An incremental approach to obtaining attribute reduction for dynamic decision systems

Liu Wenjun

doi:10.1515/math-2016-0077

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An incremental approach to obtaining attribute reduction for dynamic decision systems

Liu Wenjun

Published/Copyright: November 27, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

In the 1960s Professor Hu Guoding proposed a method of measuring information based on the idea that connotation and denotation of a concept satisfies inverse ratio rule. According to this information measure, firstly we put forward the information quantity for information systems and decision systems; then, we discuss the updating mechanism of information quantity for decision systems; finally, we give an attribute reduction algorithm for decision tables with dynamically varying attribute values.

Keywords: Rough set; Conditional information quantity; Dynamic decision system; Attribute reduction

MSC 2010: 03E99

1 Introduction

In recent years a major challenge has been created due to increasing data volumes. The prevalence of continuously collected data has led to an increasing interest in the field of data streams. For example, Internet traffic generates large streams that cannot even be stored effectively unless significant resources are spent on storage. As data sets change with time, it is very time-consuming or even infeasible to run a knowledge acquisition algorithm repeatedly. To overcome this deficiency, the researchers have recently proposed many new analytic techniques. These techniques mainly address knowledge updating from three aspects: the expansion of data [1–7], the increasing number of attributes [8–11] and the variation of data values [12, 13]. For the first two aspects, a number of incremental techniques have been developed to acquire new knowledge without recomputation. However, little research has been done on the third aspect in knowledge acquisition, which motivates this study. This paper concerns attribute reduction for data sets with dynamically varying data values.

Feature selection, a common technique for data preprocessing in many areas including machine learning, pattern recognition and data mining, has hold great significance. Among various approaches to select useful features, a special theoretical framework is Pawlak’s rough set model [14, 15]. One can use rough set theory to select a subset of features that is most suitable for a given recognition problem [16–21]. Rough feature selection is also called attribute reduction, which aims to select those features that keep the discernibility ability of the original ones [22–26]. The feature subset generated by an attribute reduction algorithm is called a reduci. In the last two decades, researchers have proposed many reduction algorithms [27–32]. However, most of these algorithms can only be applicable to static data sets. In paper [33–40], several algorithms have been proposed for dynamic data sets. Here, we continue the research on the attribute reduction algorithm of dynamic data sets.

The remainder of this paper is organized as follows. Some preliminaries about rough set theory are reviewed in Section 2. In Section 3, a new form of conditional information quantity for decision systems is introduced, the properties of this information quantity are discussed. In Section 4, the updating mechanism of information quantity for decision systems are researched. Based on the conditional information quantity, an attribute reduction algorithm for decision systems with dynamically varying attribute values is constructed in Section 5.

2 Preliminaries

In this section, we first review some basic concepts in rough set theory, which can also be referred to [14, 15]. Throughout this paper, the universe U is assumed a finite nonempty set.

In rough set theory, knowledge is regarded as the classification ability of objects. Suppose we are given a finite set U ≠ ϕ of objects we are interested in. Any subset X⊆U will be called a concept or a category in U and any family of concepts in U will be referred to as abstract knowledge about U. We will be mainly interested in the concepts which form a partition and often use equivalence relations instead of classifications, since these two concepts are mutually interchangeable and relations are easier to deal with. Suppose R is an equivalence relation over U, then by U/R we mean the family of all equivalence classes of R, and [x]_R denotes an equivalence class of R containing the element x∈U. With each subset X⊆U, we associate two subsets:

R_X=∪{Y∈U/R|Y⊆X},R¯X=∪{Y∈U/R|Y∩X≠ϕ}

called the R–lower and R–upper approximations of X respectively. When R_X=R¯X, then X is called R–definable; otherwise X is called R–undefinable.

An information system, as a basic concept in rough set theory, provides a convenient framework for the representation of objects in terms of their attribute values.

An information system is a quadruple I S = (U, A, V, f), where: U is a set of finite and nonempty objects, called the universe; A is a nonempty finite set of attributes; V is the value domain of attributes; f is an information function which assigns particular values from domains of attributes to objects, such as ∀a ∈ A, x ∈ U, f(a, x) ∈ V, where f(a,x) denotes the value of attribute a on object x.

With every subset of attributes B ⊆ A, there is an associated equivalence relation ind(B) = {(x, y) ∈ U²|∀a ∈ B, f(a, x) = f(a, y)}. This equivalence relation ind(B) divides the universe U into a family of disjoint classes, the approximation space determined by the B-equivalence relation, denoted by π_B, is defined as: π_B = {X | X ∈ U / ind(B)}, where X is called a B-equivalence block and depicts the collection of objects that are indiscernible from each other with respect to B.

One type of special information system is called a decision system, which is denoted as DS = (U, C ∪ {d},V,f), where d is the decision attribute, C is the conditional attribute set. The positive region of d with respect to C is defined as POSC(d)=⋃X∈πdC_X.DS=(U,C∪{d},V,f) is called a consistent decision system, if POS_C(d) = U, else it is called an inconsistent decision system.

The consistent degree of adecision system DS = (U, C ∪ d, V, f) is defined asγ=|POSC(d)||U|.(1)

Obviously, a decision table is consistent if and only if its consistent degree γ is 1.

3 The information quantity for information systems and decision systems

In this section, we will use a new form of condition information quantity in decision system based on the equivalence relation. Some properties of the conditional information quantity will be given.

Definition 3.1

Given an information systemI S=(U, A, V, f)andP, Q ⊆ A, π_P = {P₁, P₂,..., P_s}is finer than π_Q = {Q₁,Q₂, ...Q_t} is defined as: for everyP_i ∈ π_P, there existsQ_j ∈ π_Q, such thatP_i ⊆ Q_j, denotesπ_P ⪯ π_Q. In this case, we also say thatπ_Qis coarser thanπ_P. Ifπ_P ⪯ π_Qandπ_P ≠ π_Q, we sayπ_Pis strictly finer thanπ_Q, denotesπ_P ≺ π_Q.

Obviously, if B ⊆ A, then π_A ⪯ π_B.

Definition 3.2

LetI S = (U, A, V, f)be an information system, if π_A = {X₁, X₂,..., X_n}, the information quantity of blockX_iis defined asI(X_i) = p(X_i)(1-p(X_i)); the information quantity ofπ_Ais defined asI(πA)=∑i=1np(Xi)(1−p(Xi)),wherep(Xi)=|Xi||U|,i=1,2,⋯,n.

Theorem 3.3

Let IS =(U,A,V,f) be an information system, if π_A = {X₁, X₂,..., X_n}, the information quantity ofπ_Asatisfies the following properties:

0≤I(πA)≤1−1n.
I(πA)=1−1nifandonlyifp(Xi)=1n(i=1,2,⋯,n).
I(π_A) = 0 if and only ifπ_A = {U}.
For eachX_i, X_j ∈ π_A, I(X_i)+ I(X_j) ≥ I(X_i ∪ X_j).
IfB ⊆ A, thenI(π_B) ≤ I(π_A), that is, the finer the partition, the bigger information quantity of it.

(1) I(πA)=∑i=1np(Xi)(1−p(Xi))=1−∑i=1np2(Xi),where∑i=1np(Xi)=1.

Now, we discuss the extreme value of ∑i=1np2(Xi) under restrained condition ∑i=1np(Xi)=1,letH(λ)=∑i=1np2(Xi)+λ(∑i=1np(Xi)−1). Since

H′λ(λ)=∑i=1np(Xi)−1=0H′p(Xi)(λ)=2p(Xi)+λ=0

We have p(Xi)=1n, that is, when p(Xi)=1n,∑i=1np2(Xi) gets to its minimum value 1n, so I(π_A) gets to its maximum value 1−1n, obviously, I(π_A)≥0. So (1) and (2) hold, (3) is obvious.

(4) Since Xi∩Xj=ϕ,sop(Xi∪Xj)=p(Xi)+p(Xj),I(Xi)+I(Xj)−I(Xi∪Xj)=p(Xi)(1−p(Xi))+p(Xj)(1−p(Xj))−p(Xi∪Xj)(1−p(Xi∪Xj))=2p(Xi)p(Xj)≥0.

(5) If B ⊆ A, then π_A ⪯ π_B, so each equivalence class of π_B is made up of one or more equivalence classes of π_A. Obviously, we can get π_B through combining two equivalence classes of π_A each time. From (4), we can get I(π_A) ≥ I(π_B).

Theorem 3.4

Let I S = (U, A, V, f) be an information system, if X, Y ⊆ U, then I(X) + I(Y) ≥ I(X ∪ Y).

Let Δ = I(X) + I(Y) - I(X ∪ Y), then

Δ=p(X)[1−p(X)]+p(Y)[1−p(Y)]−p(X∪Y)[1−p(X∪Y)]=p(X)+p(Y)−p(X∪Y)+[p(X∪Y)]2−[p(X)]2−[p(Y)]2=p(X∩Y)+[p(X)+p(Y)−p(X∩Y)]2−[p(X)]2−[p(Y)]2=p(X∩Y)+2p(X)p(Y)−2p(X)p(X∩Y)−2p(Y)p(X∩Y)+[p(X∩Y)]2=[p(X∩Y)−p(X)p(X∩Y)−p(Y)p(X∩Y)+p2(X∩Y)]+2p(X)p(Y)−p(X)p(X∩Y)−p(Y)p(X∩Y)=p(X∩Y)[1−p(X)−p(Y)+p(X∩Y)]+p(X)p(Y)−p(X)p(X∩Y)+p(X)p(Y)−p(Y)p(X∩Y)=p(X∩Y)[1−p(X∪Y)]+p(X)[p(Y)−p(X∩Y)]+p(Y)[p(X)−p(X∩Y)]

Since 0≤p(X)≤1,0≤p(Y)≤1,0≤p(X∪Y)≤1,andp(X)≥p(X∩Y),p(Y)≥p(X∩Y), so Δ ≥ 0, that is I(X) + I(Y) ≥ I(X ∪ Y).

This theorem demonstrates that if we combine two blocks, their information quality is lesser.

Definition 3.5

LetDS = (U, C ∪ {d}, V, f), X ⊆ U, π_d = {Y₁, Y₂...Y_n, the information quantity of blockXwith respective toπ_dis denoted asI(πd|X)=p(X)∑j=1np(Yj|X)(1−p(Yj|X)).

Definition 3.6

LetDS = (U, C ∪ {d}, V, f), π_C = {Y₁, Y₂...Y_nthe condition information quantity ofπ_Cwith respect toπ_dis defined asI(πd|πC)=∑i=1mp(Xi)∑j=1np(Yj|Xi)(1−p(Yj|Xi)),wherep(Xi)=|Xi||U|,i=1,2,⋯,m;p(Yj|Xi)=|Yj∩Xi||Xi|,j=1,2,⋯,n.

Theorem 3.7

LetDS = (U, C ∪ {d}, V, f), π_C = {Y₁, Y₂...Y_n, the condition information quantity ofπ_Cwith respect toπ_dsatisfies the following properties:

I(πd|Xi)+I(πd|Xj)≤I(πd|(Xi∪Xj))(i,j∈{1,2,⋯,m}).
I(πd|Xi)+I(πd|Xj)=I(πd|(Xi∪Xj))ifandonlyif,foreachk∈{1,2,n},|Yk⋂Xi||Xi|=|Yk⋂Xj||Xj|(i,j∈{1,2,⋯m}).
0≤I(πd|πC)≤I(πd).
Iπd|πC=0ifandonlyifπC≼πd.
fπC=U,thenIπd|πC=Iπd.

Proof

(1)

Δ=I(πd|(Xi∪Xj))−(I(πd|Xj)+I(πd|Xj))=p(Xi⋃Xj)∑k=1np(Yk|Xi⋃Xj)(1−p(Yk|Xi⋃Xj))−p(Xi)∑k=1np(Yk|Xi)(1−p(Yk|Xi))−p(Xj)∑k=1np(Yk|Xj)(1−p(Yk|Xj))=|Xi∪Xj||U|∑k=1n|Yk∩Xi|+|Yk∩Xj||Xi∪Xj|(1−|Yk∩Xi|+|Yk∩Xj||Xi∪Xj|)−|Xi||U|∑k=1n|Yk∩Xi||Xi|(1−|Yk∩Xi||Xi|)−|Xj||U|∑k=1n|Yk∩Xj||Xj|(1−|Yk∩Xj||Xj|)

=1|U|∑!k=1n((|Yk⋂Xj|+|Yk⋂Xj|)(1−|Yk∩Xi|+|Yk∩Xj||Xi∪Xj|)−|Yk⋂Xi|+|Yk∩Xi|2|Xi|−|Yk⋂Xj|+|Yk∩Xj|2|Xj|)=1|U|∑k=1n(|Yk∩Xi|2|Xi|+|Yk∩Xj|2|Xj|−(|Yk∩Xi|+|Yk∩Xj|)2|Xi∪Xj|).

Let |Xi|=x,|Xj|=y,|Yk∩Xi|=a,|Yk∩Xj|=b, denotes

fk=|Yk∩Xi|2|Xi|+|Yk∩Xj|2|Xj|−(|Yk∩Xi|+|Yk∩Xj|)2|Xi∪Xj|,

then

fk=a2x+b2y−(a+b)2x+y=a2y(x+y)+b2x(x+y)−(a+b)2xyxy(x+y)=a2xy+a2y2+b2x2+b2xy−a2xy−b2xy−2abxyxy(x+y)=(ay−bx)2xy(x+y)≥0.

So I(πd|Xi)+I(πd|Xj)≤I(πd|(Xi∪Xj)).

(2) According to the proof of (1), I(πd|Xi)+I(πd|Xj)=I(πd|(Xi∪Xj)) if and only if ay = bx, that is, if and only if for each k∈{1,2,⋯,n},|Yk∩Xi||Xi|=|Yk∩Xj||Xj|,thenI(πd|Xi)+I(πd|Xj)=I(πd|(Xi∪Xj)).

(3) Obviously, when πc≼πd,I(πd|πc)=0. According to (1), if we combine two condition classes, the information of condition class block with respective to π_d is increased, so I(πd|πC)≤I(πd|{U})=I(πd).

(5) According to definition of I(πd|πC), we can easily determine if πC={U},thenI(πd|πC)=I(πd). We must pay attention that the inverse of (5) doesn’t hold.

Example 3.8

LetDS = (U, C ∪ {d},V, f)be a decision system, U = {u₁u₂...,u₉}, π_d = {{u₁u₄u₇},{u₂u₅u₈}, {u₃u₆u₉}} π_C = {{u₁u₂u₃}, {u₄u₅u₆}, {u₇u₈u₉}}, obviouslyI(πd|{U})=I(πd|πC)=I(πd),butπC≠{U}.

Corollary 3.9

LetDS = (U, C ∪ {d},V, f)be a decision system, if B₁ ⊆ B₂ ⊆ C thenI(πd|πB2)≤(πd|πB1).

Corollary 3.10

LetDS = (U, C ∪ {d},V, f)be a decision system, C₁, C₂ ⊆ C, ifI(πd|πC1)=k1,I(πd|πC2)=k2,then(1)I(πd|π(C1∪C2))≤min(k1,k2);(2)I(πd|π(C1∩C2))≥max(k1,k2).

Proof

Since π(C1∪C2)≼πC1≼π(C1∩C2);π(C1∪C2)≼πC2≼π(C1∩C2), from Corollary 3.9, we have: I(πd|π(C1∪C2))≤min(k1,k2);I(πd|π(C1∩C2))≥max(k1,k2)

Definition 3.11

LetDS = (U, C ∪ {d},V, f), ifX∈π_C and |λ(X)|> 1 thenXis said to be an inconsistent equivalence block; otherwise, it is said to be a consistent equivalence block, whereλ(X)={f(u,d)|u∈X}and|λ(X)|is the cardinality of λ(X).

An inconsistent equivalence block describes a group of C -indistinguishable objects that have a divergence in their decision-making, while a consistent equivalence block depicts a collection of C -definable objects that share the same decision-making.

Definition 3.12

LetDS = (U, C ∪ {d},V, f), andX ∈ π_C, the inconsistent and consistent block families ofπ_Care denoted byπCinc={X∈πC||λ(X)|>1},πCcon={X∈πC||λ(X)|=1}respectively.

The inconsistent block family collects all of the inconsistent equivalence blocks from π_C, whereas the consistent block family gathers all of the consistent equivalence blocks from π_C. It is evident that πCinc∪πCcon=πCandπCinc∩πCcon=ϕ.

Theorem 3.13

LetDS=U,C∪d,V,f,πC={X1,X2,⋯,Xm},πd={Y1,Y2,⋯,Yn},X_j ∈ π_Cis a consistence block if and only if the information of blockX_iwith respective toπ_dis zero, that is, Xi∈!πCconiffI(πd|Xi)=p(Xi)∑j=1np(Yj|Xi)(1−p(Yj|Xi))=0.

Proof

Corollary 3.14

Let DS=U,C∪d,V,f,πC={X1,X2,⋯,Xm},πd={Y1,Y2,⋯,Yn},thenI(πd|πC)=∑Xi∈πCincp(Xi)∑j=1np(Yj|Xi)(1−p(Yj|Xi)).

Corollary 3.15

LetDS=U,C∪d,V,f,πC={X1,X2,⋯,Xm},πd={Y1,Y2,⋯,Yn},DS is consistent decision system iffI(πd|πC)=0.

Corollary 3.16

In decision systemDS = (U, C ∪ {d},V, f) , a ∈ Cis calledd-dispensable ifI(πd|π(C−{a}))=I(πd|πc).

Corollary 3.17

LetDS = (U, C ∪ {d}, V, f)be a consistent decision system, a ∈ Cisd-dispensableiff∀X∈πC−{a},I(πd|X)=0.

Corollary 3.18

LetDS = (U, C ∪ {d}, V, f)be a consistent decision system, condition attributeCis indispensable with respect todiff∀a∈C,∃X∈πC−{a},I(πd|X)

Definition 3.19

LetDS = (U, C ∪ {d}, V, f)be a decision system, B⊆C, ∀ a ∈ B, the significance measure (inner measure) ofainBis defined asSiginner(a,B,D)=I(πd|π(B−{a}))−I(πd|πB).

Definition 3.20

LetDS = (U, C ∪ {d}, V, f)be a decision system, B⊆C, ∀ a ∈ C-B, the significance measure (outer measure) ofainBis defined asSigouter(a,B,D)=I(πd|πB)−I(πd|π(B∪{a})).

Theorem 3.21

In decision systemDS = (U, C ∪ {d}, V, f) , a ∈ Cisd-dispensable, thenPOSC−{a}(d)=POSc(d).

Proof

On the one hand, according to Theorem 3.7, if we combine two condition classes of decision table, the condition information quantity will increase monotonically, and only if the two condition classes X_i and X_j satisfy |Yk∩Xi||Xi|=|Yk∩Xj||Xj|, for each Y_k ∈ π_d, then the condition information quantity doesn’t change, that is, if πc={X1,X2,Xi,Xj,Xn},πB={X1,X2,Xi−1,Xi+1,Xj−1,Xj+1,Xn,Xi∪

X_j}, and I(πd|πC)=I(πd|πB), then for each Y_k ∈ π_d we have |Yk∩Xi||Xi|=|Yk∩Xj||Xj|, that is POS_B(d)= POS_C(d).

On the other hand, since π_C ⪯ π_C–{a}, then π_C–{a} can be obtained through combining the classes of π_C. According to the analysis of the above, if I(πd|πC)=I(πd|πC−{a}), we must have POS_C(d)= POS_(C–{a})(d).

Definition 3.22

LetDS = (U, C ∪ {d} V, f) be a decision system, B ⊆ Cis a relative reduct ofCrelative to decision attributedd, if

(1) I(πd|πC)=I(πd|πB)

(2) ∀B′⊂B,I(πd|πB′)≠I(πd|πB)

Theorem 3.23

LetDS = (U, C ∪ {d} V, f) be a consistent decision system, thenB ⊆ Cis a relative reduct ofCrelative to decision attributeddif and only if (1) POSC(d)=POSB(d);(2)∀B′⊂B,POSB′(d)≠POSB(d).

4 Updating mechanism of information quantity for decision systems

Given a dynamic decision table, based on the information quantity, this section presents the updating mechanisms of the information quantity for dynamically varying data values.

Theorem 4.1

LetDS = (U, C ∪ {d} V, f) be a decision system, π_d = {Y₁,Y₂,...,Y_q,Y_n, ifx ∈ X(|X|> 1), x ∈ Y_q(q ∈ {1,2,...,n}), whenxexits out of this system, the new information quantity of blockX

I(πd′|X′)=|U||X|(|U|−1)(|X|−1)I(πd|X)−2(|Xp|−1)(|U|−1)(|Xp|−|Yq∩Xp).

whereX′=X−{x},πd′={Y1;Y2;⋯;Yq′=Yq−{x};⋯;Yn}.

Proof

I(πd′|X′)=P(X′)∑j=1nP(Yj|X′)(1−P(Yj|X′))=P(X′)∑j=1,j≠q′nP(Yj|X′)(1−P(Yj|X′))+P(X′)P(Yq′|X′)(1−P(Yq′|X′))=|X|−1|U|−1∑j=1,j≠q′|Yj∩X||X|−1(1−|Yj∩X||X|−1)+|X|−1|U|−1|Yq∩X|−1|X|−1(1−|Yq∩X|−1|X|−1)=|X||U|(|X|−1)(|U|−1)⋅|X||U|∑j=1,j≠q′|Yj∩X||X|(1−|Yj∩X||X|1−1|X|)+|X||U|(|X|−1)(|U|−1)⋅|X||U|[|Yq∩X||X|⋅(1−|Yq∩X||X|)−|X|−|Yq∩X||X|2]=|X||U|(|X|−1)(|U|−1)⋅|X||U|∑j=1,j≠q′|Yj∩X||X|(1−|Yj∩X||X|)−|X||U|(|X|−1)(|U|−1)⋅|X||U|∑j=1,j≠q′|Yj∩X||X|2+|X||U|(|X|−1)(|U|−1)⋅|X||U||Yq∩X||X|.(1−|Yq∩X||X|)

−1(|X|−1)(|U|−1)(|X|−|Yq∩X|)=|U||U|−1[|X|(|X|−1)|X||U|∑j=1,j≠q′|Yj∩X||X|(1−|Yj∩X||X|)−1(|X|−1)|U|(∑j=1n|Yj∩X|−|Yq∩X|)+|X||U|⋅|Yq∩X||X|(1−|Yq∩X||X|)+1|X|−1⋅|X||U|⋅|Yq∩X||X|(1−|Yq∩X||X|)−1(|X|−1)|U|(|X|−|Yq∩X=|U||U|−1[|X||X|−1|X||U|∑j=1,j≠q|Yj∩X||X|(1−|Yj∩X||X|)+|X||X|−1⋅|X||U|⋅|Yq∩X||X|(1−|Yq∩X||X|)−2(|X|−1)|U|(|X|−|Yq∩X=|U||U|−1⋅|X||X|−1I(πd|X)−2(|X|−1)|U|(|X|−|Yq∩X|)

Theorem 4.2

LetDS = (U,C ∪ {d}, V, f) be a decision system, π_d = {Y₁,Y₂,... ,Y_q,Y_n}, ifx ∉ X, x ∈ Y_q, (q ∈ {1,2,... ,n}), whenxexits out of this system, the new information quantity of blockX

I(πd′|X)=|U||U|−1I(πd|X).

whereπd′={Y1;Y2;⋯;Yq′=Yq−{x};⋯;Yn}.

Proof

I(πd′|X)=|X||U|−1∑j=1nP(Yj|X)(1−P(Yj|X))=|U||U|−1⋅|X||U|∑j=1nP(Yj|X)(1−P(Yj|X))=|U||U|−1I(πd|X).

Theorem 4.3

LetDS = (U,C ∪ {d}, V,f) be a decision system, π_C = {X₁,X₂,... ,X_m}, π_d = {Y₁,Y₂,... ,Y_n} ifx∈Xp(|Xp|>1),x∈Yq(p∈{1;2;⋯;m};q∈{1;2;⋯n})whenxexits out of this system, the new information quantity

I(πd′|πC′)=|U||U|−1[I(πd|πC)+1|Xp|−1I(πd|Xp)−2(|Xp|−1)|U|(|Xp|−|Yq∩Xp

whereπC′={X1;X2;⋯;Xp′=Xp−{x};⋯;Yn},πd′={Y1;Y2;⋯;Yq′=Yq−{x};⋯;Yn}.

Proof. According to Theorem 4.1 and Theorem 4.2, we have

I(πd′|πC′)=∑i=1mI(πd′|Xi)=∑i=1,i≠p′mI(πd′|Xi)+I(πd′|Xp′))=|U||U|−1∑i=1,i≠pmI(πd|Xi)+|U||Xp|(|U|−1)(|Xp|−1)I(πd|Xp)

−2(|Xp|−1)(|U|−1)(|Xp|−|Yq∩Xp=|U||U|−1∑i=1mI(πd|Xi)−|U||U|−1I(πd|Xp)+|U||Xp|(|U|−1)(|Xp|−1)I(πd|Xp)−2(|Xp|−1)(|U|−1)(|Xp|−|Yq∩Xp|)=|U||U|−1I(πd|πC)+|U|(|U|−1)(|Xp|−1)I(πd|Xp)−2(|Xp|−1)(|U|−1)(|Xp|−|Yq∩Xp|)=|U||U|−1[I(πd|πC)+1|Xp|−1I(πd|Xp)−2(|Xp|−1)|U|(|Xp|−|Yq∩Xp|)]

Corollary 4.4

LetDS = (U,C ∪ {d}, V, f) be a decision system, π_C = {X₁,X₂,...X_m}, π_d = {Y₁,Y₂,...Y_n}, ifx ∈ X_p (p ∈ {1,2,... ,m} andXp∈πCcon, whenxexits out of this system, thenI(πd′|πC′)=|U||U|−1I(πd|πC).

Theorem 4.5

LetDS = (U,C ∪ {d}, V, f) be a decision system, π_d = {Y₁,Y₂,... ,Y_n}, whenxgoes into this system, ifxadds toXandY_q(q ∈ {1,2,... ,n}) then

I(πd′|X′)=|U||X|(|U|+1)(|X|+1)I(πd|X)+2(|X|+1)(|U|+1)(|X|−|Yq∩X

whereX′=X∪{x},πd′={Y1;Y2;⋯;Yq′=Yq∪{x};⋯;Yn}.

Proof

I(πd′|X′)=P(X′)∑j=1nP(Yj|X′)(1−P(Yj|X))=P(X′)∑j=1,j≠q′nP(Yj|X′)(1−P(Yj|X′))+P(X′)P(Yq′|X′)(1−P(Yq′|X′))=|X|+1|U|+1∑j=1,j≠q′|Yj∩X||X|+1(1−|Yj∩X||X|+1)+|X|+1|U|+1|Yq∩X|+1|X|+1(1−|Yq∩X|+1|X|+1)=|X||U|(|X|+1)(|U|+1)⋅|X||U|∑j=1jj≠q′|Yj∩X||X|(1−|Yj∩X||X|+1|X|)+|X||U|(|X|+1)(|U|+1)⋅|X||U|[|Yq∩X||X|⋅(1−|Yq∩X||X|)+|X|−|Yq∩X||X|2]=|X||U|(|X|+1)(|U|+1)⋅|X||U|∑j=1,j≠q′|Yj∩X||X|(1−|Yj∩X||X|)

+|X||U|(|X|+1)(|U|+1)⋅|X||U|∑j=1,j≠q′|Yj∩X||X|2+|X||U|(|X|+1)(|U|+1)⋅|X||U||Yq∩X||X|.(1−|Yq∩X||X|)+1(|X|−1)(|U|−1)(|X|−|Yq∩X|)=|U||U|+1[|X|(|X|+1)|X||U|∑j=1,j≠q′|Yj∩X||X|(1−|Yj∩X||X|)+1(|X|+1)|U|(∑j=1n|Yj∩X|−|Yq∩X|)+|X||U|⋅|Yq∩X||X|(1−|Yq∩X||X|)+1|X|−1⋅|X||U|⋅|Yq∩X||X|(1−|Yq∩X||X|)+1(|X|+1)|U|(|X|−|Yq∩X=|U||U|+1[|X||X|+1|X||U|∑j=1,j≠q|Yj∩X||X|(1−|Yj∩X||X|)+|X||X|+1⋅|X||U|⋅|Yq∩X||X|(1−|Yq∩X||X|)+2(|X|+1)|U|(|X|−|Yq∩X=|U||U|+1⋅|X||X|+1I(πd|X)+2(|X|+1)|U|(|X|−|Yq∩X|)

Theorem 4.6

LetDS = (U, C ∪ {d}, V, f) be a decision system, π_C = {X₁,X₂,... ,X_m}, π_d = {Y₁,Y₂,... ,Y_n}, whenxadds to this system, then

I(πd′|πC′)=|U||U|+1[I(πd|πC)−1|Xp|+1I(πd|Xp)+2(|Xp|+1)|U|(|Xp|−|Yq∩Xp

whereπC′={X1;X2;⋯;Xp′=Xp∪{x};⋯;Xm},πd′={Y1;Y2;⋯;Yq′=Yq∪{x};⋯;Yn}.

Proof

According to Theorem 4.5, we have

I(πd′|πC′)=∑i=1mI(πd′m|Xi)=∑i=1,i≠p′mI(πd′|Xi)+I(πd′|Xp′))=|U||U|+1∑i=1,i≠p′mI(πd|Xi)+|U||Xp|(|U|+1)(|Xp|+1)I(πd|Xp)+2(|Xp|+1)(|U|+1)(|Xp|−|Yq∩Xp=|U||U|+1∑i=1mI(πd|Xi)−|U||U|+1I(πd|Xp)+|U||Xp|(|U|+1)(|Xp|+1)I(πd|Xp)+2(|Xp|+1)(|U|+1)(|Xp|−|Yq∩Xp|)

=|U||U|+1I(πd|πC)−|U|(|U|+1)(|Xp|+1)I(πd|Xp)+2(|Xp|+1)(|U|+1)(|Xp|−|Yq∩Xp|)=|U||U|+1[I(πd|πC)−1|Xp|+1I(πd|Xp)+2(|Xp|+1)|U|(|Xp|−|Yq∩Xp

Corollary 4.7

LetDS = (U, C ∪ {d}, V, f) be a decision system, π_C = {X₁,X₂,... ,X_m}, π_d = {Y₁,Y₂,... ,Y_n}, whenxgoes into this system, ifx ∈ X_p (p ∈ {1, 2,... ,m} andXp∈πCcon, thenI(πd′|πC′)=|U||U|+1I(πd|πC).

In the following, we discuss how the information quantity is changed if attribute values of one object x are varied.

Theorem 4.8

LetDS = (U, C ∪ {d}, V, f) be a decision system, πC={X1;X2;⋯;Xp1;⋯;Xp2;⋯;Xm},πd={Y1;Y2;Yq1;⋯;Yq2;⋯;Yn}.x∈Xp1,x∈Yq1(p1∈{1;2;⋯;m};q1∈{1;2;⋯,n}).If the objectxis changed to x′, and in the new decision system, πC′={X1DX2;Xp1′=Xp1−{x};⋯;Xp2′=Xp2∪{x′};Xm},πd′={Y1;Y2;⋯;Yq1′=Yq1−{x};Yq2′=Yq2∪{x′};Yn}, then

I(πd′|πC′)=I(πd|πc)+1(|Xp1|−1)|U|I(πd|Xp1)−2(|Xp1|−1)(|U|)(|Xp1|−|Yq1∩Xp1|)−1|Xp2|+1)|U|I(πd|Xp2)+2(|Xp2|+1)(|U|)(|Xp2|−|Yq2∩Xp2

Proof

When x changes to x′, and πC={X1,X2,⋯!,Xp1,⋯,Xp2,⋯,Xm},πd={Y1,Y2,⋯,Yq1,⋯,Yq2,⋯,Yn} turn into πC′={X1,X2,⋯Xp1′=Xp1−{x},⋯,Xp2′=Xp2∪{x′},⋯,Xm},πd′={Y1,Y2,⋯,Yq1′=Yq1−{x},⋯,Yq2′=Yq2∪{x′},⋯,Yn}.

This process can be divided into two steps: first, πC={X1,X2,⋯!,Xp1,⋯,Xp2,⋯,Xm},πd={Y1,Y2,⋯,Yq1,...,Yq2,...,Yn}; turn into πC″={X1,X2,⋯,Xp1′=Xp1−{x},⋯,Xp2,⋯,Xm},πd″={Y1,Y2,⋯,Yq1′=Yq1−{x},⋯,Yq2,⋯,Yn} then πC″={X1,X2,⋯,Xp1′,⋯,Xp2,⋯,Xm},πd″={Y1,Y2,⋯,Yq1′,⋯,Yq2,⋯,Yn} turn into πC′={X1,X2,⋯Xp1′,⋯,Xp2′=Xp2∪{x′},⋯,Xm},πd′={Y1,Y2,⋯,Yq1′,⋯,Yq2′=Yq2∪{x′},⋯,Yn}

According to Theorem 4.6, we have

I(πd″|πC″)=|U||U|−1[I(πd|πC)+1|Xp1|−1I(πd|Xp1)−2(|Xp1|−1)|U|(|Xp1|−|Yq1∩Xp1I(πd′|πC′)=|U|−1(|U|−1)+1[I(πd″|πC″)−1|Xp2|+1I(πd″|Xp2)+2(|Xp2|+1)(|U|−1)(|Xp2|−|Yq2∩Xp2

Since I(πd″|Xp2)=|U|(|U|−1)I(πd|Xp2).

Thus,

I(πd′|πC′)=I(πd|πC)+1(|Xp1|−1)|U|I(πd|Xp1)+2(|Xp1|+1)(|U|)(|Xp1|−|Yq1∩Xp1|)−2(|Xp2|+1)(|U|)I(πd|Xp2)+2(|Xp2|+1)(|U|)(|Xp2|−|Yq2∩Xp2|).

Corollary 4.9

LetDS = (U, C ∪ {d}, V, f) be a decision system, πC={X1,X2,⋯,Xp1,⋯,Xp2,⋯,Xm},πd={Y1,Y2,⋯,Yq1,⋯,Yq2,⋯,Yn}. When the object x is changed tox′,x∈Xp1,x′∈Xp2ifXp2∈πCcon,Xp2∈πCcon′,thenI(πd′|πC′)=I(πd|πC)

5 Attribute reduction algorithm for decision systems with dynamically varying attribute values

Based on the updating mechanisms of the information quantity, this section introduces an attribute reduction algorithm based on information quantity for decision systems with dynamically varying attribute values. In rough set theory, core is the intersection of all reducts of a given table, and core attributes are considered as the indispensable attributes in a reduct. First, we give an algorithm to obtain the core of a dynamically decision system.

Input: A decision system DS = (U, C ∪ {d}, V, f) and object x ∈ U is changed to x′.

Output: Core attribute CoreUx′, on Ux′ where Ux′ expresses the updated objects with x ∈ U changed to x′, and CoreUx′, is the core of decision system DS=(Ux′,C∪{d},V,f).

Step 1 When x is changed to x′, compute πc={X1,X2,⋯,Xp1,⋯,Xp2,⋯,Xm},πd={Y1,Y2,⋯,Yq1,⋯,Yq2,Yn} and πC′={X1,X2,⋯,Xp1−{x},⋯,Xp2∪{x′},⋯,Xm},πd′={Y1,Y2,⋯,Yq1−{x},⋯,Yq2∪{x′},Yn}

Step 2 compute I(πd′|πC′)

Step 3 CoreUx′=ϕ, for each a ∈ C

(1) compute πC−{a}={Z1,Z2,⋯,Zt1,⋯,Zt2,⋯,Zs} and πC−{a}′={Z1,Z2,⋯,Zt1−{x},⋯,Zt2∪{x′},⋯,Zs}

(2) compute I(πd′|πC−{a}′)

(3) If I(πd′|πC−{a}′)≠I(πd′|πC′) , then Coreux,=coreUx,∪{a}

Step 4 Return CoreUx′.

Based on updating mechanisms of the information quantity, an attribute reduction algorithm for decision systems with dynamically varying attribute values is proposed in the following. In this algorithm, the existing reduction result is one of inputs, which is used to find its new reduct after data changes.

Input: A decision system DS = (U, C ∪ {d}, V,f), reduct RED_U on U, and the changed object x which is changed to x′.

Output: Attribute reduct REDUx′, on Ux′.

Step 1 B=CoreUx′.

Step 2 Compute I(πd′|πB)=I(πd′|πC′,) then REDUx′=B, turn to step 4; else turn to step 3.

Step 3 While I(πd′|πB)≠I(πd′|πC′) do

{For each a ∈ C 舑 B, computer sigUx′outera,B,d;

Select a0=max{sigUx′outera,b,d},a∈C−B;

B←B∪{a0}.

}

Step 4 For each a ∈ B do

{compute sisigUx′outera,B,d;

If sigUx′innera,B,d=0, then B←B−{a}.};

Step 5 REDUx′=B, return REDUx′ and end.

In this algorithm, firstly we obtain the core of dynamic information system; then, we add the attribute with the biggest significance gradually till the REDUx′

6 Conclusion

The incremental technique is an effective way to maintain knowledge in the dynamic environment. An attribute selection for dynamic data sets is still a challenging issue in the field of artificial intelligence. In this paper, we put forward the information quantity for information systems and decision systems according to the information measure proposed by Professor Hu Guoding, and we also discuss the updating mechanism of information quantity for decision systems. Further, we give an attribute reduction algorithm for decision tables with dynamically varying attribute values. It should be pointed out that updating mechanisms of the information quantity introduced in this paper are only applicable when data are varied one by one, whereas many real data may vary in groups in application. This gives rise to many difficulties for the proposed feature selection algorithm to deal with. In our further work, we will focus on improving the incremental algorithm for updating knowledge by varying some objects simultaneously. Furthermore, as an decision system consists of the objects, the attributes, and the domain of attributes values, all of the elements in the decision system will change as time goes by under the dynamic environment. In the future, the variation of attributes and the domain of attributes values in decision system will also be taken into consideration in terms of incremental updating knowledge.

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Received: 2016-6-2

Accepted: 2016-8-25

Published Online: 2016-11-27

Published in Print: 2016-1-1

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