J. Cent. South Univ. (2013) 20: 982−987 DOI: 10.1007/s11771-013-1574-z

Design of similarity measure for discrete data and application to

multi-dimension

LEE Myeong-ho1, WEI He(魏荷)2, LEE Sang-hyuk2, LEE Sang-min3, SHIN Seung-soo4

1. Department of e-Commerce, Semyung University, 579 Shinwol-dong, Jaechon, 390-711, Korea;

2. Department of Electrical and Electronic Engineering, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China; 3. Institute for Information and Electronics Research, Inha University, 253 Yonghyun-Dong, Nam-Gu,

Incheon, 402-751, Korea; 4. Department of Information Security, Tongmyung University, Daeyeon 3-dong, Nam-gu, Busan, 608-737, Korea

© Central South University Press and Springer-Verlag Berlin Heidelberg 2013

Abstract: Similarity measure design for discrete data group was proposed. Similarity measure design for continuous membership function was also carried out. Proposed similarity measures were designed based on fuzzy number and distance measure, and were proved. To calculate the degree of similarity of discrete data, relative degree between data and total distribution was obtained. Discrete data similarity measure was completed with combination of mentioned relative degrees. Power interconnected system with multi characteristics was considered to apply discrete similarity measure. Naturally, similarity measure was extended to multi-dimensional similarity measure case, and applied to bus clustering problem. Key words: similarity measure; multi-dimension; discrete data; relative degree; power interconnected system

Foundation item: Project(2010-0020163) supported by Key Research Institute Program through the National Research Foundation of Korea (NRF) funded

by the Ministry of Education, Science and Technology, Korea Received date: 20121105; Accepted date: 20130112 Corresponding author: SHIN Seung-soo; Tel: +82−32−860−8829; E-mail: shinss@tu.ac.kr

1 Introduction

Similarity measure design problem has been done to provide the solution of pattern recognition and clustering problem etc [17]. Research of designing similarity measure has been made by numerous researchers [12, 811], then most studies were emphasized on designing similarity measure based on membership function or fuzzy number [811] and distance measure [12]. Even similarity measure design with fuzzy number was easier than that with distance measure, it was possible only for triangular or trapezoidal fuzzy membership function. Whereas similarity measure with distance measure can be applied to unlimited membership function.

Whether the similarity measures were proposed by fuzzy number or distance measure, it provided the degree of similarity between data sets. Degree of similarity between two or more data played central role in the fields of decision making, pattern classification, etc [37]. Until now, the research of designing similarity measure has been made by numerous researchers. Two kinds of similarity measures derived from fuzzy number approach [811] and distance measure [12] were considered based on continuous fuzzy membership function. Hence, it invoked irrational result [11].

The obtained similarity measure cannot guarantee the similarity calculation of singleton distributed data set. Hence, design of similarity measure for singleton data needs different approaches to provide similarity measure to calculate the degree of similarity between discrete data sets. Since singleton or discrete data has no overlapping with other pattern, discrete characteristic was compared with whole data distribution including considered one. After calculating relational degree between discrete data with respect to total distribution similarity measure is obtained, similarity between two data points is considered to satisfy the definition of similarity measure.

In this work, preliminary results on similarity were proposed. Similarity measures based on fuzzy number and distance measure were introduced, and they were applied to discrete date. Similarity measure for discrete data was proposed, and was also proved. Power disconnected system clustering, which had two characteristics of each bus, was done with multi- dimensional similarity measure.

2 Similarity measure preliminaries

Even there are several similarity measure

definitions, following definition is common if it is used in distance measure. LIU [12] suggested axiomatic

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definition of similarity measure as follows. Definition 1 [12]: A real function s: F2R+ is

called a similarity measure, if s has the following

properties:

(S1) s(A, B)=s(B, A), A,BF(X)

(S2) s(D, DC)=0, DP(X)

(S3) s(C, C)= ,max ( , ), ( )A B F

s A B C F X

(S4) , , ( ), if ,A B C F X A B C then s(A, B)

s(A, C) and s(B, C)s(A, C)

where R+=[0, ), X is the universal set, F(X) is the class of all fuzzy sets of X, P(X) is the class of all crisp sets of X, and DC is the complement of D. By this definition, numerous similarity measures could be derived.

2.1 Similarity measure via fuzzy number

In order to understand the similarity measure design with fuzzy number, it is required to study fuzzy number, center of gravity, and axiomatic definitions of similarity measure. A generalized fuzzy number A was defined as A =(a, b, c, d, w) where 0w1 and a, b, c and d were real numbers [78]. Trapezoidal membership function

A of fuzzy number A satisfied the following conditions [8–11]:

If b=c was satisfied, then it would be natural to satisfy triangular type. Four fuzzy number operations were also found in Refs. [8–11].

CHEN and CHEN [11] presented a new method to calculate COG point of a generalized fuzzy number. They derived the new COG calculation method based on the concept of the medium curve. These COG points played an important role in the calculation of similarity measure with fuzzy number.

Now, similarity measure derivations with fuzzy number and distance measure are introduced. First one is based on fuzzy number, and the second one is designed through distance measure. They were all contained in previous results [12]; however, to evaluate the similarity, understand of back ground is needed.

In Ref. [11], degrees of similarities were derived through fuzzy number which was related with membership function, and center of gravity. CHEN [8] introduced the degree of similarity for trapezoidal or triangular fuzzy membership function of A and B as follows:

1

| |

( , ) 14

n

i ii

a b

s A B

(1) where ( , ) [0, 1]s A B . If A and B were triangular or trapezoidal fuzzy numbers, then n could be three or four, respectively. For trapezoidal membership function

fuzzy number satisfied A =(a1, a2, a3, 1) and B =(b1, b2, b3, 1).

HSIEH and CHEN [9] and LEE [10] also proposed similarity measure for the trapezoidal and triangular fuzzy membership function as follows:

1( , )

1 ( , )s A B

d A B

(2)

1/|| ||( , ) 1 4

|| ||p pA B l

s A BU

(3)

CHEN and CHEN [11] proposed similarity measure

to overcome the drawbacks of existing similarity:

( , )* *1

| |

( , ) [1 ] (1 | |)4

BA

n

i iB s si

BA

a b

s A B x x

* *

* *

min( , )

max( , )BA

BA

y x

y x

where * *, A Ax y and * *, B Bx y are the COG of fuzzy

number A and B , As and Bs are expressed by

4 1As a a and 4 1Bs b b if they are trapezoidal.

Traditional center of gravity (COG) is defined by

*( )d

( )d

AA

A

x x xx

x x

where A is the membership function of the fuzzy

number A , ( )A x indicates the membership value of

the element x in A , and generally, [0, 1]A .

( , )BAB S S is denoted by one if 0BAS S , and zero

if BAS S 0. In Eq. (4), ( , )BAB S S was used to

determine whether COG distance was considered or not.

2.2 Similarity measure with distance function To design the similarity measure via distance, it is

needed to introduce the distance measure [12]. Definition 2: A real function d: F2R+ is called a

distance measure on F, if d satisfies the following properties:

(D1) d(A, B)=d(B, A), A, BF(X)

(D2) , 0, ( )d A A A F X (D3) d(D, DC)=

,maxA B F

d(A, B), ( )D F X

(D4) A, B, ( )C F X , if A B C , then d(A, B) d(A, C) and d(B, C)d(A, C)

Hamming distance was commonly used as distance measure between fuzzy sets A and B:

1

1( , ) | ( ) ( ) |

n

A i B ii

d A B x xn

where X={x1, x2, , xn}, |k| is the absolute value of k.

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A(x) is the membership function of AF(X). Following theorem satisfied similarity measure.

Theorem 1: For any set A, BF(X), if d satisfies Hamming distance measure, then

( , ) (( ),[0] ) (( ),[1] )X Xs A B d A B d A B (4) is the similarity measure between set A and set B.

Proof: Commutativity of (S1) was clear from Eq. (4) itself. To show the property of (S2), obtain

C( , ) (( ),[0] ) (( ),[1] )

([0] ,[0] ) ([1] ,[1] ) 0

C CX X

X X X X

s D D d D D d D D

d d

Because ( ) [0]C

XD D and ( )CD D [1]X were satisfied, where, [0]X and [1]X denoted zero and one for the whole universe of discourse of X, (S2) was satisfied. (S3) was also easy to prove:

( , ) (( ),[0] ) (( ),[1] )X Xs C C d C C d C C

( ,[0] ) ( ,[1] ) 1X Xd C d C

It was logical that s(C,C) satisfied maximal value. Finally, triangular equality is obvious by definition, hence (S4) is also satisfied. Besides Theorem 1, numerous similarity measures are possible. One of the similarity measures is illustrated in Theorem 2 as below, and its proof was also found in previous result [13].

Theorem 2: For any set A, BF(X), if d satisfies Hamming distance measure, then , 1 ( , ) ( , )s A B d A A B d B A B (5)

is the similarity measure between set A and set B.

Mentioned similarity was useful for the continuous data distribution application. However, for discrete singleton data, similarity calculation cannot be obtained through Eq. (4) or Eq. (5). For example, even Fig. 1 and Fig. 2 are different data distributions, they show same

Fig. 1 Data distribution (I)

Fig. 2 Data distribution (II)

degree of similarity because of commutativity feature (S1).

That is to say, calculation of s(, ) in Fig. 1 is the same as that of s(, ) in Fig. 2 because of its measure structure

( , ) 1 ( , ) ( , )s d d

Hence, proposed similarity measures Eqs. (4) and (5) cannot guarantee degree of similarity measure between two singleton data sets. Proposed similarity measures are considered under the condition of continuous fuzzy membership function. If there are overlapping data between diamond (♦) and circle (●), it might be available.

Therefore, in order to analyze the degree of similarity between distributed singleton data, another similarity is needed.

3 Similarity measure for group data

Besides Eqs. (4) and (5), most of similarity was also

designed for continuous data distribution. Hence, we needed different approaches to get the solution.

3.1 Distributed singleton data Consider discrete singleton over considered area,

which may be one-dimensional otherwise more. There are 12 mixed data in Figs. 1 and 2, which are one-dimensional data, composed of six diamonds (♦) and circle (●), respectively. In order to discriminate two distributions, another definition is considered.

Theorem 3: For singletons a, bP(X), if d satisfied Hamming distance measure, then s(a,b)=1–|sa–sb| (6) is similarity measure between singletons a and b. In Eq. (6), sa and sb satisfy (( ), [1] )Xd a R and (( ), d b R [1] ),X respectively, where R is whole data distribution including a and b.

Proof: (S1) is clear by the definition. For (S2),

CC

C

, 1 1

(( ),[1] ) (( ), [1] ) 0

D D

X X

s D D s s

d D R d D R

When D satisfies one, (( ), [1] )=0Xd D R and

C(( ), [1] )=1Xd D R , hence following result is obtained.

When D satisfies zero, opposite results are obtained. (S3) is clear from definition,

, 1 =1

(( ), [1] ) (( ), [1] =1

C C

X X

s C C s s

d C R d C R

Finally, (S4) A, B, CF(X), if A<B<C, then

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, 1 =

1 (( ), [1] ) (( ), [1]

1 (( ), [1] ) (( ), [1]

( , )

A B

X X

X X

s A B s s

d A R d B R

d A R d C R

s A C

Because (( ), [1] ) (( ), [1] )X Xd B R d C R is satisfied,

similarly, s(B, C) s(A, C) is also satisfied. Above similarity measures were also designed using

distance measure like Hamming distance. As noted before, conventional measures Eqs. (4) and (5) are not proper for overlapping continuous data distribution. However, when distributed data have multiple characteristic, another similarity measure with Theorem 3 is required.

3.2 Distributed singleton with multiple characteristics In order to apply multi-dimensional similarity

measure, multiple similarity measure should be considered

such as1

( , ).n

ii

s A B Definitely, each similarity measure

should satisfy similarity measure definition. Considering an interconnected electrical system, it

was introduced in “The IEEE reliability test system- 1996-A Report Prepared by the Reliability Test System Task Force of the Application of Probability Methods Subcommittee” and includes 39 buses and 10 generators [14–16].

The power generated by the generators transforms to buses via transmission lines. Since the energy loss is inevitable during the transmission, delivery loss is a factor that will influence the location price of a bus.

Thus, each bus has its own information about location price and geometric location as shown in Table 1. In Table 1, there are two characteristics for 39 buses, locational price and geometrical information, hence it needs

1 21

( , ) ( , ) ( , )n

ii

s A B s A B s A B

In order to analyze 39 points with two information

such as location price and information, two measures are needed. Those are represented by s1 and s2, respectively. For designing similarity measure, it is impossible to do the similarity measure between two nodes because there is no overlap of membership function between any two individual points; technically the similarity measure will be zero for any two points.

To overcome this problem, one idea is considered. Considering points a and b, overall buses (all 39 buses) are regarded as a set R. And similarity in Theorem 3 s(a, b)=1–|sa–sb| is considered as s1(a, b), then s1(a, b) provides similarity between locational prices for each location.

Proposition 1: Let a and b be locational prices for each location, then

1 , 1

1 | (( ), 1 ) (( ), 1 ) |a b

X X

s a b s s

d a R d b R

represents the degree of similarity between a and b.

Proof: Proofs should follow verification from (S1) to (S4). With proof of Theorem 3, it is already done for distinct singleton data.

Table 1 Locational prices and location (normalized geometric information) for each bus

Bus No.

Locational price/($·kW1·h1)

Location Bus No.

Locational price/($·kW1·h1)

LocationBus No.

Locational price/($·kW1·h1)

Location

1 29.21 ( 0.9, 9 ) 14 41.74 (6.6, 6) 27 51.45 (4.6, 3.5)

2 28.53 (0.6, 6.2) 15 43.79 (6.6, 4.9) 28 55.00 (2.7, 1.5)

3 31.40 (3, 7.5) 16 45.84 (6.5, 4) 29 55.00 (2.7, 0.8)

4 32.78 (4.7, 7.5) 17 47.90 (5, 4.5) 30 28.53 (0, 6.2)

5 37.57 (7, 7.6) 18 46.40 (4.2, 6) 31 38.26 (8.3, 6.6)

6 38.26 (8.5, 7.6) 19 45.84 (6.9, 2.8) 32 40.00 (11.3, 5.8)

7 37.81 (9.6, 8.4) 20 45.84 (6.9, 1.7) 33 45.84 (8, 1.7)

8 37.35 (8.5, 9.1) 21 45.84 (8.7, 2.8) 34 45.84 (5.5, 1)

9 30.56 (6.1, 9.5) 22 45.84 (10, 2.8) 35 45.84 (10, 1.6)

10 40.00 (10.8, 5.8) 23 45.84 (11.1, 2.8) 36 45.84 (11.1, 1.6)

11 39.42 (9.7, 6.3) 24 45.84 (8.2, 4.3) 37 24.98 (0.7, 3.7)

12 40.00 (11. 1. 7.1) 25 24.98 (1.4, 4.7) 38 55.00 (2.7, 0)

13 40.58 (8.5, 5.5) 26 55.00 (2.7, 3) 39 29.88 (3.4, 9.5)

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For s2 , inverse of geometric distance is used, which

is also found in Ref. [16]. That is s2(a, b)=2/(1+d') (7) where d' denotes the geometrical distance value.

Generally, similarity measure extension for the multi-dimensional data could be possible. Furthermore, flexible design is also illustrated with weighting values:

1 1 2 2( , ) ( , ) ( , ) ( , )n ns a b s a b s a b s a b

1

( , )n

i i ii

s s a b (8)

where i denotes weighting factor. Applying s1(a, b) to locational price of 39 buses in Table 1, similarity relation is obtained in Fig. 3. The data with shadows are the ones with high similarity measure values. It is easy to see that the neighborhood nodes have a close similarity condition and the high similarity nodes are rather concentrates, thus a general trend could be developed. It is noticeable that some nodes have more shadowed area than some others, because those nodes are located in the center region where nodes are closer to each other.

Fig. 3 Similarity measure between each two nodes

Together with s2(a, b) in Eq. (7), data grouping is

obtained in Figs. 4 and 5. Three data groups are considered for clustering. Neglecting locational prices, 1=0, clustering is obtained in Fig. 4. As considering more for locational prices to 2=0.2, there is some change of grouping, as shown in Fig. 5. This result is obtained by applying fuzzy C-mean clustering method [17]:

3 3 392

1 2 31 1 1

( , , , ) mi ij ij

i i j

J U c c c J u d

(9)

where uij denotes value between 0 and 1, ci is also the center of fuzzy group i, dij satisfies the Euclidean distance between the i-th cluster center and the j-th data point xj, and m is defined by the weighting value. With Lagrange multiplier, necessary conditions for Eq. (9) to reach a minimum are found in Ref. [18].

Hence, locational price similarity between point a and b can be obtained with s1(a, b)=1–|sa–sb|

And distance similarity is considered through s2(a, b)=2/(1+d')

Total similarity is expressed by

1 1 2 2( , ) ( , ) ( , )s a b s a b s a b (10) where 1 and 2 are weighting factors.

Fig. 4 Data grouping (1=0, 2=1)

Fig. 5 Data grouping1=0.2, 2=0.8)

Considering discrete singleton data, proposed

similarity measure Eq. (10) provides proper result, whereas similarity Eq. (4) is efficient to continuous data distribution or much overlapping data.

4 Conclusions

1) Similarity measure for continuous data is

proposed. Similarity measure can be designed through various approaches such as fuzzy number and distance measure. Fuzzy number guarantees easy constructing of measure, however it has an limitation for membership function. Whereas similarity measure with distance measure provides unlimited membership function

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including non-convex. 2) Similarity measure calculation of discrete data is

impossible with proposed measure because it has design based on continuous membership function. In order to calculate the degree of similarity of discrete data, relation of singleton and total distribution is considered. Similarity measure for discrete data is obtained with obtained data relation.

3) Multi-dimension similarity measure is also proposed. Each similarity measure satisfies the definition of similarity measure. Proposed multi-dimensional similarity constitutes two similarity measures. One measure is considered through relational degree between discrete data and total distribution, and the other is proportional to the inverse of distance between buses.

4) Multi-dimensional similarity is applied to the clustering problem of power interconnected system. It is proposed as three groups. Multi-dimensional similarity measure represents different grouping results under the variation of weighting factors of each similarity measure. References

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(Edited by YANG Bing)