
2
3
1
1
x5
0.7
—
0
0
2
We may also first reduce the algorithm and then reduce further decision rules in the reduced algorithm. As in Example 6.10 the algorithm has one reduced form
Conclusion
In this paper, we have proposed the concept of co-information system and consistent co decision system based on the soft metric. Reduction algorithms for both systems have been investigated. An example from zoo data was used to verify the effectiveness of the proposed algorithms. Moreover we applied a system of reduction in decision logic. In our future work, the following topic can be considered:
We would apply an algorithm based on soft metric to multi attribute group decision making to deal with uncertainty
We would propose the algorithm of parameter reduction of inconsistent and incomplete decision tables based on our models.
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