Modified Cosine Similarity Measure between ... - Semantic Scholar
Modified Cosine Similarity Measure between Intuitionistic Fuzzy Sets Chao-Ming Hwang1 and Miin-Shen Yang2,* 1
2
Department of Applied Mathematics, Chinese Culture University, Taipei, Taiwan Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taiwan [email protected]
Abstract. Similarity of intuitionistic fuzzy sets (IFSs) is an important measure to indicate the similarity degree between IFSs. Recently, Ye (2011) proposed a similarity measure between IFSs based on the cosine concept. Although this cosine similarity measure has good concept and merit, the measure is not satisfied the definition of a similarity between IFSs and not presented well for analyzing IFS data. In this paper, we modify the cosine similarity measure between IFSs. This modified similarity measure between IFSs is not only to satisfy the definition of a similarity between IFSs, but also to improve the efficiency of the Ye’s measure. An example is used to demonstrate this phenomenon. Keywords: Fuzzy set, Intuitionistic fuzzy set, Similarity measure, Cosine similarity measure.
1
Introduction
Fuzzy sets, first introduced by Zadeh [1], give an approach for treating fuzziness. In fuzzy sets, the degree of nonbelongingness is just the complement to 1 of the membership degree. However, humans who expresses the degree of membership of a given element in a fuzzy set very often does not express a corresponding degree of nonmembership as the complement to 1. Thus, Atanassov [2] introduced the concept of an intuitionistic fuzzy set (IFS) which is a generalization of a fuzzy set. Since an IFS can present the degrees of membership and nonmembership with a degree of hesitancy, the knowledge and semantic representation become more meaningful and applicable [3-4]. These IFSs have been widely studied and applied in various areas such as decision making problems [5], medical diagnosis [6], pattern recognition [7]. Similarity measures are an important tool for determining the degree of similarity between two objects. Different similarity measures between fuzzy sets have been proposed and similarity measures between IFSs are also widely studied in the literature. Dengfeng and Chuntian [8] proposed some similarity measures between IFSs used in pattern recognition problem. Liang and Shi [9] proposed similarity measures between IFSs in which they used numerical comparisons to show that Liang and Shi’s similarity measures are more reasonable than those of Dengfeng and Chuntian. Mitchell [10] interpreted IFSs as ensembles of ordered fuzzy sets from the *
Corresponding author.
J. Lei et al. (Eds.): AICI 2012, LNAI 7530, pp. 285–293, 2012. © Springer-Verlag Berlin Heidelberg 2012
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statistical point. Hung and Yang [11] proposed several similarity measures between IFSs based on Hausdorff distance which are well used with linguistic variables. Xu and Chen [12] gave a comprehensive overview of distance and similarity measures between IFSs. Recently, Ye [13] proposed a cosine similarity measure between IFSs. Although this cosine similarity measure has good concept and merit, the measure cannot satisfy the definition of a similarity between IFSs. In this paper, we modify this cosine similarity measure such that the modified measure can satisfy the definition of a similarity between IFSs. Some examples are used to demonstrate the efficiency of the modified cosine similarity measure.
2
Intuitionistic Fuzzy Set and Similarity Measures
Let X = {x1 , x2 ,..., xn } be the universe of discrete discourses. Consider two intuitionistic fuzzy sets (IFSs) A~ and B~ in X . We first describe the aspects of IFSs discussed by Atanassov [2] as follows. Definition 1. (Atanassov [2]) An intuitionistic fuzzy set (IFS) A = {( x, μ A ( x),ν A ( x)) x ∈ X )} where μ A : X → [0,1] and
A in X is defined as ν A : X → [0,1] with
the condition 0 ≤ μ A + ν A ( x ) ≤ 1, ∀x ∈ X .
~ A in X , the numbers μ A ( x) and ν A ( x) denote the degree of membership and non-membership of x to A , respectively, and the number ~ π A ( x) = 1 − μ A ( x) −ν A ( x) denotes a hesitancy degree of x to A . In this paper, we use IFSs(X) to denote the class of all IFSs of X .
For each IFS
Definition 2. If A~ and B~ are two IFSs of X, then ~ ~ (i) A ⊆ B if and only if ∀ x ∈ X , u A ( x) ≤ uB ( x) and
v A~ ( x) ≥ vB~ ( x) .
(ii) A = B if and only if ∀ x ∈ X , u A ( x) = uB ( x) and
v A~ ( x ) = vB~ ( x ) .
~
~
Measuring a similarity between IFSs is important in IFSs researches. Some methods had been proposed to calculate the similarity degree between IFSs where Li et al. [14] introduced the following definition.
× IFSs(X) →[0,1] .
~ ~ S ( A, B ) is ~ ~ satisfies said to be the degree of similarity between A~ and B~ in IFSs(X) if S ( A , B)
Definition 3. (Li et al. [14]) A mapping S: IFSs(X)
the following properties:
~
~ (S1) 0 ≤ S ( A, B ) ≤ 1;
~ ~ ~ ~ S ( A, B ) = 1 iff A = B ; ~ ~ ~ ~ ; (S3) S ( A, B ) = S ( B , A) ~ ~ ~ ~ ~ ~ ~ in IFSs(X). (S4) S ( A, C ) ≤ S ( A, B ) and S ( A , C ) ≤ S ( B , C ) if A ⊆B⊆C (S2)
(S5)
~ ~ S ( A, A C ) = 0
~
iff A is a crisp set.
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Consider two IFSs A~ and B~ in IFSs(X), Dengfeng and Chuntian [8] proposed a similarity measure between them as follows:
1 S DC ( A , B ) = 1 − p n
n
p
| m i =1
A
(i) − mB (i) | p
where m A (i ) = (u A ( xi ) + 1 − v A ( xi )) / 2, mB (i) = (u B ( xi ) + 1 − vB ( xi )) / 2, 1 ≤ p < ∞.
S DC ( A , B ) = SC ( A , B ) where SC ( A , B ) was in Chen [15] as
When p = 1, follow:
1 n SC ( A , B ) = 1 − (u A ( xi ) − v A ( xi )) − (uB ( xi ) − vB ( xi )) 2n i =1 , B ) , S ( A , B ) and Hong and King [16] proposed new similarity measures S H ( A L S ( A, B) (also see Li et al. [14]) as follows: O
S H ( A , B ) = 1 − S L ( A , B ) = 1 −
1 n u ( xi ) − uB ( xi ) + vA ( xi ) − vB ( xi ) 2n i =1 A
1 n 1 n (u A ( xi ) − vA ( xi )) − (u B ( xi ) − vB ( xi )) − u A ( xi ) − u B ( xi ) + v A ( xi ) − vB ( xi ) 4n i =1 4n i =1 1/ 2
1 n SO ( A , B ) = 1 − (u A ( xi ) − u B ( xi )) 2 + (v A ( xi ) − vB ( xi )) 2 2 n i =1
Liang and Shi [9] proposed a similarity measure between A~ and B~ as follows:
1 p n (ϕtAB (i )) p where ϕtAB (i ) =| u A ( xi ) − u B ( xi ) | /2 , (i ) + ϕ fAB n i =1 ϕ fAB (i ) =| (1 − v A ( xi )) − (1 − vB ( xi )) | /2 and 1 ≤ p < ∞. To get more information on Sep ( A , B ) = 1 −
p
IFSs, Liang and Shi [9] gave another similarity measure as follows:
1 S sp ( A , B ) = 1 − p n and
n
p
(ϕ i =1
s1
(i ) + ϕ s 2 (i )) p where
ϕ s 2 (i ) =| mA 2 ( xi ) − mB 2 ( xi ) | /2
mA 2 (i ) = (mA (i ) + 1 − v A ( xi )) / 2
,
ϕ s1 (i ) =| mA1 ( xi ) − mB1 ( xi ) | /2
with m A 1 (i ) =
(u A ( xi ) + mA (i )) / 2 ,
mB 1 (i ) = (u B ( xi ) + mB (i )) / 2
and
mA 2 (i ) = (mB (i ) + 1 − vB ( xi )) / 2 . Mitchell [10] interpreted IFSs as ensembles of ordered fuzzy sets from statistical a viewpoint and proposed a similarity measure between A~ and B~ as follows:
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C.-M. Hwang and M.-S. Yang
1 S HB ( A , B ) = ( ρu ( A , B ) + ρ v ( A , B )) 2
where
ρu ( A , B ) = 1 −
1 n
n
p
p
| u i =1
A
( xi ) − uB ( xi ) | p ,
1 p n | vA ( xi ) − vB ( xi ) | p . Hung and Yang [11] proposed several p n i =1 similarity measures of IFSs based on Hausdorff distance. For two IFSs A~ and B~ in IFSs(X), they first defined I A ( xi ) = [u A ( xi ),1 − v A ( xi )] and
ρv ( A , B ) = 1 −
I B ( xi ) = [u B ( xi ),1 − vB ( xi )], i = 1,..., n.
The
Hausdorff
distance
H ( I A ( xi ), I B ( xi )) between I A ( xi ) and I B ( xi ) was then defined as follows: H ( I A ( xi ), I B ( xi )) = max{| u A ( xi ) − u B ( xi ) |,|1 − v A ( xi ) − (1 − vB ( xi )) | ~ ~ ~ ~ They defined the distance A and B d H ( A, B ) between
with
n
1 H ( I A ( xi ), I B ( xi )) . In Hung and Yang [11], they proposed three n i =1 similarity measures of A~ and B~ as follows: 1 e− d H ( A, B ) − e −1 1 − d H ( A , B ) . 2 3 S HY ( A , B ) = 1 − d H ( A , B ), S HY ( A , B ) = , S HY ( A , B ) = −1 1− e 1 + d H ( A , B ) d H ( A , B ) =
3
Modified Cosine Similarity Measure between IFSs
By considering the information carried by the membership and nonmembership degrees in intuitionistic fuzzy sets (IFSs) as a vector representation, Ye [13] proposed a cosine similarity measure between IFSs A~ and B~ based on the cosine concept as follows:
u A~ ( xi ) × u B~ ( xi ) + v A~ ( xi ) × v B~ ( xi ) ~ ~ 1 n C IFS ( A, B ) = n i =1 (u ~ ( xi )) 2 + (v ~ ( xi )) 2 × (u ~ ( xi )) 2 + (v ~ ( xi )) 2 B B A A ~ ~ satisfies the conditions: (P1) In Ye [13], he also proved that C IFS ( A , B) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 ≤ C IFS ( A , B ) ≤ 1 ; (P2) C IFS ( A , B ) = C IFS ( B , A ) ; (P3) C IFS ( A , B ) = 1 if A = B . However, the above three conditions are only for the correlation coefficient. If we consider the general definition of a similarity between IFSs as shown in Definition 3 ~ ~ of Section 2, Ye’s [13] cosine similarity measure C IFS ( A , B ) between IFSs does not satisfy. For example, if we give A = { x , 0.1, 0.1} and B = { x, 0.4 , 0.4}, then ~ ~ C IFS ( A , B ) = 1 . That is, the condition (S2) in Definition 3 is not satisfied. Furthermore, ~ ~ if A = { x , 0.0 , 0.1} and B = { x, 0.1, 0.0} are given, then C ( A , B ) = 0 . This IFS
~ ~
is also unreasonable. We find that Ye’s [13] cosine similarity measure C IFS ( A , B ) is only to consider one side of the information between membership and nonmembership degrees, but not consider the middle-side and opposite-side
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289
information between them. If we also consider the middle information with (u A ( xi ) + 1 − v A ( xi )) 2 and (uB ( xi ) + 1 − vB ( xi )) 2 and the opposite-side information with (1 − u A ( xi )), (1 − u B ( xi )), (1 − v A ( xi )) and (1 − vB ( xi )) , we could get a good similarity measure between IFSs A~ and B~ through the cosine concept. We next utilize them to propose a new similarity measure between IFSs. We first define the two items
C
∗ IFS
1 n ( A , B ) = n i =1 =
∗ ∗∗ CIFS ( A , B ) and CIFS ( A , B ) as follows:
1 + u A ( xi ) − v A ( xi ) 1 + uB ( xi ) − vB ( xi ) ( × ) + vA ( xi ) × vB ( xi ) 2 2 1 + u A ( xi ) − v A ( xi ) 2 1 + uB ( xi ) − vB ( xi ) 2 ( ) + (v A ( xi )) 2 × ( ) + (vB ( xi )) 2 2 2 ((1 + u A ( xi ) − v A ( xi )) × (1 + u B ( xi ) − vB ( xi )) + 4v A ( xi ) × vB ( xi )
1 n n i =1 (1 + u ( xi ) − v ( xi )) 2 + (2v ( xi )) 2 × (1 + u ( xi ) − v ( xi )) 2 + (2v ( xi )) 2 A A A B B B
(1 − u A ( xi )) × (1 − u B ( xi )) + (1 − v A ( xi )) × (1 − vB ( xi )) 1 n ∗∗ CIFS ( A , B ) = n i =1 (1 − u ( xi )) 2 + (1 − v ( xi )) 2 × (1 − u ( xi )) 2 + (1 − v ( xi )) 2 A A B B We propose a new similarity measure between IFSs A~ and B~ as follows:
~ ~ 1 ~ ~ ~ ~ ~ ~ ∗ ∗∗ S IFS ( A, B ) = (C IFS ( A, B ) + C IFS ( A, B ) + C IFS ( A, B )) 3 As defined by equation (1), we can show that Proposition 1. The measure
(1)
~ ~ S IFS ( A, B ) is a similarity measure.
~ ~ S IFS ( A, B ) is a similarity measure between IFSs
~ ~ A and B .
Proof: It is trivial to claim S IFS ( A , B )) = S IFS ( B , A )) that is the condition (S3) of Definition 3. To claim the conditions (S1) and (S3) of Definition 3, we first consider the following induction. For any a ≥ 0, b ≥ 0, x ≥ 0, y ≥ 0 , we have that
(ay − bx)2 ≥ 0 iff (ax + by ) 2 ≤ (a 2 + b 2 )( x 2 + y 2 ) iff 0 ≤ ( ax + by ) ≤ a 2 + b 2 x 2 + y 2 ax + by iff 0 ≤ ≤ 1 . Since u A ( xi ), u B ( xi ), v A ( xi ) and 2 2 2 2 a +b x + y
vB ( xi ) are all
∗ between 0 and 1, it is easy to show that 0 ≤ CIFS ( A , B ) ≤ 1 , 0 ≤ CIFS ( A , B ) ≤ 1 and
∗∗ 0 ≤ CIFS ( A , B ) ≤ 1
based
on
the
fact
0≤
ax + by a + b2 x2 + y 2 2
≤1
a ≥ 0, b ≥ 0, x ≥ 0, y ≥ 0 . , B )) ≤ 1 that is the condition (S1) of Definition 3. We then have 0 ≤ S IFS ( A
for
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Similarly, we consider the following induction:
~ ~ ∗ ∗∗ S IFS ( A, B ) = 1 iff (C IFS ( A , B ) + C IFS ( A , B ) + C IFS ( A , B )) = 3 iff ∗ , B ) = 1 and C ∗∗ ( A , B ) = 1 iff C ( A , B ) = 1 , C ( A IFS
IFS
IFS
u A~ ( xi ) × vB~ ( xi ) = v A~ ( xi ) × u B~ ( xi ) , ((1 + u A~ ( xi ) − vA~ ( xi )) × vB~ ( xi ) = vA~ ( xi ) × (1 + uB~ ( xi ) − vB~ ( xi )) , (1 − u A~ ( xi )) × (1 − vB~ ( xi )) = (1 − v A~ ( xi )) × (1 − u B~ ( xi ))
~ ~ u A ( xi ) = u B ( xi ) and vB ( xi ) = v A ( xi ) iff A = B . ~ ~ ~ ~ Thus, we claim S ( A, B ) = 1 iff A = B that is the condition (S2) in Definition 3. iff
~ ~ satisfies the condition (S4) in Definition 3 as We next prove that S IFS ( A , B) follows. ~ ~ ~ , then for each x ∈ X , we have that u ( x) ≤ u ( x) ≤ u ( x ) and If A ⊆ B ⊆C B A C v A ( x) ≥ vB ( x) ≥ vC ( x) . We consider a function f with f ( x) = ( ax + by )
x2 + y2 . Then, we have that d f ( x) = y (ay − bx) . In this case, dx ( x 2 + y 2 )3/2
a = u A ( xi ),
if
b = v A ( xi ),
x = uB ( xi ) and y = vB ( xi ) , then d f ( x) < 0 . That is, f (x) is a dx decreasing
function
g ( y ) = ( ax + by )
of
x.
Let
us
consider
another
function
g
with
x2 + y2 .
d x(bx − ay ) . In this case, if a = u ~ ( x ), b = v ~ ( x ), g ( y) = 2 i i A A dy ( x + y 2 )3/2 x = uB ( xi ) and y = vB ( xi ) , then d g ( y ) > 0 . That is, g ( y ) is an increasing dy
Then, we have
function of y. Hence, we have
CIFS ( A , C ) =
≤
u A ( xi ) × uC ( xi ) + vA ( xi ) × vC ( xi )
((u A ( xi )) 2 + (v A ( xi )) 2 ) × ((uC ( xi ))2 + (vC ( xi ))2 ) u A ( xi ) × u B ( xi ) + v A ( xi ) × vB ( xi ) ((u A ( xi )) + (v A ( xi )) ) × ((uB ( xi )) + (vB ( xi )) ) 2
2
2
2
= CIFS ( A , B )
Modified Cosine Similarity Measure between Intuitionistic Fuzzy Sets
Similarly, we can claim that
291
∗ CIFS ( A , C ) ≤ CIFS ( B , C ) . We know that CIFS and
∗∗ CIFS have a similar form as CIFS with
ax + by a + b2 x2 + y2
so
that,
2
∗ ∗ CIFS ( A , C ) ≤ CIFS ( A , B ) , ∗ ∗ ∗∗ ∗∗ ∗∗ ∗∗ CIFS ( A , C ) ≤ CIFS ( B , C ), CIFS ( A , C ) ≤ CIFS ( A , B ), CIFS ( A , C ) ≤ CIFS ( B , C ).
for
A ⊆ B ⊆ C ,
Hence, we have
we
can
claim
~ ~ ~ ~ S IFS ( A, C ) ≤ S IFS ( A, B )
that
and
~ ~ ~ ~ S IFS ( A, C ) ≤ S IFS ( B , C )
if
~ ~ ~ A ⊆ B ⊆ C . This is the condition (S4) in Definition 3. We next prove
~ ~
that S IFS ( A, B ) satisfies the condition (S5) in Definition 3 as follows. ∗ c c ~ ~ ∗∗ c S IFS ( A, ( A) c ) = 0 iff C IFS ( A , ( A ) ) = 0, CIFS ( A ,( A ) ) = 0, CIFS ( A , ( A ) ) = 0 iff u ~ ( x ) × v ~ ( x ) = 0 , ((1 + u A ( xi ) − vA ( xi )) × (1 + vA ( xi ) − u A ( xi )) + 4v A ( xi ) × u A ( xi ) = 0, i i A A
(1 − u A ( xi )) × (1 − v A ( xi )) = 0 iff u A ( xi ) = 1, vA ( xi ) = 0
~ vA ( xi ) = 1 iff A is a crisp set. Thus, the proof is completed.
or
u A ( xi ) = 0 , □
If we follow the proof of Proposition 1, we can find that the cosine similarity measure CIFS between IFSs proposed by Ye [13] can satisfy (S4), but it cannot satisfy (S2) and (S5). We next use an example to demonstrate this phenomenon and also make comparisons of our modified similarity measure with some existing measures. Example 1. We consider the data used in Ye [13] where the data were first used in Li et al. [14]. The six data sets are as follows: X 1 = { A = (x, 0.3, 0.3), B = (x, 0.4, 0.4)}, X 2 = { A = (x, 0.3, 0.4), B = (x, 0.4, 0.3)},
X 3 = {A = (x, 1, 0), B = (x, 0, 0)}, X = { A = (x, 0.5, 0.5), B = (x, 0, 0)}, 4
X 5 = { A = (x, 0.4, 0.2), B = (x, 0.5, 0.3)}, X 6 = { A = (x, 0.4, 0.2), B = (x, 0.5, 0.2)}.
and B are The degrees of some existing similarity measures between the two IFSs A shown in Table 1. We find that, for the 1st data set X 1 with A ≠ B , Ye’s [13]
~
~ similarity measure C IFS ( A , B ) is given with CIFS ( A , B ) = 1 . Note that similar cases ~ ~
from C IFS ( A , B ) are also occurred for the 3rd X 3 and 4th X 4 data sets. Obviously,
CIFS ( A , B ) does not satisfy the condition (S2) of Definition 3 for a similarity measure. However, most other similarity measures give the result with CIFS ( A , B ) ≠ 1 . On the other hand, by comparing the 5th data set X 5 and the 6th data set X 6 , the similarity S ( X 6 ) for X 6 should be larger than the similarity S ( X 5 ) for X 5 . We find that the similarity measures S H , SO , S HB , Sep and our
S IFS present the correct
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case with S ( X 6 ) > S ( X 5 ) , but others do not. Our modified cosine similarity measure
S IFS actually corrects the drawbacks of the Ye’s measure CIFS . and B Table 1. Similarity measures between IFSs A
X1
X2
X3
X4
X5
X6
SC
1
0.9
0.5
1
1
0.95
SH
0.9
0.9
0.5
0.5
0.9
0.95
SL
0.95
0.9
0.5
0.75
0.95
0.95
SO
0.9
0.9
0.3
0.5
0.9
0.93
S DC
1
0.9
0.5
1
1
0.95
S HB
0.9
0.9
0.5
0.5
0.9
0.95
S
p e
0.9
0.9
0.5
0.5
0.9
0.95
S
p s
0.95
0.9
0.5
0.75
0.95
0.95
S
1 HY
0.9
0.9
0
0.5
0.9
0.9
S
2 HY
0.85
0.85
0
0.38
0.85
0.85
S
3 HY
0.82
0.82
0
0.33
0.82
0.82
CIFS
1
0.96
0
0
0.9971
0.9965
S IFS
0.997
0.859
0.902
0.902
0.995
0.997
Data sets
4
Conclusions
Although Ye’s [13] cosine similarity measure has good concept and merit, the measure is not satisfied the definition of a similarity between IFSs. In this paper we analyzed the drawback of Ye’s [13] similarity measure and then modify it. We showed that the modified measure satisfies the definition of a similarity between IFSs. The example presented better results of our modified similarity measure than Ye’s [13] measure and some other existing measures.
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4. Atanassov, K., Georgeiv, G.: Intuitionistic Fuzzy Prolog. Fuzzy Sets and Systems 53, 121–128 (1993) 5. Pankowska, A., Wygralak, M.: General IF-Sets with Triangular Norms and Their Applications to Group Decision Making. Information Sciences 176, 2713–2754 (2006) 6. De, S.K., Biswas, R., Roy, A.R.: An Application of Intuitionistic Fuzzy Sets in Medical Diagnosis. Fuzzy Sets and Systems 117, 209–213 (2001) 7. Hung, W.L., Yang, M.S.: On the J-Divergence of Intuitionistic Fuzzy Sets with Its Application to Pattern Recognition. Information Sciences 178, 1641–1650 (2008) 8. Dengfeng, L., Chuntian, C.: New Similarity Measures of Intuitionistic Fuzzy Sets and Application to Pattern Recognition. Pattern Recognition Letters 23, 221–225 (2002) 9. Liang, Z., Shi, P.: Similarity Measures on Intuitionistic Fuzzy Sets. Pattern Recognition Letters 24, 2687–2693 (2003) 10. Mitchell, H.B.: On the Dengfeng-Chuntian Similarity Measure and Its Application to Pattern Recognition. Pattern Recognition Letters 24, 3101–3104 (2003) 11. Hung, W.L., Yang, M.S.: Similarity Measures of Intuitionistic Fuzzy Sets Based on Hausdorff Distance. Pattern Recognition Letters 25, 1603–1611 (2004) 12. Xu, Z.S., Chen, J.: An Overview of Distance and Similarity Measures of Intuitionistic Fuzzy Sets. International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems 16, 529–555 (2008) 13. Ye, J.: Cosine Similarity Measures for Intuitionistic Fuzzy Sets and Their Applications. Mathematical and Computer Modelling 53, 91–97 (2011) 14. Li, Y., Olson, D.L., Qin, Z.: Similarity Measures between Intuitionistic Fuzzy (Vague) Sets: A Comparative Analysis. Pattern Recognition Letters 28, 2687–2693 (2007) 15. Chen, S.M.: Measures of Similarity between Vague Sets. Fuzzy Sets and Systems 74, 217–223 (1995) 16. Hong, D.H., Kim, C.: A Note on Similarity Measures between Vague Sets and between Elements. Information Science 115, 83–96 (1999)
2
Department of Applied Mathematics, Chinese Culture University, Taipei, Taiwan Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taiwan [email protected]
Abstract. Similarity of intuitionistic fuzzy sets (IFSs) is an important measure to indicate the similarity degree between IFSs. Recently, Ye (2011) proposed a similarity measure between IFSs based on the cosine concept. Although this cosine similarity measure has good concept and merit, the measure is not satisfied the definition of a similarity between IFSs and not presented well for analyzing IFS data. In this paper, we modify the cosine similarity measure between IFSs. This modified similarity measure between IFSs is not only to satisfy the definition of a similarity between IFSs, but also to improve the efficiency of the Ye’s measure. An example is used to demonstrate this phenomenon. Keywords: Fuzzy set, Intuitionistic fuzzy set, Similarity measure, Cosine similarity measure.
1
Introduction
Fuzzy sets, first introduced by Zadeh [1], give an approach for treating fuzziness. In fuzzy sets, the degree of nonbelongingness is just the complement to 1 of the membership degree. However, humans who expresses the degree of membership of a given element in a fuzzy set very often does not express a corresponding degree of nonmembership as the complement to 1. Thus, Atanassov [2] introduced the concept of an intuitionistic fuzzy set (IFS) which is a generalization of a fuzzy set. Since an IFS can present the degrees of membership and nonmembership with a degree of hesitancy, the knowledge and semantic representation become more meaningful and applicable [3-4]. These IFSs have been widely studied and applied in various areas such as decision making problems [5], medical diagnosis [6], pattern recognition [7]. Similarity measures are an important tool for determining the degree of similarity between two objects. Different similarity measures between fuzzy sets have been proposed and similarity measures between IFSs are also widely studied in the literature. Dengfeng and Chuntian [8] proposed some similarity measures between IFSs used in pattern recognition problem. Liang and Shi [9] proposed similarity measures between IFSs in which they used numerical comparisons to show that Liang and Shi’s similarity measures are more reasonable than those of Dengfeng and Chuntian. Mitchell [10] interpreted IFSs as ensembles of ordered fuzzy sets from the *
Corresponding author.
J. Lei et al. (Eds.): AICI 2012, LNAI 7530, pp. 285–293, 2012. © Springer-Verlag Berlin Heidelberg 2012
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C.-M. Hwang and M.-S. Yang
statistical point. Hung and Yang [11] proposed several similarity measures between IFSs based on Hausdorff distance which are well used with linguistic variables. Xu and Chen [12] gave a comprehensive overview of distance and similarity measures between IFSs. Recently, Ye [13] proposed a cosine similarity measure between IFSs. Although this cosine similarity measure has good concept and merit, the measure cannot satisfy the definition of a similarity between IFSs. In this paper, we modify this cosine similarity measure such that the modified measure can satisfy the definition of a similarity between IFSs. Some examples are used to demonstrate the efficiency of the modified cosine similarity measure.
2
Intuitionistic Fuzzy Set and Similarity Measures
Let X = {x1 , x2 ,..., xn } be the universe of discrete discourses. Consider two intuitionistic fuzzy sets (IFSs) A~ and B~ in X . We first describe the aspects of IFSs discussed by Atanassov [2] as follows. Definition 1. (Atanassov [2]) An intuitionistic fuzzy set (IFS) A = {( x, μ A ( x),ν A ( x)) x ∈ X )} where μ A : X → [0,1] and
A in X is defined as ν A : X → [0,1] with
the condition 0 ≤ μ A + ν A ( x ) ≤ 1, ∀x ∈ X .
~ A in X , the numbers μ A ( x) and ν A ( x) denote the degree of membership and non-membership of x to A , respectively, and the number ~ π A ( x) = 1 − μ A ( x) −ν A ( x) denotes a hesitancy degree of x to A . In this paper, we use IFSs(X) to denote the class of all IFSs of X .
For each IFS
Definition 2. If A~ and B~ are two IFSs of X, then ~ ~ (i) A ⊆ B if and only if ∀ x ∈ X , u A ( x) ≤ uB ( x) and
v A~ ( x) ≥ vB~ ( x) .
(ii) A = B if and only if ∀ x ∈ X , u A ( x) = uB ( x) and
v A~ ( x ) = vB~ ( x ) .
~
~
Measuring a similarity between IFSs is important in IFSs researches. Some methods had been proposed to calculate the similarity degree between IFSs where Li et al. [14] introduced the following definition.
× IFSs(X) →[0,1] .
~ ~ S ( A, B ) is ~ ~ satisfies said to be the degree of similarity between A~ and B~ in IFSs(X) if S ( A , B)
Definition 3. (Li et al. [14]) A mapping S: IFSs(X)
the following properties:
~
~ (S1) 0 ≤ S ( A, B ) ≤ 1;
~ ~ ~ ~ S ( A, B ) = 1 iff A = B ; ~ ~ ~ ~ ; (S3) S ( A, B ) = S ( B , A) ~ ~ ~ ~ ~ ~ ~ in IFSs(X). (S4) S ( A, C ) ≤ S ( A, B ) and S ( A , C ) ≤ S ( B , C ) if A ⊆B⊆C (S2)
(S5)
~ ~ S ( A, A C ) = 0
~
iff A is a crisp set.
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287
Consider two IFSs A~ and B~ in IFSs(X), Dengfeng and Chuntian [8] proposed a similarity measure between them as follows:
1 S DC ( A , B ) = 1 − p n
n
p
| m i =1
A
(i) − mB (i) | p
where m A (i ) = (u A ( xi ) + 1 − v A ( xi )) / 2, mB (i) = (u B ( xi ) + 1 − vB ( xi )) / 2, 1 ≤ p < ∞.
S DC ( A , B ) = SC ( A , B ) where SC ( A , B ) was in Chen [15] as
When p = 1, follow:
1 n SC ( A , B ) = 1 − (u A ( xi ) − v A ( xi )) − (uB ( xi ) − vB ( xi )) 2n i =1 , B ) , S ( A , B ) and Hong and King [16] proposed new similarity measures S H ( A L S ( A, B) (also see Li et al. [14]) as follows: O
S H ( A , B ) = 1 − S L ( A , B ) = 1 −
1 n u ( xi ) − uB ( xi ) + vA ( xi ) − vB ( xi ) 2n i =1 A
1 n 1 n (u A ( xi ) − vA ( xi )) − (u B ( xi ) − vB ( xi )) − u A ( xi ) − u B ( xi ) + v A ( xi ) − vB ( xi ) 4n i =1 4n i =1 1/ 2
1 n SO ( A , B ) = 1 − (u A ( xi ) − u B ( xi )) 2 + (v A ( xi ) − vB ( xi )) 2 2 n i =1
Liang and Shi [9] proposed a similarity measure between A~ and B~ as follows:
1 p n (ϕtAB (i )) p where ϕtAB (i ) =| u A ( xi ) − u B ( xi ) | /2 , (i ) + ϕ fAB n i =1 ϕ fAB (i ) =| (1 − v A ( xi )) − (1 − vB ( xi )) | /2 and 1 ≤ p < ∞. To get more information on Sep ( A , B ) = 1 −
p
IFSs, Liang and Shi [9] gave another similarity measure as follows:
1 S sp ( A , B ) = 1 − p n and
n
p
(ϕ i =1
s1
(i ) + ϕ s 2 (i )) p where
ϕ s 2 (i ) =| mA 2 ( xi ) − mB 2 ( xi ) | /2
mA 2 (i ) = (mA (i ) + 1 − v A ( xi )) / 2
,
ϕ s1 (i ) =| mA1 ( xi ) − mB1 ( xi ) | /2
with m A 1 (i ) =
(u A ( xi ) + mA (i )) / 2 ,
mB 1 (i ) = (u B ( xi ) + mB (i )) / 2
and
mA 2 (i ) = (mB (i ) + 1 − vB ( xi )) / 2 . Mitchell [10] interpreted IFSs as ensembles of ordered fuzzy sets from statistical a viewpoint and proposed a similarity measure between A~ and B~ as follows:
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1 S HB ( A , B ) = ( ρu ( A , B ) + ρ v ( A , B )) 2
where
ρu ( A , B ) = 1 −
1 n
n
p
p
| u i =1
A
( xi ) − uB ( xi ) | p ,
1 p n | vA ( xi ) − vB ( xi ) | p . Hung and Yang [11] proposed several p n i =1 similarity measures of IFSs based on Hausdorff distance. For two IFSs A~ and B~ in IFSs(X), they first defined I A ( xi ) = [u A ( xi ),1 − v A ( xi )] and
ρv ( A , B ) = 1 −
I B ( xi ) = [u B ( xi ),1 − vB ( xi )], i = 1,..., n.
The
Hausdorff
distance
H ( I A ( xi ), I B ( xi )) between I A ( xi ) and I B ( xi ) was then defined as follows: H ( I A ( xi ), I B ( xi )) = max{| u A ( xi ) − u B ( xi ) |,|1 − v A ( xi ) − (1 − vB ( xi )) | ~ ~ ~ ~ They defined the distance A and B d H ( A, B ) between
with
n
1 H ( I A ( xi ), I B ( xi )) . In Hung and Yang [11], they proposed three n i =1 similarity measures of A~ and B~ as follows: 1 e− d H ( A, B ) − e −1 1 − d H ( A , B ) . 2 3 S HY ( A , B ) = 1 − d H ( A , B ), S HY ( A , B ) = , S HY ( A , B ) = −1 1− e 1 + d H ( A , B ) d H ( A , B ) =
3
Modified Cosine Similarity Measure between IFSs
By considering the information carried by the membership and nonmembership degrees in intuitionistic fuzzy sets (IFSs) as a vector representation, Ye [13] proposed a cosine similarity measure between IFSs A~ and B~ based on the cosine concept as follows:
u A~ ( xi ) × u B~ ( xi ) + v A~ ( xi ) × v B~ ( xi ) ~ ~ 1 n C IFS ( A, B ) = n i =1 (u ~ ( xi )) 2 + (v ~ ( xi )) 2 × (u ~ ( xi )) 2 + (v ~ ( xi )) 2 B B A A ~ ~ satisfies the conditions: (P1) In Ye [13], he also proved that C IFS ( A , B) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 ≤ C IFS ( A , B ) ≤ 1 ; (P2) C IFS ( A , B ) = C IFS ( B , A ) ; (P3) C IFS ( A , B ) = 1 if A = B . However, the above three conditions are only for the correlation coefficient. If we consider the general definition of a similarity between IFSs as shown in Definition 3 ~ ~ of Section 2, Ye’s [13] cosine similarity measure C IFS ( A , B ) between IFSs does not satisfy. For example, if we give A = { x , 0.1, 0.1} and B = { x, 0.4 , 0.4}, then ~ ~ C IFS ( A , B ) = 1 . That is, the condition (S2) in Definition 3 is not satisfied. Furthermore, ~ ~ if A = { x , 0.0 , 0.1} and B = { x, 0.1, 0.0} are given, then C ( A , B ) = 0 . This IFS
~ ~
is also unreasonable. We find that Ye’s [13] cosine similarity measure C IFS ( A , B ) is only to consider one side of the information between membership and nonmembership degrees, but not consider the middle-side and opposite-side
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289
information between them. If we also consider the middle information with (u A ( xi ) + 1 − v A ( xi )) 2 and (uB ( xi ) + 1 − vB ( xi )) 2 and the opposite-side information with (1 − u A ( xi )), (1 − u B ( xi )), (1 − v A ( xi )) and (1 − vB ( xi )) , we could get a good similarity measure between IFSs A~ and B~ through the cosine concept. We next utilize them to propose a new similarity measure between IFSs. We first define the two items
C
∗ IFS
1 n ( A , B ) = n i =1 =
∗ ∗∗ CIFS ( A , B ) and CIFS ( A , B ) as follows:
1 + u A ( xi ) − v A ( xi ) 1 + uB ( xi ) − vB ( xi ) ( × ) + vA ( xi ) × vB ( xi ) 2 2 1 + u A ( xi ) − v A ( xi ) 2 1 + uB ( xi ) − vB ( xi ) 2 ( ) + (v A ( xi )) 2 × ( ) + (vB ( xi )) 2 2 2 ((1 + u A ( xi ) − v A ( xi )) × (1 + u B ( xi ) − vB ( xi )) + 4v A ( xi ) × vB ( xi )
1 n n i =1 (1 + u ( xi ) − v ( xi )) 2 + (2v ( xi )) 2 × (1 + u ( xi ) − v ( xi )) 2 + (2v ( xi )) 2 A A A B B B
(1 − u A ( xi )) × (1 − u B ( xi )) + (1 − v A ( xi )) × (1 − vB ( xi )) 1 n ∗∗ CIFS ( A , B ) = n i =1 (1 − u ( xi )) 2 + (1 − v ( xi )) 2 × (1 − u ( xi )) 2 + (1 − v ( xi )) 2 A A B B We propose a new similarity measure between IFSs A~ and B~ as follows:
~ ~ 1 ~ ~ ~ ~ ~ ~ ∗ ∗∗ S IFS ( A, B ) = (C IFS ( A, B ) + C IFS ( A, B ) + C IFS ( A, B )) 3 As defined by equation (1), we can show that Proposition 1. The measure
(1)
~ ~ S IFS ( A, B ) is a similarity measure.
~ ~ S IFS ( A, B ) is a similarity measure between IFSs
~ ~ A and B .
Proof: It is trivial to claim S IFS ( A , B )) = S IFS ( B , A )) that is the condition (S3) of Definition 3. To claim the conditions (S1) and (S3) of Definition 3, we first consider the following induction. For any a ≥ 0, b ≥ 0, x ≥ 0, y ≥ 0 , we have that
(ay − bx)2 ≥ 0 iff (ax + by ) 2 ≤ (a 2 + b 2 )( x 2 + y 2 ) iff 0 ≤ ( ax + by ) ≤ a 2 + b 2 x 2 + y 2 ax + by iff 0 ≤ ≤ 1 . Since u A ( xi ), u B ( xi ), v A ( xi ) and 2 2 2 2 a +b x + y
vB ( xi ) are all
∗ between 0 and 1, it is easy to show that 0 ≤ CIFS ( A , B ) ≤ 1 , 0 ≤ CIFS ( A , B ) ≤ 1 and
∗∗ 0 ≤ CIFS ( A , B ) ≤ 1
based
on
the
fact
0≤
ax + by a + b2 x2 + y 2 2
≤1
a ≥ 0, b ≥ 0, x ≥ 0, y ≥ 0 . , B )) ≤ 1 that is the condition (S1) of Definition 3. We then have 0 ≤ S IFS ( A
for
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Similarly, we consider the following induction:
~ ~ ∗ ∗∗ S IFS ( A, B ) = 1 iff (C IFS ( A , B ) + C IFS ( A , B ) + C IFS ( A , B )) = 3 iff ∗ , B ) = 1 and C ∗∗ ( A , B ) = 1 iff C ( A , B ) = 1 , C ( A IFS
IFS
IFS
u A~ ( xi ) × vB~ ( xi ) = v A~ ( xi ) × u B~ ( xi ) , ((1 + u A~ ( xi ) − vA~ ( xi )) × vB~ ( xi ) = vA~ ( xi ) × (1 + uB~ ( xi ) − vB~ ( xi )) , (1 − u A~ ( xi )) × (1 − vB~ ( xi )) = (1 − v A~ ( xi )) × (1 − u B~ ( xi ))
~ ~ u A ( xi ) = u B ( xi ) and vB ( xi ) = v A ( xi ) iff A = B . ~ ~ ~ ~ Thus, we claim S ( A, B ) = 1 iff A = B that is the condition (S2) in Definition 3. iff
~ ~ satisfies the condition (S4) in Definition 3 as We next prove that S IFS ( A , B) follows. ~ ~ ~ , then for each x ∈ X , we have that u ( x) ≤ u ( x) ≤ u ( x ) and If A ⊆ B ⊆C B A C v A ( x) ≥ vB ( x) ≥ vC ( x) . We consider a function f with f ( x) = ( ax + by )
x2 + y2 . Then, we have that d f ( x) = y (ay − bx) . In this case, dx ( x 2 + y 2 )3/2
a = u A ( xi ),
if
b = v A ( xi ),
x = uB ( xi ) and y = vB ( xi ) , then d f ( x) < 0 . That is, f (x) is a dx decreasing
function
g ( y ) = ( ax + by )
of
x.
Let
us
consider
another
function
g
with
x2 + y2 .
d x(bx − ay ) . In this case, if a = u ~ ( x ), b = v ~ ( x ), g ( y) = 2 i i A A dy ( x + y 2 )3/2 x = uB ( xi ) and y = vB ( xi ) , then d g ( y ) > 0 . That is, g ( y ) is an increasing dy
Then, we have
function of y. Hence, we have
CIFS ( A , C ) =
≤
u A ( xi ) × uC ( xi ) + vA ( xi ) × vC ( xi )
((u A ( xi )) 2 + (v A ( xi )) 2 ) × ((uC ( xi ))2 + (vC ( xi ))2 ) u A ( xi ) × u B ( xi ) + v A ( xi ) × vB ( xi ) ((u A ( xi )) + (v A ( xi )) ) × ((uB ( xi )) + (vB ( xi )) ) 2
2
2
2
= CIFS ( A , B )
Modified Cosine Similarity Measure between Intuitionistic Fuzzy Sets
Similarly, we can claim that
291
∗ CIFS ( A , C ) ≤ CIFS ( B , C ) . We know that CIFS and
∗∗ CIFS have a similar form as CIFS with
ax + by a + b2 x2 + y2
so
that,
2
∗ ∗ CIFS ( A , C ) ≤ CIFS ( A , B ) , ∗ ∗ ∗∗ ∗∗ ∗∗ ∗∗ CIFS ( A , C ) ≤ CIFS ( B , C ), CIFS ( A , C ) ≤ CIFS ( A , B ), CIFS ( A , C ) ≤ CIFS ( B , C ).
for
A ⊆ B ⊆ C ,
Hence, we have
we
can
claim
~ ~ ~ ~ S IFS ( A, C ) ≤ S IFS ( A, B )
that
and
~ ~ ~ ~ S IFS ( A, C ) ≤ S IFS ( B , C )
if
~ ~ ~ A ⊆ B ⊆ C . This is the condition (S4) in Definition 3. We next prove
~ ~
that S IFS ( A, B ) satisfies the condition (S5) in Definition 3 as follows. ∗ c c ~ ~ ∗∗ c S IFS ( A, ( A) c ) = 0 iff C IFS ( A , ( A ) ) = 0, CIFS ( A ,( A ) ) = 0, CIFS ( A , ( A ) ) = 0 iff u ~ ( x ) × v ~ ( x ) = 0 , ((1 + u A ( xi ) − vA ( xi )) × (1 + vA ( xi ) − u A ( xi )) + 4v A ( xi ) × u A ( xi ) = 0, i i A A
(1 − u A ( xi )) × (1 − v A ( xi )) = 0 iff u A ( xi ) = 1, vA ( xi ) = 0
~ vA ( xi ) = 1 iff A is a crisp set. Thus, the proof is completed.
or
u A ( xi ) = 0 , □
If we follow the proof of Proposition 1, we can find that the cosine similarity measure CIFS between IFSs proposed by Ye [13] can satisfy (S4), but it cannot satisfy (S2) and (S5). We next use an example to demonstrate this phenomenon and also make comparisons of our modified similarity measure with some existing measures. Example 1. We consider the data used in Ye [13] where the data were first used in Li et al. [14]. The six data sets are as follows: X 1 = { A = (x, 0.3, 0.3), B = (x, 0.4, 0.4)}, X 2 = { A = (x, 0.3, 0.4), B = (x, 0.4, 0.3)},
X 3 = {A = (x, 1, 0), B = (x, 0, 0)}, X = { A = (x, 0.5, 0.5), B = (x, 0, 0)}, 4
X 5 = { A = (x, 0.4, 0.2), B = (x, 0.5, 0.3)}, X 6 = { A = (x, 0.4, 0.2), B = (x, 0.5, 0.2)}.
and B are The degrees of some existing similarity measures between the two IFSs A shown in Table 1. We find that, for the 1st data set X 1 with A ≠ B , Ye’s [13]
~
~ similarity measure C IFS ( A , B ) is given with CIFS ( A , B ) = 1 . Note that similar cases ~ ~
from C IFS ( A , B ) are also occurred for the 3rd X 3 and 4th X 4 data sets. Obviously,
CIFS ( A , B ) does not satisfy the condition (S2) of Definition 3 for a similarity measure. However, most other similarity measures give the result with CIFS ( A , B ) ≠ 1 . On the other hand, by comparing the 5th data set X 5 and the 6th data set X 6 , the similarity S ( X 6 ) for X 6 should be larger than the similarity S ( X 5 ) for X 5 . We find that the similarity measures S H , SO , S HB , Sep and our
S IFS present the correct
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case with S ( X 6 ) > S ( X 5 ) , but others do not. Our modified cosine similarity measure
S IFS actually corrects the drawbacks of the Ye’s measure CIFS . and B Table 1. Similarity measures between IFSs A
X1
X2
X3
X4
X5
X6
SC
1
0.9
0.5
1
1
0.95
SH
0.9
0.9
0.5
0.5
0.9
0.95
SL
0.95
0.9
0.5
0.75
0.95
0.95
SO
0.9
0.9
0.3
0.5
0.9
0.93
S DC
1
0.9
0.5
1
1
0.95
S HB
0.9
0.9
0.5
0.5
0.9
0.95
S
p e
0.9
0.9
0.5
0.5
0.9
0.95
S
p s
0.95
0.9
0.5
0.75
0.95
0.95
S
1 HY
0.9
0.9
0
0.5
0.9
0.9
S
2 HY
0.85
0.85
0
0.38
0.85
0.85
S
3 HY
0.82
0.82
0
0.33
0.82
0.82
CIFS
1
0.96
0
0
0.9971
0.9965
S IFS
0.997
0.859
0.902
0.902
0.995
0.997
Data sets
4
Conclusions
Although Ye’s [13] cosine similarity measure has good concept and merit, the measure is not satisfied the definition of a similarity between IFSs. In this paper we analyzed the drawback of Ye’s [13] similarity measure and then modify it. We showed that the modified measure satisfies the definition of a similarity between IFSs. The example presented better results of our modified similarity measure than Ye’s [13] measure and some other existing measures.
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