New Similarity and Inclusion Measures between Type-2 Fuzzy Sets Chao-Ming Hwang
Miin-Shen Yang*
Wen-Liang Hung
Department of Applied Mathematics, Chinese Culture University, Taipei, Taiwan
Department of Applied Mathematics, Chung Yung Christian University, Chung-Li 32023, Taiwan * E-mail:
[email protected]
Graduate Institute of Computer Science, National Hsinchu University of Education, Hsin-Chu, Taiwan II.
Abstract—In this paper, new similarity and inclusion measures between type-2 fuzzy sets are proposed. It is known that similarity measures of type-2 fuzzy sets are used to indicate the similarity degree between type-2 fuzzy sets and inclusion measures for type-2 fuzzy sets are the degrees to which a type2 fuzzy set is a subset of another type-2 fuzzy set. In the paper, we compare these proposed measures with several existing methods for type-2 fuzzy sets. Numerical results show that the proposed measures are better than existing measures.
We first give a brief review of type-2 fuzzy sets and then review and discuss similarity and inclusion measures between type-2 fuzzy sets. A. Type-2 Fuzzy Sets A type-2 fuzzy set was first proposed by Zadeh [1] as an extension of a fuzzy set. Mendel and John [8] presented a Representation Theorem for type-2 fuzzy sets and showed that it can be used to derive formulas for the union, intersection and complement of type-2 fuzzy sets without using the Extension Principle. We give a brief review of type-2 fuzzy sets as follows. Definition 1 (Muzimoto and Tanaka [2]) A type-2 fuzzy set A in a set X is the fuzzy set which is characterized by a
Keywords-type-2 fuzzy sets; similarity measure; inclusion measure.
I.
INTROUCTION
Zadeh [1] proposed type-2 fuzzy sets as an extension of (type-1) fuzzy sets whose membership values are fuzzy sets on the interval [0,1]. There were further studies and applications of type-2 fuzzy sets, such as Mizumoto and Tanaka [2], Yager [3], Mendel et al. [4], Jammeh et al. [5], Mendoza et al. [6] and Wagner and Hagras [7]. The membership function of a type-2 fuzzy set provides additional degree of freedom for modeling uncertainties so that type-2 fuzzy sets can better improve certain kinds of inference than do fuzzy sets with increasing imprecision, uncertainty and fuzziness in information. Mendel and John [8] presented a new Representation Theorem for type-2 fuzzy sets and showed that it can be used to derive formula for the union, intersection and complement of type-2 fuzzy sets without using the Extension Principle. Similarity measure could be the foundation for analogical reasoning between two fuzzy concepts. Inclusion measures between fuzzy sets are the degrees to which a fuzzy set is a subset of another fuzzy set. Although several similarity and inclusion measures for type-2 fuzzy sets have been proposed in previous studies, no one has considered in the use of the Sugeno integral to define those for type-2 fuzzy sets. In this paper, we propose new similarity and inclusion measures between type-2 fuzzy sets based on the Sugeno integral. Some examples are used to compare the proposed measures with previous methods. Numerical results show that the proposed measures are better than those existing methods. _________________________ * Corresponding author:
[email protected]
978-1-61284-076-5/11/$26.00 ©2011 IEEE
PRELIMINARIES
fuzzy membership function the value
μ A (x)
μA
as
μ A : X → [0,1] J with
being called a fuzzy grade and being a
fuzzy set in [0,1] (or in the subset J of [0,1]) ( x ∈ X ). Definition 2 (Mendel and John [8]) A type-2 fuzzy set,
~
denoted A , is characterized by a type-2 membership function μ A~ ( x , u ) with x ∈ X and u ∈ J x ⊆ [0,1] , i.e.,
~ A = {((x, u ), μ A~ (x, u ))∀x ∈ X , ∀u ∈ J x ⊆ [0,1]} ~ in which 0 ≤ μ A~ ( x, u ) ≤ 1 . A can be also expressed as
A = ∑ x∈X ∫
u∈J x
μ A ( x, u ) / ( x, u ) , J x ⊆ [0,1]
Σ ∫ denotes union over all admissible x and u . For a universe of continuous discourses, Σ is replaced by ∫ . Definition 3 (Mendel and John [8]) At each value of x , say x = x ′ , the 2-D plane whose axes are u and μ A~ ( x ′, u ) where
is called a vertical slice of
μ A~ ( x, u )
. A secondary
membership function is a vertical slice of μ A~ ( x , u ) . It is
μ A~ ( x = x ′, u )
for
x ′ ∈ X and ∀u ∈ J x′ ⊆ [0,1] , i.e.,
μ A ( x = x′, u ) ≡ μ A ( x′) = ∫
u∈J x′
in which
82
f x′ (u ) / u , J x′ ⊆ [0,1]
~ 0 ≤ f x′ (u ) ≤ 1 . A can be also re-expressed as
~ A = {( x, μ A~ ( x )) ∀x ∈ X } or, as
Yang and Lin [13] proposed a similarity measure between
~
type-2 fuzzy sets A and
A = ∑ x∈X μ A ( x) / x = ∑ x∈X ⎡ ∫ f (u ) / u ⎤ / x, J x ⊆ [0,1] ⎣⎢ u∈J x x ⎦⎥ Definition 4 (Mendel and John [8]) The domain of a secondary membership function is called the primary membership of x. In the representation of primary membership of x, where
1 SYL ( A , B ) = ∑ x∈X n
~ A , J x is the
The notations
(I1) I ( A , A ) = 1 (I2) A ⊆ B ⇔ I ( A , B ) = 1
C
(S2) S ( D, D ) = 0 , ∀D ∈ P ( X ) (the power set of X); (S3) S ( E , E ) = max A , B∈F ( X ) S ( A , B ) , 2 ~ ∀E ∈ F2 ( X ) . ~ ~ ~ ~ ~ ~ (S4) For any A, B , C ∈ F2 ( X ), if A ⊆ B ⊆ C , then
~ ~ ~ (I3) For any A , B , C ∈ F2 ( X ), if A ⊆ B ⊆ C , then I (C , A ) ≤ I ( B , A ) and I (C , A ) ≤ I (C , B ) . Based on information from the primary membership and secondary membership functions, Yang and Lin [13] proposed an inclusion measure between type-2 fuzzy sets
S ( A , B ) ≥ S ( A , C ) and S ( B , C ) ≥ S ( A , C ) .
~ ~ A and B as
Note that, Mizumoto and Tanaka [2] had defined
~ ~ ~ A ⊆ B for any A , B ∈ F2 ( X ) . For A ⊆ B ⊆ C in the (S4) of Definition 5, we use the definition of Mizumoto
1 IYL ( A , B ) = ∑ x∈X n
and Tanaka [2]. For two fuzzy sets A and B in the finite set X = {x1 , x2 ," , xn }, Pappis and Karacapilidis [9]
x∈X
A
( x), μ B ( x)}
∑ max{μ
A
Σ and ∫ in the similarity SYL ( A , B ) are
Afterward, Sinha and Dogherty [15] considered an indicator I(A,B) for an inclusion measure between two fuzzy sets A and B and then gave several axioms that I(A,B) needs to satisfy (also see Cornelis et al. [16]). Zeng and Li [17] gave the axioms for the mapping I : F1 ( X ) × F1 ( X ) → [0,1] to be an inclusion measure of fuzzy sets A and B. Based on Zeng and Li’s [17] axioms for an inclusion measure I ( A, B ) , we give a definition of an inclusion measure for type-2 fuzzy sets as follows. Definition 6 A real function I : F2 ( X ) × F2 ( X ) →[0,1] is called an inclusion measure for type-2 fuzzy sets, if I satisfies the following axioms:
~ ~ (S1) S ( A , B ) = S ( B , A ) , ∀A, B ∈ F2 ( X ) .
∑ min{μ
~ B are with
continuous discourses, Σ is replaced by the integral ∫ . Note that Zadeh [14] gave a definition of fuzzy inclusion for any two fuzzy sets A and B in F1 ( X ) with A ⊆ B ⇔ μ A ( x) ≤ μ B ( x ), ∀x ∈ X .
called a similarity measure for type-2 fuzzy sets, if S satisfies the following axioms:
with
~
summation and integral, respectively. For a universe of
→ [0,1] to be a similarity measure between A and B in F1 ( X ) . Based on the similar concept of Xuechang’s [11], we give a definition of a similarity measure for type-2 fuzzy sets as follows. Definition 5 A real function S : F2 ( X ) × F2 ( X ) → [ 0,1] is
measure
max{u ⋅ f x (u ), u ⋅ g x (u )}du
B = ∑ ⎡ ∫ g x (u ) / u ⎤ / x, J x ⊆ [0,1] . ⎢ u∈J x ⎥⎦ x∈X ⎣
F2 ( X ) is the class of all type-2 fuzzy sets of X; For any two fuzzy sets A and B in F1 ( X ) , Xuechang [11] gave the axioms for a mapping S : F1 ( X ) × F1 ( X )
similarity
u∈J x
min{u ⋅ f x (u ), u ⋅ g x (u )}du
A = ∑ ⎡ ∫ f (u ) / u ⎤ / x, J x ⊆ [0,1] and ⎢ u∈J x x ⎦⎥ x∈X ⎣
J x ⊆ [0,1] , ∀x ∈ X .
fuzzy sets of X;
a
u ∈J x
where the two type-2 fuzzy sets A and
B. Similarity and Inclusion Measures between Type-2 Fuzzy Sets A similarity measure between fuzzy sets is an important way to measure the degree of similarity between two fuzzy concepts. Pappis and Karacapilidis [9] proposed three similarity measures for fuzzy sets. Afterwards, many researches and applications of similarity measures for fuzzy sets were given (see [10]-[12]). Throughout this paper, the following notations are used. X is the universe of discourse; F1 ( X ) is the class of all
proposed
∫ ∫
~ B as
III.
S ( A, B) =
( x), μ B ( x)}.
∫
u∈J x
min{u ⋅ f x (u ), u ⋅ g x (u )}du
∫
u∈J x
{u ⋅ f x (u )}du
NEW SIMILARITY AND INCLUSION MEASURES BETWEEN TYPE-2 FUZZY SETS
In this section, we first review fuzzy measures and the Sugeno integral (see Murofushi and Sugeno [18]). Let X be a nonempty set and let F be a σ − field of subsets
x∈X
Based on the similar idea of Pappis and Karacapilidis [9],
83
of X . Let m : F → [ 0 , 1] be a nonnegative and realvalued set function defined on F . Definition 7 The measure m is called a fuzzy measure on ( X , F ) if it satisfies the following conditions: (F1)
(a)
i,0
⊆ (b) If A
m(φ ) = 0 ( vanishing at φ ).
(F2) For any E ∈ F and
F , if
1 I ( A , B ) = ∑ n 1≤i ≤ n
Let F be the class of all finite nonnegative measurable
(X ,
B , then
0 ≤ f x (u ) ≤ g x (u ) ≤ 1, ∀x ∈ X , u ∈ J x ⊆ [0,1]
E ⊂ F , then F∈ m(E) ≤ m(F) ( monotonicity ).
functions defined on
min{∫ f x f xi du, ∫ f x f xi du} ~ ~ 1 i ,0 i,0 I ( A, A) = ∑ =1 1 ≤ i ≤ n n ∫ f x f xi du
) . For any given f ∈ F ,
F
we write fα = { x f ( x) ≥ α } where
α ∈[ 0 , ∞ ] .
The
=
fα is called α − cut . For simplicity, we consider the range of the function with the closed interval [ 0 , 1] . sets
Definition 8 Let
A∈
and
F
of f on A with respect to
f
∫
A
A
by
fdm , is
1 I ( A , B ) = ∑ n 1≤i ≤ n
A = X , the fuzzy integral may be also denoted
f xi ,0
∫
f xi ,0
g xi ,0
I (C , A ) =
(1)
f xi du
f xi ,0
min{∫
hxi du , ∫
hxi ,o
∫
hxi ,0
f xi ,0
