Consistency p-measure and fuzzy p-measure in decision tables
Yong Chan Kim1and Jung Mi Ko2
Department of Mathematics, Gangneung-Wonju National University, Gangneung, 201-702, Korea
Abstract
In this paper, we introduce the notions of inconsistency p-measure, consistency p-measure and fuzziness p-measure. We discuss various properties of them. We investigate the degree of inconsistency and roughness p-measure in a decision table.
Key words : Decision tables; Inclusion degree; (In)consistency p-measure; Fuzziness p-measure
1. Introduction and preliminaries
Rough set theory was introduced by Pawlak [4-5] to generalize the classical set theory. There has been a rapid growth in interest in rough set theory in recent years [1,2,9- 14]. Its applications are decision system modeling and analysis of complex systems, fuzzy sets, neural networks, evolutionary computing, data mining and knowledge dis- covery, pattern recognition, etc. Recently, Qian [9-10] fo- cused on how to measure the consistencies of a decision table and fuzziness of rough set.
In this paper, we introduce the notions of inconsis- tency p-measure, consistency p-measure and fuzziness p- measure which is generalized p = 2 in a sense Qian [9].
We discuss various properties of them. We investigate the degree of inconsistency and roughness p-measures in a de- cision table.
Definition 1.1. [4-8] The pair (X, AT ) is called an infor- mation system if X is a non-empty finite set of objects and AT is a non-empty finite set of attributes. The triple (X, AT, d) is called a decision table if (X, AT ) is an in- formation system and d is a decision attribute set. For a ∈ P ⊂ AT ∪ d, a map a : X → Va, where Va is the value set of a. We define an equivalence relation X/P = {[x]P | x ∈ X} where
[x]P = {y ∈ X | a(x) = a(y), ∀a ∈ P }.
Definition 1.2. [9] Let (X, ≤) be a poset. A function D : X ×X → [0, 1] is called a measure of inclusion degree if it satisfies the following conditions:
(1) 0 ≤ D(a, b) ≤ 1,
(2) if a ≤ b, then D(a, b) = 1,
(3) if c ≤ a ≤ b, then D(a, c) ≥ D(b, c).
If we modify the condition (3) as
(3)* if a ≤ b, then D(a, c) ≥ D(b, c) for all a ∈ X, D is called a S1-strong inclusion degree.
If D is an inclusion degree and satisfies the condition:
(4) if b ≤ c, then D(a, b) ≤ D(a, c) for all a ∈ X, D is called a S2-strong inclusion degree.
Example 1.3. Let X be a finite set. We define functions Di : P (X) × P (X) → [0, 1] for i = 1, 2, 3, 4 as follows:
D1(A, B) =
( |A∩B|
|A| if A 6= ∅,
1 if A = ∅.
D2(A, B) =
( 1 if A ⊂ B,
|Bc|
|X| if A 6⊂ B.
D3(A, B) =
( 1 if A ⊂ B,
|A|
|X| if A 6⊂ B.
D4(A, B) = |Ac∪ B|
|X| .
(1) D1 is a S2-strong inclusion degree but not a S1- strong inclusion degree because
D1(B, C) = |B ∩ C|
|B| =2
3 6≤ D1(A, C) = |A ∩ C|
|A| = 1 2
where A = {2, 5}, B = {2, 3, 5}, C = {1, 2, 3, 4}.
(2) D2 is a S1-strong inclusion degree but not a S2- strong inclusion degree because, X = {1, 2, 3, 4, 5},
D2(A, B) = |Bc|
|X| = 3
5 6≤ D2(A, C) = |Cc|
|X| =2 5
+
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where A = {1, 2, 3, 4}, B = {1, 2}, C = {1, 2, 3}.
(3) D3 is a S2-strong inclusion degree but not a S1- strong inclusion degree because, X = {1, 2, 3, 4, 5},
D3(A, C) = |A|
|X| = 2
5 6≥ D3(B, C) = |B|
|X| =3 5
where A = {1, 2}, B = {1, 2, 3}, C = {1}.
(4) D4is a S1and S2-strong inclusion degree.
2. Consistency p-measure and fuzzy p-measure in decision tables
Let F (X) = {A : X → [0, 1] is function} be given.
For A ∈ F (X), x ∈ X, δA(x) = A(x).
Definition 2.1. [9] A function e : F (X) → [0, 1] is called an entropy on F (X) if it satisfies the following conditions:
(1) e(A) = 0 iff A ∈ P (X);
(2) e(A) =W
B∈F (X)e(B) iff A = 0.5;
(3) if A ≤ B for A ≥ 12, then e(B) ≤ e(A);
(4) if B ≤ A for A ≤ 12, then e(B) ≤ e(A);
(5) e(A) = e(Ac), for A ∈ F (X).
Definition 2.2. Let (X, AT, d) be a decision table. E ∈ X/AT an equivalence class and X/d = {[x]d| x ∈ X}.
(1) A inconsistency p-measure of E with respect to d is defined as:
Mp(δE) = 1
|X|
X
x∈X
(−2p|δE(x) −1
2|p+ 1) (p > 0)
where δE(x) = |E∩[x]|E|d|.
(2) A consistency p-measure of E with respect to d is defined as:
Cp(δE) = 1 − 1
|X|
X
x∈X
(−2p|δE(x) −1
2|p+ 1) (p > 0).
(3) A consistency p-measure of AT with respect to d is defined as:
C(AT, d) = Xm
j=1
|Ej|
|X|
³
1 − Mp(δE)
´
(p > 0)
where X/AT = {Ej | j = 1, ..., m}.
If A ⊂ B, A is consistent with respect to B. If δE(x) = 1, then E is consistent with respect to [x]d, that is, E ⊂ [x]d.
Let F (X) be the class of all fuzzy sets of X. For A ∈ F (X), x ∈ X, δA(x) = A(x).
Remark 2.3. In above definition, p = 2 we obtain defini- tions in a sense Quan [9] as follows:
(1) An inconsistency 2-measure of E with respect to d is defined as:
M2(δE) = 4
|X|
X
x∈X
δE(x)(1 − δE(x))
(2) A consistency 2-measure of E with respect to d is defined as:
C(δE) = 1 − 4
|X|
X
x∈X
δE(x)(1 − δE(x))
(3) A consistency 2-measure of AT with respect to d is defined as:
C(AT, d) = Xm
j=1
|Ej|
|X|
³ 1 − 4
|X|
X
x∈X
δE(x)(1 − δE(x))´
Theorem 2.4. An inconsistency p-measure Mp: F (X) → [0, 1] is an entropy on F (X).
Proof. (1) If A ∈ P (X), then δA(x) = A(x) = 0 or δA(x) = A(x) = 1. Since −2p|δA(x) − 12|p + 1 = 0, then Mp(A) = 0.
Let Mp(A) = |X|1 P
x∈X(−2p|δA(x) − 12|p+ 1) =
1
|X|
P
x∈X(−2p|A(x) −12|p+ 1) = 0. Then A(x1) = 1;
i.e, x ∈ A or A(x) = 0; i.e, x 6∈ A. Hence A ∈ P (X).
(2) Since 0 ≤ δA(x) = A(x) ≤ 1, we have Mp(A) = _
B∈F (X)
1
|X|
X
x∈X
(−2p|δB(x) −1
2|p+ 1) = 1
Thus, δB(x) = B(x) = 12 for all x ∈ X. Conversely, Mp(0.5) =W
B∈F (X)Mp(B)=1.
(3) If δA(x) ≤ δB(x) for δA(x) ≥ 12, then Mp is a decreasing function. Hence M (B) ≤ M (A).
(4) If δA(x) ≤ δB(x) for δB(x) ≤ 12, then Mp is an increasing function. Hence Mp(A) ≤ Mp(B).
(5)
Mp(Ac) = |X|1 P
x∈X(−2p|δAc(x) −12|p+ 1)
=|X|1 P
x∈X(−2p|Ac(x) − 12|p+ 1)
=|X|1 P
x∈X(−2p|1 − A(x) −12|p+ 1)
=|X|1 P
x∈X(−2p|A(x) − 12|p+ 1)
= Mp(A).
Theorem 2.5. Let (X, AT, d) be a decision table. If A is a consistence set, then an inconsistency p-measure Mp(δA) = 0.
Proof. Let A ∈ X/AT and X/d = {[x]d| x ∈ X}. Since A is a consistence set, then, for x ∈ A, A ⊂ [x]d. Hence
δA(x) = |A∩[x]|A|d| = 1. For x ∈ X − A, A ∩ [x]d = ∅.
Hence δA(x) = |A∩[x]|A|d|= 0. Therefore
Mp(δA) = 1
|X|
X
x∈X
(−2p|δA(x) −1
2|p+ 1) = 0.
Theorem 2.6. Let (X, AT, d) be a consistent decision ta- ble. Then Cp(AT, d) = 1.
Proof. Let X/AT = {X1, ..., Xm} and X/d = {[x]d | x ∈ X}. Since (X, AT, d) is consistent, every Xi ∈ X/AT is consistence, Xi ⊂ [x]d. Hence δXi =
|Xi|∩[x]d|
|Xi| = 1. For x ∈ X − Xi, Xi ∩ [x]d = ∅. Thus, δXi = |Xi|X|∩[x]d|
i| = 0. Therefore, for x ∈ X, we obtain
−2p|δXj(x) −12|p+ 1 = 0. Hence Cp(AT, d)
=Pm
j=1
|Xj|
|X|(1 − Mp(δXj))
=Pm
j=1
|Xj|
|X|
³
1 −|X|1 P
x∈X(−2p|δXj(x) −12|p+ 1)
´
=Pm
j=1
|Xj|
|X| = 1
Theorem 2.7. Let (X, AT ) be an information system.
Then
D(P, Q) = X
A∈X/P
|A|
|X|
³ 1− 1
|X|
X
x∈X
(−2p|δA(x)−1 2|p+1)´
where δA(x) = |A∩[x]A Q| with δA(x) ≥ 12, is an inclusion degree on (P (AT ), ≤).
Proof. (1) Let P, Q ∈ P (AT ). We have D(P, Q)
=P
A∈X/P
|A|
|X|
³
1 −|X|1 P
x∈X(−2p|δA(x) − 12|p+ 1)
´
=P
A∈X/P
|A|
|X|
³ 1
|X|
P
x∈X(2p|δA(x) −12|p
´
Since 1 ≥ δA(x) ≥ 12, we have 0 ≤ D(P, Q) ≤ 1.
(2) Let P ≤ Q. Then, for A ∈ X/P , there exists B ∈ X/Q such that A ⊂ B. For x ∈ X, if x ∈ B, then δA(x) = |A∩[x]|A|Q| = |A||A| = 1. If x 6∈ B, then δA(x) = |A∩[x]|A|Q| = |A||∅| = 0. Hence D(P, Q) = P
A∈X/P
|A|
|x| = 1.
(3) Let P ≤ Q ≤ R. Then, for [x]P ∈ X/P , there ex- ist [x]Q∈ X/Q and [x]R∈ X/Q such that [x]P ⊂ [x]Q⊂
[x]R. Since δA(x) ≥ 12, we have D(P, R)
=P
A∈X/P |A|
|X|
³ 1
|X|
P
x∈X(2p|δA(x) −12|p
´
=P
A∈X/P
|A|
|X|
³ 1
|X|
P
x∈X(2p||A∩[x]|A|p|−12|p
´
=P
A∈X/P
|A|
|X|
³ 1
|X|
P
x∈X(2p||[x]|A|P|−12|p
´
≤P
A∈X/P
|A|
|X|
³ 1
|X|
P
x∈X(2p||[x]|A|Q|−12|p
´
= D(Q, R)
Example 2.8. Let (X, AT, d) be a decision table as fol- lows:
car Price Mileage Size MaxSpeed d
x1 High Low Full Low Good
x2 Low High Full Low Good
x3 Low Low Compact Low Poor x4 High High Full High Good x5 High High Full High Excel
x6 Low High Full Low Good
Then X/AT = {E1 = {x1}, E2 = {x2, x6}, E3 = {x3}, E4= {x4, x5}}
X/d = {{x1, x2, x4, x6}, {x3}, {x5}}.
Furthermore, we have
δE1(x1) = δE1(x2) = δE1(x4) = δE1(x6) = 1 δE1(x3) = δE1(x5) = 0
δE2(x1) = δE2(x2) = δE2(x4) = δE2(x6) = 1 δE2(x3) = δE2(x5) = 0
δE3(x1) = δE3(x2) = δE3(x4) = δE3(x6) = 0 δE3(x3) = δE3(x5) = 0
δE4(x1) = δE4(x2) = δE4(x4) = δE4(x6) = 12 δE4(x5) =12, δE4(x3) = 0
(1) Since E1, E2⊂ {x1, x2, x4, x6} ∈ X/d and E3⊂ {x3} ∈ X/d, then E1, E2and E3are consistence sets, then an inconsistency p-measure Mp(δEi) = 0 for i = 1, 2, 3.
Mp(δE4) = 1
|X|
X6
i=1
(−2p|δE4(xi) −1
2|p+ 1) = 5 6,
Cp(δE4) = 1 − 1
|X|
X6
i=1
(−2p|δE4(xi) −1
2|p+ 1) = 1 6, Cp(AT, d) =
X4
j=1
|Ej| 6
³
1 − Mp(δEj)´
=13 18. (2) Let A ∈ F (X) as
A(x1) = 0.7, A(x2) = 0.9, A(x3) = 0.5, A(x4) = 0.2, A(x5) = 0.4, A(x6) = 0.1.
we have
Mp(A) = 16P6
i=1(−2p|δA(xi) −12|p+ 1)
= 1 −162p((0.2)p+ 2(0.4)p+ (0.3)p+ (0.1)p).
(3) We obtain equivalence relations for Price and Mileage,respectively as follows;
X/P = {A1= {x1, x4, x5}, A2= {x2, x3, x6}}
X/M = {B1= {x1, x3}, B2= {x2, x4, x5, x6}}, δA1(x1) =13, δA1(x2) =23, δA1(x3) = 13 δA1(x4) =23, δA1(x5) =23, δA1(x6) = 0 δA2(x1) =13, δA2(x2) =23, δA2(x3) = 13 δA2(x4) =23, δA2(x5) =23, δA2(x6) = 23. We can obtain the inclusion degrees of D(P, M ) and D(P, M ) as follows;
D(P, M )
=P
A∈X/P
|A|
|X|
³
1 −|X|1 P
x∈X(1 − 2p|δA(x) − 12|p)
´
= 36(1 −56(1 −31p)) +36(1 − (1 −31p)
= 121 −121(31p)
δB1(xi) = δB2(xi) =1
2, ∀i ∈ {1, 2, 3, 4, 5, 6}
D(M, P )
=P
A∈X/M
|A|
|X|
³
1 − |X|1 P
x∈X(1 − 2p|δA(x) −12|p)´
= 26(1 −66) +46(1 − 66) = 0
We interpret that the attribute of Mileage does not include that of Price. Similarly, we can obtain the inclusion degrees between attributes.
Definition 2.9. Let (X, AT, d) be a decision table and Y ⊂ X.
(1) A fuzziness p-measure of the rough set Y is defined as:
Mp(µY) = 1
|X|
X
x∈X
(−2p|µY(x) − 1
2|p+ 1) (p > 0)
where µY(x) = |Y ∩[x]|[x] AT|
AT| .
(2) If X/d = {Z1, ..., Zn} is an equivalence relation of decision, a fuzziness p-measure of a rough decision is defined as:
Mp(µd) = 1
|X|
X
x∈X
(−2p|µd(x) −1
2|p+ 1) (p > 0)
where µd(x) = |Z|[x]i∩[x]AT|
AT| for x ∈ Zi.
Theorem 2.10. (1) For a crisp set χA, M (χA) = 0.
(2) M (Ac) = M (A) for A ∈ F (X).
(3) If X/d = {Z1, ..., Zn} is an equivalence rela- tion of decision and χAa crisp set, a fuzziness p-measure M (µd) = 0.
Proof. (1) If A ∈ P (X), then µA(x) = A(x) = 0 or µA(x) = A(x) = 1. Thus, M (χA) = 0.
(2)
Mp(Ac) = |X|1 P
x∈X(−2p|µAc(x) − 12|p+ 1)
=|X|1 P
x∈X(−2p|Ac(x) − 12|p+ 1)
=|X|1 P
x∈X(−2p|1 − A(x) −12|p+ 1)
=|X|1 P
x∈X(−2p|µA(x) −12|p+ 1)
= Mp(A).
(3) It is easy from (1).
Theorem 2.11. Let (X, AT ) be a information system.
Then
D2(P, Q) = 1− 1
|X|
X
x∈X
(−2p|µQ(x)−1
2|p+1) (p > 0) where µQ(x) = |Yi[x]∩[x]P|
P with µQ(x) ≥ 12 and x ∈ Yi ∈ X/Q. Then D2(P, Q) is a S2-inclusion degree on (P (AT ), ≤).
Proof. (1) Let P, Q ∈ P (AT ). We have D2(P, Q) = 1 −|X|1 P
x∈X(−2p|µQ(x) − 12|p+ 1)
= |X|1 P
x∈X2p|µQ(x) −12|p Since 1 ≥ µQ(x) ≥12, we have 0 ≤ D2(P, Q) ≤ 1.
(2) Let P ≤ Q. Then, for A ∈ X/P , there ex- ists B ∈ X/Q such that A ⊂ B. For x ∈ Yi and Yi ∈ X/Q , then µQ(x) = |Y|[x]i∩[x]P|
P| = |[x]|[x]P|
P| = 1. Hence D2(P, Q) = |X|1 P
x∈X2p|µQ(x) −12|p= 1.
(3) Let P ≤ Q ≤ R. Then, for [x]P ∈ X/P , there ex- ist [x]Q∈ X/Q and [x]R∈ X/Q such that [x]P ⊂ [x]Q⊂ [x]R. Since µA(x) ≥ 12, we have
D2(P, R) = |X|1 P|X|
i=1,A∈X/P2p|µA(x) −12|p
= |X|1 P|X|
i=1,A∈X/P2p||A∩[x]|[x] R|
R|| −12|p
≤ |X|1 P|X|
i=1,B∈X/Q2p||B∩[x]|[x] R|
R|| −12|p
= |X|1 P|X|
i=1,B∈X/Q2p|µB(x) − 12|p
= D2(Q, R)
Example 2.12. Let (X, AT, d) be a decision table as in Example 2.8. Since
µd(x4) = |{x1, x2, x4, x6} ∧ {x4, x5}|
|{x4, x5}| =1 2, we have
µd(x1) = µd(x2) = µd(x3) = µd(x6) = 1, µd(x5) =1 2
Mp(µd) = 1
|X|
X6
i=1
(−2p|µd(xi) −1
2|p+ 1) = 1 3.
(1) The values of consistency and fuzziness p-measures induced by Price.
X/P = {{[x1, x4, x5}, {x2, x3, x6}}
X/d = {{x1, x2, x4, x6}, {x3}, {x5}}
(A = {x1, x4, x5}, δA(x) = A∩[x]A d) Mp(δA) =|X|1 P
x∈X(−2p|δA(x) −12|p+ 1´
=16³
3 − 2p(2(16)p+ (12)p)´ (B = {x2, x3, x6}, δB(x) = B∩[x]B d) Mp(δB) =|X|1 P
x∈X(−2p|δB(x) −12|p+ 1
´
=16³
3 − 2p(2(16)p+ (12)p)´ Cp(P, d) =P
A∈X/P
|A|
|X|
³
1 − Mp(δA)´
=12³
1 − Mp(δA)´ +12³
1 − Mp(δB)´ .
=12+16³
2p(2(16)p+ (12)p)´ . µd(x1) = µd(x2) = µd(x4) = µd(x6) = 23 µd(x3) = µd(x5) =13
Mp(µd) =|X|1 P6
i=1(−2p|µd(xi) −12|p+ 1)
= 1 − (23)p.
(2) The values of consistency and fuzziness p-measures induced by Mileage.
X/M = {{x1, x3}, {x2, x4, x5, x6}}
(A = {x1, x3}, δA(x) = A∩[x]A d) Mp(δA) = |X|1 P
x∈X(−2p|δA(x) −12|p+ 1
´
= 23
(B = {x2, x4, x5, x6}, δB(x) =B∩[x]B d) Mp(δB) = |X|1 P
x∈X(−2p|δB(x) −12|p+ 1´
= 23(1 − (12)p) Cp(M, d) =P
A∈X/P |A|
|X|
³
1 − Mp(δA)
´
= 13
³
1 − Mp(δA)
´ +23
³
1 − Mp(δB)
´ .
= 13+49(12)p.
µd(x1) = µd(x3) = 12, µd(x5) = 14 µd(x2) = µd(x4) = µd(x6) =34 Mp(µd) = |X|1 P6
i=1(−2p|µd(xi) −12|p+ 1)
= 1 − 23(12)p.
(3) The values of consistency and fuzziness p-measures induced by Size.
X/S = {{x1, x2, x4, x5, x6}, {x3}}
(A = {x1, x2, x4, x5, x6}, δA(x) = A∩[x]A d) Mp(δA) = 56(1 −45p)
(B = {x3}, δB(x) = B∩[x]B d) Mp(δB) = 0
Cp(S, d) = 56
³
1 − Mp(δA)
´ +16
³
1 − Mp(δB)
´ .
= 1136+2536(45)p.
µd(x1) = µd(x2) = µd(x4) = µd(x6) =45 µd(x3) = 1, µd(x5) = 15
Mp(µd) = |X|1 P6
i=1(−2p|µd(xi) −12|p+ 1)
= 56(1 −45p).
(4) The values of consistency and fuzziness p-measures induced by Max-speed.
X/E = {{x1, x2, x3, x6}, {x4, x5}}
(A = {x1, x2, x3, x6}, δA(x) = A∩[x]A d) Mp(δA) =23(1 − (12)p)
(B = {x4, x5}, δB(x) = B∩[x]B d) Mp(δB) =56
Cp(E, d) = 23
³
1 − Mp(δA)
´ +13
³
1 − Mp(δB)
´ .
=16+49(12)p.
µd(x1) = µd(x2) = µd(x4) = µd(x6) = 34 µd(x3) =14, µd(x5) =12
Mp(µd) =|X|1 P6
i=1(−2p|µd(xi) −12|p+ 1)
=56(1 −12p).
car Cp(, d) Mp(µd)
X/P 12+16³
2p(2(16)p+ (12)p)´
1 − (23)p X/M 13+49(12)p 1 −23(12)p
X/S 1136+2536(45)p 56(1 − 45p) X/E 16+49(12)p 56(1 − 12p).
(5) Since
X/P = {A1= {x1, x4, x5}, A2= {x2, x3, x6}}
X/M = {B1= {x1, x3}, B2= {x2, x4, x5, x6}}
µM(x1) =13, µM(x2) =23, µM(x3) = 13 µM(x4) =23, µM(x5) =23, µM(x6) = 23. D2(P, M ) = 1 −|X|1 P
x∈X(1 − 2p|µM(x) −12|p)
= 31p. µP(xi) =1
2, ∀i ∈ {1, 2, 3, 4, 5, 6}
D2(M, P ) = 1 − 1
|X|
X
x∈X
(1 − 2p|µM(x) − 1 2|p) = 0.
Similarly, we can obtain the inclusion degrees between at- tributes.
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Yong Chan Kim
He received the M.S and Ph.D. degrees in Department of Mathematics from Yonsei University, in 1984 and 1991, respectively. From 1991 to present, he is a professor in De- partment of Mathematics, Gangneung-Wonju University.
His research interests are fuzzy logic and fuzzy topology.
Jung Mi Ko
She received the M.S and Ph.D. degrees in Department of Mathematics from Yonsei University, in 1983 and 1988, re- spectively. From 1988 to present, she is a professor in De- partment of Mathematics, Gangneung-Wonju University.
Her research interests are fuzzy logic.
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