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Intuitionistic rough fuzziness and its generalizations D.Latha, Md.Abdul Rahim, G.Ganesan

Intuitionistic rough fuzziness and its generalizations D.Latha *1, Md.Abdul Rahim*2, G.Ganesan*3 Adarsh Post Graduate College of Computer Sciences, Mahabubnagar, Andhra Pradesh, India, *2Department of Mathematics, Lords Institute of Engineering and Technology, Hyderabad, Andhra Pradesh, India, *3Department of Mathematics, Jayaprakash Narayan College of Engineering, Mahabubnagar, Andhra Pradesh, India, [email protected], [email protected], [email protected]

*1 Corresponding Author

Abstract

2.1 Rough Sets

In this paper, we introduce four kinds of rough approximations for a given intuitionistic fuzzy set.

Keywords Fuzzy set, Intuitionistic Approximations

Fuzzy

Set,

Rough

1. Introduction Rough sets and fuzzy sets find wide applications in information retrieval; bioinformatics etc. Knowing this importance, Dubois and Prade hybridized both approaches and introduced rough fuzzy sets. Atanassov had given his contribution towards intuitionistic fuzzy sets by treating fuzzy approach in two dimensions namely membership and non membership functions. Ganesan and Raghavendra Rao discussed generalizations of rough fuzziness in [3]. In this paper, we introduce four different approximations using the approaches defined in [3]. This paper is organized into six sections. Section two deals with mathematical preliminaries, which include the basics of rough and fuzzy sets as well as rough fuzzy concepts. Section three and four deal with generalizations in rough fuzziness as in [3] and the basics of intuitionistic fuzzy sets respectively. Section five introduces four different approaches of approximations under intuitionistic fuzziness.

2. Mathematical Preliminaries In this section, we review the concepts of rough sets, fuzzy sets and intuitionistic fuzzy sets.

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In 1982, Pawlak introduced rough sets [8,9,4] which are useful in deriving various tools for decision making problems which find applications in the fields bioinformatics, consumer behavior etc. This tool is mathematically defined as follows: For a given finite universe of discourse U and an indiscernibility relation R, the approximation space U/R={X1,X2,…Xn} is defined. For any subset A of U, the lower and upper approximations are given by

RA = ∪ { X i ∈U / R : X i ⊆ A} and

RA = ∪ { X i ∈ U / R : X i ∩ A ≠ Φ} respectively. The lower and upper approximations represent the certainty and possibility of the elements of A with respect to R respectively. The positive and negative regions are defined as RA and U − RA respectively. The boundary BdR(A) is the difference between the upper and lower approximations of A under R. If Bdr(A)=Φ, then A is said to be R-Exact. From the definition of Pawlak, the following properties can be observed. a)

R ( A ∪ B ) = RA ∪ RB

R ( A ∩ B ) ⊆ RA ∩ RB c) R( A ∪ B) ⊇ RA ∪ RB d) R( A ∩ B) = RA ∩ RB

b)

e)

U − RA = R (U − A)

f)

U − RA = R (U − A)

g) A ⊆ B ⇒ RA ⊆ RB; RA ⊆ RB In 1996, Yao and Lin [6] discussed various aspects of generalizations of rough computing. One of the major aspects among their discussions is on granular

Journal of Convergence Information Technology Vol. 3 No. 3, September 2008

computing. Let R be any arbitrary relation defined on a finite universe of discourse U. Given any two elements x and y in U with xRy, we say that x is a predecessor of y and y is a successor of x. For any x∈U, its successor neighborhood is defined as RS(x)={y∈U: xRy}. For any subset A of U, the lower and upper approximations are defined as

RA = ∪ { x ∈U : RS ( x) ⊆ A} and

RA = ∪ { x ∈ U : RS ( x) ∩ A ≠ Φ} respectively. Similar to rough sets, Zadeh’s fuzzy sets are applicable for solving problems when Boolean concepts fail to succeed. This theory generalizes the characteristic function [i.e. a function with the codomain consists of two elements 0 and 1] into the membership function [i.e., the function with the codomain consisting of closed interval [0,1]].

2.2 Fuzzy Sets

termed of its characteristic function

⎧1 ⎩0

if if

χ A defined by

x∈ A x∉ A

This representation lacks in specifying the significance of an element in the set which made Zadeh to work on grades of memberships for the elements of a set. These membership grades allow values ranging from 0 to 1 which find applications in real problems such as recognition, clustering etc.. By Zadeh, a fuzzy subset A of U={x1,x2,…} is defined as

⎧ μ A ( x1 ) μ A ( x2 ) ⎫ μ (x ) + + .... + A n + ...⎬ where the ⎨ x2 xn ⎩ x1 ⎭ membership function μA is defined from U to [0,1]. For any two fuzzy subsets A and B of U, their union and intersection are defined as ⎧ max( μ A ( x1 ), μ B ( x1 )) max( μ A ( x2 ), μ B ( x2 )) ⎫ + + .....⎬ ⎨ x1 x2 ⎩ ⎭ and

⎧ min( μ A ( x1 ), μ B ( x1 )) min( μ A ( x2 ), μ B ( x2 )) ⎫ + + .....⎬ ⎨ x1 x2 ⎩ ⎭

respectively.

defined as

⎧1 − μ A ( x1 ) 1 − μ A ( x2 ) ⎫ + + ......⎬ . ⎨ x1 x2 ⎩ ⎭

2.3. Rough Fuzzy Sets Researchers Dubois and Prade [2] found that the combined theories of rough and fuzzy sets are essential for several applications and they worked towards hybridization of both concepts and they introduced rough fuzzy sets and fuzzy rough sets. In late eighties, Dubois and Prade published their work on rough fuzzy sets [2]. For a given finite universe of discourse with an indiscernibility relation R, the approximation space U/R is denoted by U/R={X1,X2,…Xn}. For a fuzzy subset A of U, define μ R A ( X i ) = in f μ ( x j ) : x j ∈ X i and j

In 1965, Zadeh introduced fuzzy sets [7,5] by generalizing Classical Sets. For a given universe of discourse U, a classical subset A of U is expressed in

χ A ( x) = ⎨

For a fuzzy subset A of U, its complement AC is

{

}

μ R A ( X i ) = su p {μ ( x j ) : x j ∈ X i } ∀ i = 1, 2 , ...n j

The lower and upper rough fuzzy approximations are defined as

μ (X )⎫ ⎧μ (X ) μ (X ) RA = ⎨ RA 1 + RA 2 + .... + RA n ⎬ X2 Xn ⎭ ⎩ X1 and

μ (X )⎫ ⎧μ (X ) μ (X ) RA = ⎨ RA 1 + RA 2 + .... + RA n ⎬ X2 Xn ⎭ ⎩ X1

respectively. The following properties are observed from rough fuzzy sets. a)

R ( A ∪ B ) = RA ∪ RB

R ( A ∩ B ) ⊆ RA ∩ RB c) R( A ∪ B) ⊇ RA ∪ RB d) R( A ∩ B) = RA ∩ RB

b)

e)

U − RA = R (U − A)

f)

U − RA = R (U − A)

g) A ⊆ B ⇒ RA ⊆ RB; RA ⊆ RB Now, we describe the generalizations of rough fuzziness over the overlapping granules which were discussed by G.Ganesan and C.Raghavendra Rao in [3].

3. Generalized Rough Fuzzy Sets In 2005, G.Ganesan and C.Raghavendra Rao discussed generalizations [3] on rough fuzziness defined over overlapping granules. Consider a finite

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Intuitionistic rough fuzziness and its generalizations D.Latha, Md.Abdul Rahim, G.Ganesan

universe of discourse U={ a1,a2,…at}and R is a relation defined on U. Let R/U={ X1,X2,…Xn}. Let A be any fuzzy subset of U. Define the functions αi,βij,ρi and δij as follows:

α i = inf {μ A (a j ) : a j ∈ X i } ⎧α β ij = ⎨ i

if

aj ∈ Xi otherwise

⎩0 ρi = sup {μ A (a j ) : a j ∈ X i } ⎧ ρi ⎩1

δ ij = ⎨

if

and

and

aj ∈ Xi otherwise

4. Intuitionistic Fuzzy Sets

The generalized lower and upper rough fuzzy approximations are defined as max( β11 , β12 ,..., β1t ) max( β 21 , β 22 ,..., β 2 t ) RA = {

+

X2

+ ..

max( β n1 , β n 2 ,..., β nt ) } Xn

.. +

and

RA = { .+

X1

0.4 0.38 0.38 0.4 Joe = { + + + + Disease _1 Disease _2 Disease _3 Disease _ 4 0.52 0.38 0.38 + + } Disease _5 Disease _6 Disease _7 and 0.4 0.52 0.38 0.52 Joe = { + + + + Disease _1 Disease _2 Disease _3 Disease _ 4 0.52 0.52 0.4 + + } Disease _5 Disease _6 Disease _7 .

min(δ11 , δ12 ,..., δ1t ) min(δ 21 , δ 22 ,..., δ 2t ) + + ... X1 X2

min(δ n1 , δ n 2 ,..., δ nt ) } Xn

respectively. This definition is illustrated with the following example. Example 3.1: Consider the following information between a certain set of symptoms and the related diseases:

By the theory of fuzzy sets, the non membership function assigns a value which is the difference between one and the membership value. However, for some applications, a relaxation of this rule was required. In 1986, Atanassov K.T. initiated work in this direction and introduced intuitionistic fuzzy sets [1]. His approach takes a dimensional view of fuzzy sets in terms of membership grade and non membership grade. For any intuitionistic fuzzy subset A of U={x1,x2,…}, let μA and γA be the membership and non membership functions defined from the universe of discourse U to [0,1] with 0≤μA(x)+ γA(x)≤1 for all x in U. Here, the intuitionistic fuzzy subset A of U is represented by

( μ A ( x1 ), γ A ( x1 )) ( μ A ( x 2 ), γ A ( x 2 )) + + .. x1 x2 . ( μ A ( x n ), γ A ( x n )) .. + + ...} xn

{

If A and B are intuitionistic fuzzy subsets of U with μA(x)≤μB(x) and γB(x)≤ γA(x) for all x∈U, then we say that A is subset of B. For any two fuzzy subsets A and B of U, their union and intersection are defined as (max( μ A ( x1 ), μ B ( x1 )), min(γ A ( x1 ), γ B ( x1 ))) { + x1 B

Symptom_1: Disease_1, Disease_4, Disease_7 Symptom_2:Disease_2, Disease_4, Disease_5, Disease_6 Symptom_3:Disease_2, Disease_3, Disease_6, Disease_7 Let a patient say Joe with the possible symptoms of all the three say

⎧ ⎫ 0.4 0.52 0.38 + + ⎨ ⎬ ⎩ symptom _1 symptom _ 2 symptom _ 3 ⎭

arrives for the information. Using available information, α1=α4=0.4, α2=α3=α6=α7=0.38, α5=0.52, ρ1=ρ7=0.4, ρ2=ρ4=ρ5=ρ6=0.52 and ρ3=0.38 can be defined. By computing βij and δij, we arrive at the lower and upper approximations as

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B

(max( μ A ( x2 ), μ B ( x2 )), min(γ A ( x2 ), γ B ( x2 ))) + .....} x2 and (m in( μ A ( x1 ), μ B ( x1 )), m ax( γ A ( x1 ), γ B ( x1 ))) {

x1

+

(m in( μ A ( x 2 ), μ B ( x 2 )), m ax( γ A ( x 2 ), γ B ( x 2 ))) + .....} x2

respectively. Now, we introduce the generalizations as discussed in section 3, under intuitionistic fuzziness.

Journal of Convergence Information Technology Vol. 3 No. 3, September 2008

5. Generalized Intuitionistic Rough Fuzzy Sets Unlike rough fuzziness, due to the membership and non membership parameters, it is possible to define four kinds of approximations as described below:

Certainty

MCNC

MCNP

Possibility

MPNC

MPNP

MCNC: It gives approximation on certainty in both membership and non membership values. b) MCNP: It deals with the approximation on certainty of membership values and possibility on non membership values. c) MPNC: It deals with the approximation on possibility of membership values and certainty on non membership values. d) MPNP: It gives approximation on possibility in both membership and non membership values. Now, we describe these four approximations in detail.

a)

Consider a finite universe of discourse U={ a1,a2,…at}and R is a relation defined on U. Let R/U={ X1,X2,…Xn}. Let A be any intuitionistic fuzzy subset of U. Define the functions αi,βij,ρi and δij as follows:

⎧α β ij = ⎨ i ⎩0

if

and

aj ∈ Xi otherwise

ρi = sup {γ A (a j ) : a j ∈ X i }

aj ∈ Xi otherwise

M i = m a x ( β i 1 , β i 2 , . . . , β i t ) and N M i = m i n ( δ i 1 , δ i 2 , ..., δ i t ) The MCNC approximation of A is defined as

⎧ ( M 1 , NM 1 ) ( M 2 , NM 2 ) MCNCR ( A) = ⎨ + + X1 X2 ⎩ ( M n , NM n ) ⎫ .... + ⎬ Xn ⎭

The above definition is illustrated with the following example. Example 5.1.1: Consider the information as given in example 3.1 Suppose that a patient say Joe with the possible symptoms of all the three say

⎧ (0.4, 0.5) (0.52, 0.4) (0.38, 0.56) ⎫ + + ⎨ ⎬ ⎩ symptom _1 symptom _ 2 symptom _ 3 ⎭

arrives for the information about possible diseases. Using the information, α1=α4=0.4, α2=α3=α6=α7=0.38, α5=0.52 and ρ1=ρ4=0.5, ρ2=ρ3=ρ6=ρ7=0.56, ρ5=0.5 can be defined. Hence, by computing βij and δij, we arrive MCNC approximation as

(0.4,0.5) (0.38,0.56) (0.38,0.56) MCNCR (Joe) ={ + + + Disease _1 Disease _2 Disease _3 (0.4,0.5) (0.52,0.4) (0.38,0.56) (0.38,0.56) } + + + Disease _4 Disease _5 Disease _6 Disease _7

The following graph represents the results of the above.

5.1 MCNC Approximations

α i = inf {μ A (a j ) : a j ∈ X i }

if

Define

Membership&Non membership Grades

Membership

Non Membership Certainty Possibility

⎧ ρi ⎩1

δ ij = ⎨

0.6 0.5 0.4 0.3 0.2 0.1 0

Membership Non Membership

1 2 3 4 5 6 7 Disease

and

Figure 5.1.1 [a]

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Intuitionistic rough fuzziness and its generalizations D.Latha, Md.Abdul Rahim, G.Ganesan

(0.4,0.5) (0.38,0.4) (0.38,0.56) MCNPR (Joe) = { + + + Disease _1 Disease _2 Disease _3 (0.4,0.4) (0.52,0.4) (0.38,0.4) (0.38,0.5) + + + } Disease _4 Disease _5 Disease _6 Disease _7

5.2 MCNP Approximations

The graphical representation of the result is as follows:

Consider a finite universe of discourse U={ a1,a2,…at}and R is a relation defined on U. Let R/U={ X1,X2,…Xn}. Let A be any intuitionistic fuzzy subset of U. Define the functions αi,βij,ρi and δij as follows:

α i = inf {μ A (a j ) : a j ∈ X i } ⎧α β ij = ⎨ i

⎧ ρi ⎩1

and

aj ∈ Xi otherwise

if

⎩0 ρi = inf {γ A (a j ) : a j ∈ X i }

δ ij = ⎨

Membership & Non Membership Grades

Properties 5.1.2. Using the definition of MCNC, the following properties can be inferred. a) MCNC(A∪B)⊇MCNC(A)∪MCNC(B) b) MCNC(A∩B)= MCNC(A)∪MCNC(B) c) If A⊆B then MCNC(A)⊆MCNC(B)

Define

M i = m a x ( β i 1 , β i 2 , . . . , β i t ) and N M i = m i n ( δ i1 , δ i 2 , . . . , δ it ) The MCNP approximation of A is defined as

⎧ ( M 1 , NM 1 ) ( M 2 , NM 2 ) + + .... MCNPR ( A) = ⎨ X1 X2 ⎩ +

( M n , NM n ) ⎫ ⎬ Xn ⎭

The above definition is illustrated with the following example. Example 5.2.1: As given in example 3.1, suppose that the patient say Joe with the possible symptoms of all the three say

⎧ (0.4, 0.5) (0.52, 0.4) (0.38, 0.56) ⎫ + + ⎨ ⎬ ⎩ symptom _1 symptom _ 2 symptom _ 3 ⎭

Membership Non Membership

1 2 3 4 5 6 7

and

aj ∈ Xi otherwise

if

0.6 0.5 0.4 0.3 0.2 0.1 0 Disease

Figure 5.2.1 [a] Properties 5.2.2: Using the definition of MCNP, the following properties can be inferred. a) MCNP(A∪B)⊇MCNP(A)∪MCNP(B) b) MCNP(A∩B)⊆ MCNP(A)∪MCNP(B) c) If A⊆B then MCNP(A)⊆MCNP(B)

5.3 MPNC Approximations Consider a finite universe of discourse U={a1,a2,…at}and R is a relation defined on U. Let R/U={X1,X2,…Xn}. Let A be any intuitionistic fuzzy subset of U. Define the functions αi,βij,ρi and δij as follows:

α i = sup {μ A (a j ) : a j ∈ X i } ⎧α β ij = ⎨ i

if

⎧ ρi ⎩1

if

and

aj ∈ Xi otherwise

⎩0 ρi = sup {γ A (a j ) : a j ∈ X i } and

δ ij = ⎨

aj ∈ Xi otherwise

arrives for the information about possible diseases. Define Using the information, α1=α4=0.4, α2=α3=α6=α7=0.38, N M i = m i n ( δ i 1 , δ i 2 , . . . , δ i t ) and α5=0.52 and ρ1=ρ7=0.5, ρ2=ρ4=ρ5=ρ6=0.4, ρ3=0.56 can be defined. By computing βij and δij, we arrive MCNP ⎧max(βi1,..., βit ) if max(βi1,..., βit ) + NMi < 1 approximation as Mi = ⎨

⎩

1− NMi

otherwise

The MPNC approximation of A is defined as

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Journal of Convergence Information Technology Vol. 3 No. 3, September 2008

.... +

(M n , N M n ) Xn

(M 1, N M 1) (M 2 , N M 2 ) + + X1 X2 ⎫ ⎬ ⎭

Example 5.3.1 : As given in example 3.1, suppose that the patient say Joe with the possible symptoms of all the three say

⎧ (0.4, 0.5) (0.52, 0.4) (0.38, 0.56) ⎫ + + ⎨ ⎬ ⎩ symptom _1 symptom _ 2 symptom _ 3 ⎭

arrives for the information about possible diseases. Using the information, α1=α7=0.4, α2=α4=α5=α6=0.52, α3=0.38 and ρ1=ρ4=0.5, ρ2=ρ3=ρ6=ρ7=0.56, ρ5=0.4 can be defined. By computing βij and δij, we arrive MPNC approximation as

(0.4,0.5) (0.44,0.56) (0.38,0.56) MPNCR (Joe) = { + + + Disease _1 Disease _2 Disease _3 (0.5,0.5) (0.52,0.4) (0.44,0.56) (0.4,0.56) } + + + Disease _4 Disease _5 Disease _6 Disease _7

Membership & Non Membership Grades

⎧α β ij = ⎨ i

representation

0.6 0.5 0.4 0.3 0.2 0.1 0

of

this

Membership Non Membership

1 2 3 4 5 6 7 Disease

Figure 5.3.1 (a) Properties 5.3.2. Using the definition of MPNC, the following properties can be inferred. a) MPNC(A∪B)⊇MPNC(A)∪MPNC(B) b) MPNC(A∩B)⊆ MPNC(A)∪MPNC(B) c) If A⊆B then MPNC(A)⊆MPNC(B)

5.4 MPNP Approximations Consider a finite universe of discourse U={ a1,a2,…at}and R is a relation defined on U. Let R/U={ X1,X2,…Xn}. Let A be any intuitionistic fuzzy subset of U. Define the functions αi,βij,ρi and δij as follows:

and

aj ∈ Xi

if

⎩0

The above definition is illustrated with the following example.

The diagrammatical approximation is

α i = sup {μ A (a j ) : a j ∈ X i } otherwise

ρi = inf {γ A (a j ) : a j ∈ X i } ⎧ ρi ⎩1 Define

and

aj ∈ Xi otherwise

if

δ ij = ⎨

M i = m a x ( β i1 , β i 2 , . . . , β it ) N M i = m i n ( δ i1 , δ i 2 , . . . , δ

and it

)

The MPNC approximation of A is defined as ⎧ (M1 , NM1 ) (M 2 , NM 2 ) (M , NM n ) ⎫ MPNCR ( A) = ⎨ + + .... + n ⎬ X1 X2 Xn ⎩ ⎭

The above definition is illustrated with the following example. Example 5.4.1: As given in example 3.1, suppose that the patient say Joe with the possible symptoms of all the three say

⎧ (0.4, 0.5) (0.52, 0.4) (0.38, 0.56) ⎫ + + ⎨ ⎬ ⎩ symptom _1 symptom _ 2 symptom _ 3 ⎭

arrives for the information about possible diseases. Using the information, α1=α7=0.4, α2=α4=α5=α6=0.52, α3=0.38 and ρ1=ρ7=0.5, ρ2=ρ4=ρ5=ρ6=0.4, ρ3=0.4 can be defined. By computing βij and δij, we arrive MPNP approximation as

(0.4,0.5) (0.52,0.4) (0.38,0.56) MPNPR (Joe) = { + + + Disease _1 Disease _2 Disease _3 (0.52,0.4) (0.52,0.4) (0.52,0.4) (0.4,0.5) } + + + Disease _4 Disease _5 Disease _6 Disease _7

The graphical representation of this approximation is given below: M e m b e r s h ip & N o n M e m b e r s h ip V a lu e s

⎧ M PN C R ( A) = ⎨ ⎩

0.6 0.5 0.4 0.3 0.2 0.1 0

Membership Non Membership

1

2

3

4

5

6

7

Disease

Figure 5.4.1 (a) Properties 5.4.2. Using the definition of MPNP, the following properties can be inferred. a) MPNP(A∪B)= MPNP(A)∪MPNP(B)

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Intuitionistic rough fuzziness and its generalizations D.Latha, Md.Abdul Rahim, G.Ganesan

b) MPNP(A∩B)⊆ MPNP(A)∪MPNP(B) c) If A⊆B then MPNP(A)⊆MPNP(B) d) U-MPNP(A)=MCNC(U-A) e) U-MCNC(A)=MPNP(A)

5.5 Relationship between the approximations Using min and max properties, it is possible to define the following lattice which yields the user to perform any these four approximations according to the accuracy.

[4] Ganesan G, Raghavendra Rao C., Feature Selection using Fuzzy Decision Attributes, INFORMATION, Vol 9, No3, pp:381-394, 2006 [5] George J.Klir. Bo Yuan, “Fuzzy Sets and Fuzzy Logic Theory and Applications”, Prentice-Hall of India Pvt. Ltd., 1997 [6] Yao YY , Lin , ‘Generalization of Rough Sets using Modal Logic’, Intelligent Automation and Soft Computing, An International Journal, 2, pp 103-120, 1996 [7] Zadeh L., Fuzzy Sets, Journal of Information and Control, 8, pp. 338-353, 1965.

MPNP

[8] Zdzislaw Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11, 341-356, 1982.

MPNC

MCNP

MCNC From the above structure, for any intuitionistic fuzzy set A, it is inferred that M C N C ( A ) ⊆ M P N C ( A ) and

⊆

M P N P ( A ) M C N C ( A ) ⊆ ⊆

M C N P ( A )

M P N P ( A )

6. Conclusion In this paper, we discussed four approaches of approximating a given fuzzy input under crisp knowledge. These approximations allow the users to handle various approximations based on their choice and needs. As intuitionistic fuzzy sets are highly useful in problems based on bioinformatics, data mining etc., these tools provide research directions in the emerging areas.

7. References [1]Atanassov, K.T., “Intuitionistic fuzzy sets" Fuzzy sets and Systems 20, 87-96,1986. [2] Dubois D, Prade H, Rough Fuzzy sets and Fuzzy Rough Sets, International Journal of General Systems, 17, pp 191209, 1989 [3] Ganesan G, Raghavendra Rao C., Generalized Rough Fuzzy Sets, Journal Electronic Modeling, No 6, pp:29-35, 2005.

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[9] Zdzislaw Pawlak, ”Rough Sets-Theoretical Aspects and Reasoning about Data”, Institute of Computer Science, Warsaw, 1991