Representation of Rough Sets Based onIntuitionistic Fuzzy Special Sets

Zheng Pei, Li Zhang, and Honghua Chen

School of Mathematics & Computer, Xihua University,Chengdu, Sichuan, 610039, China

pqyz@263.net

Abstract. Intuitionistic fuzzy special sets is a special case of intuitionis-tic fuzzy sets. In this paper, under the framework of information systems,the relationship between intuitionistic fuzzy special sets and rough sets isanalyzed. Based on basic intuitionistic fuzzy special sets of informationsystems, intuitionistic fuzzy special σ-algebra are generated, and roughsets are embedded in the intuitionistic fuzzy special σ-algebra. Naturally,distances (e.g., Hamming distance or Euclidean distance) of intuitionisticfuzzy special sets in intuitionistic fuzzy special σ-algebra can be used toevaluate predication rules of information systems which is an importantsubject of rough set theory.

1 Introduction

As generalization of fuzzy sets, intuitionistic fuzzy sets use the degree of mem-bership and nonmembership of object x [1]-[10]. In some cases, intuitionisticfuzzy sets has more advantages than classical fuzzy sets in describing uncertainconcepts. Rough sets theory (RST) proposed by Z. Pawlak is an important the-ory for data mining [11], [12]. In the process of uncertain information, fuzzy setstheory and rough sets theory both have advantages, respectively.

Formally, intuitionistic fuzzy subset A is A = {(x, μA(x), νA(x))|x ∈ X}, inwhich, X is domain, μA : X → [0, 1] and νA : X → [0, 1] are membershipfunction and nonmembership function of object x in A such that ∀x ∈ X , 0 ≤μA(x)+νA(x) ≤ 1. Intuitionistic fuzzy special subset (IFSS) is A = {X, A1, A2},in which, X �= ∅, A1 ⊆ X , A2 ⊆ X and A1 ∩ A2 = ∅.

1. ∀B ⊂ X , define B′ = 〈X, B, Bc〉 (Bc is the complement of B in X), thenB ←→ B′. This means that IFSS is extension of classical subset.

2. Let A = {X, A1, A2} be IFSS, define

μA1(x) ={

1 if x ∈ A10 otherwise, νA2(x) =

{1 if x ∈ A20 otherwise, (1)

then {(x, μA1(x), νA2(x))|x ∈ X} such that ∀x ∈ X , due to A1 ∩ A2 = ∅,μA1(x)+ νA2(x) = 1 or 0, hence, IFSS is a special case of intuitionistic fuzzysets.

P. Melin et al. (Eds.): IFSA 2007, LNAI 4529, pp. 114–121, 2007.c© Springer-Verlag Berlin Heidelberg 2007

Representation of Rough Sets Based on Intuitionistic Fuzzy Special Sets 115

In IFSS, the following operations can be defined [4]: Let A = 〈X, A1, A2〉, B =〈X, B1, B2〉 and {Ai|i ∈ J}, where Ai = 〈X, A1

i , A2i 〉 be IFSS on X , then

1. A ⊂ B if and only if A1 ⊂ B1 and A2 ⊃ B2;2. A = B if and only if A ⊂ B and A ⊃ B;3. A = 〈X, A2, A1〉;4. ∅− = 〈X, ∅, X〉, X− = 〈X, X, ∅〉;5. ∪Ai = 〈X, ∪A1

i , ∩A2i 〉, ∩Ai = 〈X, ∩A1

i , ∪A2i 〉;

6. A − B = A ∩ B.

Based on the above operations of IFSS, intuitionistic fuzzy special σ-algebra canbe defined as following

Definition 1. [4] Intuitionistic fuzzy special σ-algebra Φ on X is such that

a) X− ∈ Φ;b) If A ∈ Φ, then A ∈ Φ;c) If ∀n ∈ N , A1, · · · , An ∈ Φ, then ∪n

i=1Ai ∈ Φ.

2 Representation of Rough Set Based on IFSS

Rough sets are defined on information systems. Formally, an information systemis expressed as a quaternion denoted as (U, A, V, f), where U is a non-empty setof objects, A is a non-empty finite set of attributes, V =

⋃a∈A Va and Va is the

domain of a, f : U × A → V is information function. In (U, A, V, f), ∀a ∈ Aand xi, xj ∈ U , define xi ∼a xj if and only if f(x1, a) = f(x2, a), then ∼a is anequivalence relation on U . As we known, intersection of equivalence relations isalso equivalence relation, hence, ∼A, which is intersection of all ∼a (a ∈ A), isan equivalence relation on U , denotes U/ ∼A= {Uk|k = 1, · · · , n}, where Uk isan equivalence class. Based on U/ ∼A, ∀X ⊆ U , define

X =⋃

{Uk ∈ U/ ∼A |Uk ⊆ X}, X =⋃

{Uk ∈ U/ ∼A |Uk ∩ X �= ∅}, (2)

when X �= X, (X, X) is ∼A rough set. In (2), due to X ⊆ X, X ∩ (U − X) = ∅is obviously. Hence, 〈U, X, U − X〉 is IFSS based on ∼A, this means that in theframework of U/ ∼A, ∀X ⊆ U , there is the following one to one mapping

(X, X) ←→ 〈U, X, U − X〉. (3)

Definition 2. Let (U, A, V, f) be an information system, ∀X ⊆ U , 〈U, X, U−X〉is called IFSS representation of (X, X).

As a special case, (U i, U i) ←→ 〈U, U iθ, (U

iθ)

c〉 = 〈U, U iθ,

⋃j �=i U j

θ 〉.

Definition 3. Let (U, A, V, f) be an information system, U/ ∼A= {Uk|k =1, · · · , n}. If 〈U, Uk1 , Uk2〉 such that Uk1 , Uk2 ∈ U/ ∼A and k1 �= k2, then〈U, Uk1 , Uk2〉 is called basic IFSS based on ∼A. Denote B∼A = {〈U, Uk1 , Uk2〉|Uk1 ,Uk2 ∈ U/ ∼A, k1 �= k2}.

116 Z. Pei, L. Zhang, and H. Chen

In the Definition, for a fixed (U, A, V, f), if |U/ ∼A | = n, then |B∼A | = n(n−1).Let a class of recursive set Φ(B∼A) such that

1. ∀〈U, Uk1 , Uk2〉 ∈ B∼A , 〈U, Uk1 , Uk2〉 ∈ Φ(B∼A);2. If 〈U, A1, B1〉, 〈U, A2, B2〉 ∈ Φ(B∼A), then

〈U, A1, B1〉 ∩ 〈U, A2, B2〉 = 〈U, A1 ∩ A2, B1 ∪ B2〉 ∈ Φ(B∼A).

3. If 〈U, A1, B1〉 ∈ Φ(B∼A), then 〈U, A1, B1〉 = 〈U, B1, A1〉 ∈ Φ(B∼A).

Property 1. ∀〈U, A1, B1〉 ∈ Φ(B∼A), A1 ∩ B1 = ∅, i.e., 〈U, A1, B1〉 is IFSS.

Proof. According to structure of 〈U, A1, B1〉 ∈ Φ(B∼A) and 〈U, Uk1 , Uk2〉 =〈U, Uk2,Uk1 〉 ∈ B∼A

a) If there exists 〈U, Uk1 , Uk2〉 ∈ B∼A such that 〈U, A1, B1〉 = 〈U, Uk1 , Uk2〉,then A1 ∩ B1 = ∅, 〈U, A1, B1〉 is IFSS.

b) If there exist 〈U, Uk1 , Uk′1〉, · · · , 〈U, Ukm , Uk′

m〉 ∈ B∼A such that 〈U, A1, B1〉=

⋂mi=1〈U, Uki , Uk′

i〉, ∀i, ki �= k′i, then

A1 ∩ B1 = (m⋂

i=1

Uki)⋂

(m⋃

i=1

Uk′i) =

m⋃j=1

((m⋂

i=1

Uki) ∩ Uk′j ),

in (⋂m

i=1 Uki) ∩ Uk′j , due to Ukj ∩ Uk′

j = ∅, hence, every (⋂m

i=1 Uki) ∩ Uk′j = ∅,

〈U, A1, B1〉 is IFSS.c) If there exist 〈U, A1, B1〉, 〈U, A2, B2〉 ∈ Φ(B∼A) such that 〈U, A1, B1〉 =⋂m

i=1〈U, Uki , Uk′i〉, 〈U, A2, B2〉 =

⋂rj=1〈U, U lj , U l′j 〉 and 〈U, A3, B3〉 = 〈U, A1, B1〉

∩〈U, A2, B2〉, then

A3 ∩ B3 = (A1 ∩ A2)⋂

(B1 ∪ B2) = ((A1 ∩ A2) ∩ B1)⋃

((A1 ∩ A2) ∩ B2),

according to b), A3 ∩ B3 = ∅, 〈U, A3, B3〉 is IFSS.d) If there exist 〈U, A1, B1〉, 〈U, A2, B2〉 ∈ Φ(B∼A) such that 〈U, A3, B3〉 =

〈U, A1, B1〉 ∩ 〈U, A2, B2〉 or 〈U, A3, B3〉 = 〈U, A1, B1〉 ∩ 〈U, A2, B2〉, then

A3 ∩ B3 = (B1 ∩ A2)⋂

(A1 ∪ B2) = ((B1 ∩ A2) ∩ A1)⋃

((B1 ∩ A2) ∩ B2), or

A3 ∩ B3 = (B1 ∩ B2)⋂

(A1 ∪ A2) = ((B1 ∩ B2) ∩ A1)⋃

((B1 ∩ B2) ∩ A2),

according to c), A3 ∩ B3 = ∅, 〈U, A3, B3〉 is IFSS.

Theorem 1. Φ(B∼A) is an intuitionistic fuzzy special σ-algebra generated byB∼A .

Proof. According to Φ(B∼A), a) and c) of Definition 1 need to be proved. Fixedk1, k

′1 ∈ N , then ∀k2(�= k1) ∈ N and ∀k′

2(�= k′1) ∈ N , 〈U, Uk1 , Uk2〉 ∈ B∼A and

〈U, Uk′1 , Uk′

2〉 ∈ B∼A , hence,n⋂

k2 �=k1,k2=1

〈U, Uk1 , Uk2〉 = 〈U, Uk1 ,n⋃

k2 �=k1,k2=1

Uk2〉 ∈ Φ(B∼A),

n⋂k′2 �=k′

1,k′2=1

〈U, Uk′1 , Uk′

2〉 = 〈U, Uk′1 ,

n⋃k′2 �=k′

1,k′2=1

Uk′2〉 ∈ Φ(B∼A).

Representation of Rough Sets Based on Intuitionistic Fuzzy Special Sets 117

Due to Uk1 ∩ Uk′1 = ∅ and

(n⋃

k2 �=k1,k2=1

Uk2)⋃

(n⋃

k′2 �=k′

1,k′2=1

Uk′2) = U,

〈U, ∅, U〉 = 〈U, Uk1 ,

n⋃k2 �=k1,k2=1

Uk2〉⋂

〈U, Uk′1 ,

n⋃k′2 �=k′

1,k′2=1

Uk′2〉 ∈ Φ(B∼A),

U− = 〈U, U, ∅〉 = 〈U, ∅, U〉 ∈ Φ(B∼A).

For c) of Definition 1, due to

〈U, Uk1 , Uk2〉⋃

〈U, Uk′1 , Uk′

2〉 = 〈U, Uk1 , Uk2〉⋂

〈U, Uk′1 , Uk′

2〉 ∈ Φ(B∼A).

Property 2. In Φ(B∼A),

1. 〈U, ∅, ∅〉 ∈ Φ(B∼A);2. ∀U i ∈ U/ ∼A, 〈U, U i, (U i)c〉 ∈ Φ(B∼A);3. ∀X ⊆ U , 〈U, ∅, X〉, 〈U, ∅, X〉 ∈ Φ(B∼A);4. ∀X ⊆ U , 〈U, X, U − X〉 ∈ Φ(B∼A).

Proof. Let X = {Uk′1 , · · · , Uk′

m}, then

〈U, ∅, X〉 =m⋂

i=1

〈U, Uki , Uk′i〉 = 〈U,

m⋂i=1

Uki ,

m⋃i=1

Uk′i〉, (4)

in which, ∃i and j such that ki �= kj . The others can be proved similarly.

Based on intuitionistic fuzzy special σ-algebra Φ(B∼A), measures on Φ(B∼A)can be defined.

Definition 4. [13] ∀A = 〈U, A1, A2〉 ∈ Φ(B∼A), define μ : Φ(BUθ) → [0, ∞) asfollowing:

μ(A) = 1 +|A1||U | − |A2|

|U | , (5)

then μ is a measure on Φ(B∼A), where |X | is cardinality of X.

For 〈U, A1, B1〉, 〈U, A2, B2〉 ∈ Φ(B∼A), the distance between 〈U, A1, B1〉 and〈U, A2, B2〉 can be defined similarly as in intuitionistic fuzzy sets [14]-[20], e.g.,Hamming distance and Euclidean distance,

d1 =∑x∈U

|μA1(x) − μA2(x)| + |νB1(x) − νB2(x)| + |πC1(x) − πC2(x)|, (6)

d2 =√∑

x∈U

(μA1(x) − μA2(x))2 + (νB1(x) − νB2(x))2 + (πC1(x) − πC2(x))2(7)

where, πC1(x) = 1−μA1(x)− νB1(x) and πC2(x) = 1−μA2(x)− νB2(x). In intu-itionistic fuzzy sets, μ∗, μ∗ and π∗ are fuzzy sets. In intuitionistic fuzzy specialsets, μ∗, μ∗ and π∗ are characteristic functions of ∗, i.e., the forms of (1).

118 Z. Pei, L. Zhang, and H. Chen

3 Prediction Based on the Distance of IFSS

As a special case of information system, decision information systems 〈U, Ω, V, f〉are widely used in application. In decision information systems, attributes Ω aredivided by two parts: one is called condition attribute set, denoted by Q; theother is called decision attribute set, denoted by D. From real world applicationpoint of view, the prediction problem is expressed as following [12]

– given a decision attribute d ∈ D, which is the “best” attribute set C ⊆ Qto predict the d-value of an object x ∈ U , given the values of x under thefeatures contained in Q?

The prediction problem raises two questions:

– Which subsets C of Q are candidates to be such a “best attribute set”?– What should a metric look like to determine and select the “best attribute

set”?

In this paper, the distance of IFSS is used to solve the prediction problem.Suppose that all information of decision information system I = 〈U, Q∪D, V, f〉are known, let C = {c1, · · · , ck} ⊆ Q and d ∈ D, decision rule is expressed as

R : c1 ∧ · · · ∧ ck −→ d. (8)

According to the above discussions, equivalence relations ∼C , ∼d and ∼C∪{d}on U can be obtained in 〈U, C ∪ {d}, V, f〉, denote

U/ ∼C = {U1C , · · · , Um

C }, U/ ∼d= {U1d , · · · , Un

d }, (9)U/ ∼C∪{d} = {U1

C∪{d}, · · · , UpC∪{d}}. (10)

For every Um′

C ∈ U/ ∼C and Un′

d ∈ U/ ∼d, based on U/ ∼C∪{d}, their rough set,i.e., (Um′

C , Um′C ) and (Un′

d , Un′d ), can be obtained, respectively. By (3), IFSS repre-

sentations of (Um′

C , Um′C ) and (Un′

d , Un′d ), i.e., 〈U, Um′

C , U −Um′C 〉 and 〈U, Un′

d , U −Un′

d 〉, can be obtained, respectively. Based on U/ ∼C∪{d}, (9) can be rewrittenas following

U/ ∼C = {〈U, U1C , U − U1

C〉, · · · , 〈U, UmC , U − Um

C 〉}, (11)

U/ ∼d = {〈U, U1d , U − U1

d 〉, · · · , 〈U, Und , U − Un

d 〉}. (12)

For every 〈U, Um′

C , U − Um′C 〉 ∈ U/ ∼C and 〈U, Un′

d , U − Un′d 〉 ∈ U/ ∼d, using

(6) and (7), their Hamming distance dHm′n′ and Euclidean distance dE

m′n′ can becalculated, respectively, denote

dHC,d =

∑mm′=1

∑nn′=1 dH

m′n′

mn, (13)

dEC,d =

∑mm′=1

∑nn′=1 dE

m′n′

mn. (14)

Representation of Rough Sets Based on Intuitionistic Fuzzy Special Sets 119

(13) and (14) are average values of Hamming distances and Euclidean distancesabout rule R : c1 ∧ · · · ∧ ck −→ d, respectively.

From the standpoint of logical systems, c1∧· · ·∧ck −→ d is logical proposition.Considering information system 〈U, C ∪{d}, V, f〉, c1 ∧ · · · ∧ ck −→ d is meaningwhen all objects, which have attributes c1, · · · , ck−1 and ck, are asserted to ownattribute d. From the standpoint of information theory, if c1 ∧ · · · ∧ ck −→ d

is considered as knowledge, then 〈U, Um′

C , U − Um′C 〉 ∈ U/ ∼C and 〈U, Un′

d , U −Un′

d 〉 ∈ U/ ∼d can be regarded as certain information about objects have or notattributes {c1, · · · , ck} and {d} based on the knowledge, respectively. Hence, fromobjects point of view, dH

m′n′ (or dEm′n′) express similarity degree of the certain

information between conditions and conclusion of c1 ∧ · · · ∧ ck −→ d. In thispaper, dH

C,d (or dEC,d) are selected as evaluation index of rule c1 ∧ · · · ∧ ck −→ d,

i.e., for a fixed decision attribute d, the “best” attribute set C ⊆ Q is such that

dHC,d = min{dH

C1,d, · · · , dHCs,d}, (15)

dEC,d = min{dE

C1,d, · · · , dECs,d}, (16)

in which, C1, · · · , Cs ⊆ Q are candidates to predict d.Example 1. Table 1 [12] is a heart disease diagnosis information system, thecondition attributes are S: smoker and BMI: avoirdupois, the decision attributeis HD: heart disease.

Table 1. Heart disease diagnosis information system.

No S BMI HD1 no normal no2 no obese no3 no normal no4 no obese no5 yes normal yes6 yes normal yes7 yes obese no8 yes obese yes9 no normal no

In Table 1, the following equivalence classes can be obtained: {S} : {1, 2, 3, 4,9}, {5, 6, 7, 8}; {BMI} : {1, 3, 5, 6, 9}, {2, 4, 7, 8}; {S, BMI} : {1, 3, 9}, {2, 4},{5, 6}, {7, 8}; {HD} : {1, 2, 3, 4, 7, 9}, {5, 6, 8}.

There are three candidates to predict HD, i.e.,

S −→ HD, BMI −→ HD, S ∧ BMI −→ HD.

For S −→ HD, the equivalence classes are {1, 2, 3, 4, 9}, {5, 6, 8} and {7},according to (11) and (12), the following can be obtained

U/ ∼S = {〈U, {1, 2, 3, 4, 9}, {5, 6, 7, 8}〉, 〈U, {5, 6, 7, 8}, {1, 2, 3, 4, 9}〉},U/ ∼HD = {〈U, {1, 2, 3, 4, 7, 9}, {5, 6, 8}〉, 〈U, {5, 6, 8}, {1, 2, 3, 4, 7, 9}〉},

120 Z. Pei, L. Zhang, and H. Chen

By (6) and (7),

dH11(〈U, {1, 2, 3, 4, 9}, {5, 6, 7, 8}〉, 〈U, {1, 2, 3, 4, 7, 9}, {5, 6, 8}〉) = 2,

dE12(〈U, {1, 2, 3, 4, 9}, {5, 6, 7, 8}〉, 〈U, {1, 2, 3, 4, 7, 9}, {5, 6, 8}〉) =

√2,

the others can be calculated similarly,

dHS,HD =

dH11 + dH

12 + dH21 + dH

22

4= 9,

dES,HD =

dH11 + dH

12 + dH21 + dH

22

4=

√2 + 42

.

For BMI −→ HD and S ∧ BMI −→ HD, average values of Hammingdistances and Euclidean distances can be obtained similarly, respectively, seeTabel 2.

Table 2. Average values of prediction rules

Hamming distance Euclidean distance{S} → {HD} 9

√2+42

{BMI} → {HD} 212

2√

2+2√

10+√

144

{S, BMI} → {HD} 9 3√

2+4√

3+2√

6+√

10+48

According to Table2, if using Hamming distance, then the “best” attribute set,which is used to predict HD, is {S} or {S, BMI}. If using Euclidean distance,then the “best” attribute set is {S}. The conclusion is same as in [12].

4 Conclusion

In this paper, under the framework of information systems, intuitionistic fuzzyspecial σ-algebra are generated by basic IFSS of information systems, and roughsets are embedded in the intuitionistic fuzzy special σ-algebra. Hamming dis-tance (or Euclidean distance) of intuitionistic fuzzy special sets in intuitionisticfuzzy special σ-algebra are used to evaluate predication rules of informationsystems.

Acknowledgments

This work was supported by the youth foundation of sichuan province (grantno. 06ZQ026-037) and the education department foundation of sichuan province(grant no. 2005A121, 2006A084).

Representation of Rough Sets Based on Intuitionistic Fuzzy Special Sets 121

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