A new model of evaluating concept similarity

Knowledge-Based Systems 21 (2008) 842–846

Contents lists available at ScienceDirect

Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

A new model of evaluating concept similarity q Lidong Wang *, Xiaodong Liu Research Center of Information and Control, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Article history: Received 18 November 2007 Accepted 28 March 2008 Available online 4 April 2008 Keywords: Concept similarity Concept lattice Meet irreducible Join irreducible Rough set

a b s t r a c t In this paper, a new similarity model is proposed, which based on rough set to evaluate the similarity degree of the two concepts of concept lattice. The proposed method combines featural and structural information into decision and has a higher correlation with human judgement, which can be viewed as the development of Tversky’s similarity model. Compared with other similarity models this approach is convenient to measure the similarity of the concepts of the large contexts, by which we can avoid constructing Hasse diagram and looking through all concepts of the context. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Formal concept analysis (FCA) proposed by Wille [8], which provides a theoretical framework for the design and discovery concept hierarchies from relational information system. FCA has been used for various applications in fields: conceptual clustering method [4,10], information retrieval and knowledge discovery [16,19], etc. It is especially suitable for exploration of symbolic knowledge (concepts) contained in a formal context, such as a corpus, a database, or an ontology [17]. Currently, FCA techniques are revealing interesting in supporting difficult activities that are becoming fundamental in the development of the semantic web [1,3,5,7]. Assessing concept similarity is one of such activities which is growing in importance within ontology engineering and, in particular, ontology merging and ontology alignment [6,7,9]. With the rapid development of the semantic web, it is likely that the number of ontologies will greatly increase during the next few years, which leads to the arising demand for rapid and accurate assessing concept similarity [22,23]. So, assessing the similarity between concepts has attracted much attention of the researchers (see e.g., [2,6,7,13,17,22,23]). Rough set originated by Pawlak [11], which is another important method to deal with knowledge processing. The basic operators in rough set theory are approximation operations. Using the concepts of lower and upper approximations, knowledge hidden in information tables may be expressed in the form of decision q

Supported by the Natural Science Foundation of China under Grant No. 60575039 and the National Key Basic Research and Development Program of China under Grant No. 2002CB312200-06. * Corresponding author. Tel.: +86 0411 84701479. E-mail address: [email protected] (L. Wang).

0950-7051/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2008.03.042

rules. In recent years, many efforts have been made to compare and combine the two theories [14,20]. Combination of FCA and rough set theory provide some new approaches for data analysis and knowledge discovery [13,15]. In this paper, a new measure model based on rough set is proposed, which preserve more structural and featural information of concept lattice. The advantage of the proposed model consists in the measure is convenient to be calculated from objects and attributes classes sets, especially for the large context. The paper is organized as follows. In the next section the notion of a concept lattice is recalled. In Section 3, the related similarity model is briefly summarized. Successively, in Section 4, a new method for assessing concept similarity is proposed. Finally, a conclusion is drawn in Section 5. 2. Fundamentals of FCA and rough set In FCA, a concept is defined within a context. A context is a triple ðX; M; IÞ, where X and M are two finite sets called objects and attributes, respectively, and I is a binary relation between X and M. In particular, for x 2 X and m 2 M, denote xIm to express that an object x is in a relation I with an attribute m. Given two sets A # X, B # M, we define A0 ¼ fm 2 M j xIm B0 ¼ fx 2 X j xIm

for all for all

x 2 Xg; m 2 Mg

to express that the set of attributes common to the objects in A and the set of objects which have all attributes in B, respectively. Definition 1. ([8]) A formal concept of the context ðX; M; IÞ is a pair ðA; BÞ with A # X; B # M satisfy A0 ¼ B and B0 ¼ A. We call A the

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L. Wang, X. Liu / Knowledge-Based Systems 21 (2008) 842–846

extent and B the intent of the concept ðA; BÞ. LðX; M; IÞ denotes the set of all concepts of the context ðX; M; IÞ. For any ðA1 ; B1 Þ, ðA2 ; B2 Þ 2 LðX; M; IÞ, define ðA1 ; B1 Þ 6 ðA2 ; B2 Þ () A1 # A2 (which is equivalent to B2 # B1 ). Theorem 1. ([8])Let ðX; M; IÞ be a context. Then ðLðX; M; IÞ; 6Þ is a complete lattice in which suprema and infima are given by !00 ! !00 ! ^ _ \ [ [ \ ; ðAt ; Bt Þ ¼ At ; Bt ðAt ; Bt Þ ¼ At ; Bt : t2T

t2T

t2T

t2T

t2T

t2T

In the concept lattice, a concept is meet irreducible if it cannot be written as meet ð^Þ of other concepts. A concept is join irreducible if it cannot be written as join ð_Þ of other concepts.

Example 1. The concept lattice that generated from the context of Table 1 is shown in Fig. 1. There are 12 join-irreducible concepts and 12 meet irreducible generated from Table 1. For instance, concept labeled by 5 is meet irreducible, concept labeled by 25 is join irreducible. The rough set theory is based on equivalence relations, in which approximation operations are a key concept. Let U be a finite set, the domain of discourse, and R an equivalence relation on U, R # U  U. R will generate a partition U=R ¼ fY 1 ; Y 2 ; . . . ; Y m g on U, where Y 1 ; Y 2 ; . . . ; Y m are the equivalence classes generated by the equivalence relation R, and called elementary sets of R. For any H # U, the lower and upper approximations of H are, respectively, defined as follows:

Table 1 Part of the formal contexts of the ontologies A and B [17] m1 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21

A production A processes A prod system A intensive A fattening A growth A feeding system A males A brachiaria A pasture usage A brachiaria 2 A feeding system 2 B production B prod systems B feeding B concentrate food B calves B postweaning B preweaning B elephantGrass B intensive

m2

x x x x x x x x x x x x x x x x x x x x x

m3

x x x x x x x x x

m4

m5

m6

m7

m8

x x x x x

x x

x

x x x

x

x x

x x

x x x x

x x x

x x x x x x x

m9

m10

m11

m12

m13

x x x x x x x x x x x x

m14

m15

x

x x x x x x x x x x x x x

x x

x

x x x

x x x x

Where mi ði ¼ 1; . . . ; 15Þ represent attributes Production, AnimallHusbandryMethods, AnimallFeeding, Growth, Sex, Poaceae, FeedingSystems, GrazingSystems, BeefCattle, DairyCattle, Postweaning, Preweaning, Pennisetum, DevelopmentalStages, Braquiaria, respectively.

m1 1

x13, x14 m7

m 10 2

x15, x16

x17 x19 m 12

16

3

m2

7

11

x18

17

x1, x2, x3 m9

4

m 14 5

6

8

9

10

x4

13

14

x6 m4

19

20

x12

m 15 22

23

x10

m8

12

m6

18

15

x5, x7 m3

24

x8 m5

26

x9

m 11

x20, x21 m 13

21

x11

25

27 Fig. 1. The concept lattice generated from Table 1.

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L. Wang, X. Liu / Knowledge-Based Systems 21 (2008) 842–846

RLA ðHÞ ¼ [fY i 2 U=R j Y i # Hg;

RUA ðHÞ ¼ [fY i 2 U=R j Y i \ H–;g:

3. Related works of Tversky’s model By using set theory, Tversky [18] defined a similarity measure in terms of a matching process Sða; bÞ ¼

f ðB1 \ B2 Þ f ðB1 \ B2 Þ þ af ðB1  B2 Þ þ bf ðB2  B1 Þ

ð3:1Þ

where a, b P 0, f measures the contribution of any particular (common or distinctive) feature to the similarity between objects, B1 and B2 are the set of features of a and b, respectively. In this approach, objects are represented as collections of features, and similarity is described as a feature matching process. In [12], Rodriguez and Egenhofer have proposed an assessment of semantic similarity among entity classes in different ontologies based on the normalization of Tversky’s similarity model [18]. The similarity model is a direct extension of Tversky’s model Sða; bÞ ¼

j B1 \ B2 j j B1 \ B2 j þaða; bÞ j B1  B2 j þð1  aða; bÞÞ j B2  B1 j ð3:2Þ

where the function j  j represents the cardinality of a set, aða; bÞ ¼

depthðaÞ , depthðaÞþdepthðbÞ

if

depthðaÞ 6 depthðbÞ;

aða; bÞ ¼

depthðaÞ 1  depthðaÞþdepthðbÞ , if depthðaÞ > depthðbÞ.

Another modification of Tversky’s model is presented by de Souza and Davis [17] to use structural elements of the lattice meet-irreducible elements as features. The concept lattice operations – join ð_Þ and meet ð^Þ are used to compute the meet-irreducible elements ðB1 _ B2 Þ^ . The similarity model is as follows: Sða; bÞ ¼

j ðB1 _ B2 Þ^ j j ðB1 _ B2 Þ^ j þa j ðB1  B2 Þ^ j þð1  aÞ j ðB2  B1 Þ^ j

ð3:3Þ

To further improve the performance of the FCA-based similarity measure, Zhao et al. proposed extension of the measure with the employment of rough set theory [22,23]. The attribute-based similarity measure between two concept sets a and b based on Tversky’s model can be described as follows: Sða; bÞ ¼

j ðB1 _ B2 ÞLA j j ðB1 _ B2 ÞLA j þa j B1LA  B2LA j þð1  aÞ j B2LA  B1LA j

sidered. Moreover, we give an effective method to calculate this similarity measure by using objects classes and attributes classes. Meet-irreducible elements play an important role in concept lattice. Every meet-irreducible element corresponds to one new attribute being added [17]. It is isolated in concept lattice. Every concept of concept lattice can be written as meet of meet-irreducible elements [8,21]. However, join-irreducible elements can also play the same important role with as meet-irreducible elements in concept lattice. Every concept of concept lattice can also be written as join of join-irreducible elements [8,21]. Meet-irreducible elements and join-irreducible elements contain main structure information to the lattice. The similarity measure contained irreducible elements is feasible. Inspired by [13,23], define a relation S over M as following, for any mi , mj 2 M, mi Smj ;

iff

Imi # Imj

ð4:5Þ

where Im ¼ fx 2 X j xImg. It is easy to get that S is a partial relation over M. Denote partial class ½m ¼ fg 2 M j mSgg, which called attributes classes. Similarity, define a relation T on X as following, for any xi , xj 2 X, xi Txj ;

iff

xi I # xj I

ð4:6Þ

where xI ¼ fm 2 M j xImg. T is a partial relation on X. Denote partial class ½x ¼ fh 2 X j xThg, which called objects classes. Lemma 1. ([21]) Let ðX; M; IÞ be formal context, for any x 2 X, m 2 M, both ð½x; ð½xÞ0 Þ and ðð½mÞ0 ; ½mÞ are concepts of ðX; M; IÞ: Denote P ¼ ðð½mÞ0 ; ½mÞ, Q ¼ ð½x; ð½xÞ0 Þ. Denote MI is the set of all meet-irreducible concepts of ðX; M; IÞ except special nodes with empty element in intent; UI is the set of all join-irreducible concepts of ðX; M; IÞ except special nodes with empty element in extent. Lemma 2. ([21]) Let ðX; M; IÞ be formal context, then MI # P, UI # Q . From above discussion, P and Q contain all join-irreducible elements and meet-irreducible elements of ðX; M; IÞ, respectively. Moreover, P and Q are also easy to obtained. In the following, we will use P and Q as candidate sets of low approximation to define similarity measure. By using P and Q, we define low approximation operators of object A and attribute B as follows:

ð3:4Þ

A^LA ¼ extentð_fðx; yÞ 2 P j x # AgÞ

ð4:7Þ

where BLA ¼ intentð^fðx; yÞ 2 L j y # BgÞ, L is the set of all concepts of ðX; M; IÞ.

B^LA ¼ intentð^fðx; yÞ 2 Q j y # BgÞ

ð4:8Þ

4. Similarity measure based on rough set In previous section, we recall some related works of Tversky’s similarity model. In de Souza and Davis’s model, meet irreducible elements are employed. The model is both structural and featural at the same time. In [17,22,23], authors gave a method to identify these elements in the lattice, join-irreducible elements are linked downwards by just one edge, whereas meet irreducible elements are linked upwards by just one edge in Hasse Diagram. Although this method is visual, it seem not convenient to construct a Hasse diagram for large context. In [22,23], Zhao et al. proposed a novel rough similarity measure method based on rough concept lattice theory, in which all concepts are viewed as candidate set in the processing of obtaining low approximation. But it is also not convenient to looking through all concepts for large context. In this section, inspired by [6,7,17,22,23], we will propose a new model to evaluate the similarity of concepts of concept lattice, in which both meet irreducible elements and join irreducible are con-

By using A^LA and B^LA , the similarity of concepts ðA1 ; B1 Þ and ðA2 ; B2 Þ can be defined as follows: j ðA1 \ A2 Þ^LA j 1 j þ 2 j A^1LA  A^2LA j þ 12 j A^2LA  A^1LA j j ðB1 \ B2 Þ^LA j ð4:9Þ þ ð1  xÞ ^ 1 j ðB1 \ B2 ÞLA j þ 2 j B^1LA  B^2LA j þ 12 j B^2LA  B^1LA j

S^LA ððA1 ;B1 Þ;ðA2 ; B2 ÞÞ ¼ x

j ðA1 \ A2 Þ^LA

where x is a weight such that 0 6 x 6 1, that can be established by the user to enrich the flexibility of the method. B^1LA  B^2LA represents the attribute sets in B^1LA but not in B^2LA , and B^2LA  B^1LA presents those attributes in B^2LA but not in B^1LA . Similarly, A^1LA  A^2LA represents the object sets in A^1LA but not in A^2LA , and A^2LA  A^1LA presents those objects in A^2LA but not in A^1LA . Here, the new rough lower approximation operators are introduced. In which the influence from extent of concept is also considered. The more objects and attributes two concepts share, the more similar they are. The proposed method combines featural and structural information into decision and has a higher correlation with human judgement and can be viewed the as development of Tversky’s similarity model.

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L. Wang, X. Liu / Knowledge-Based Systems 21 (2008) 842–846 Table 2 The partial classes of objects and attributes of Table 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

x2X

½x

m2M

½m

x1 ; x2 ; x3 x4 x5 ; x7 x6 x8 x9 x10 x11 x12 x13 ; x14 x15 ; x16 x17 x18 x19 x20 ; x21

fx1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 ; x8 ; x9 ; x10 ; x11 ; x12 g fx4 ; x5 ; x6 ; x7 ; x8 ; x9 ; x10 ; x11 ; x12 g fx5 ; x7 ; x8 ; x9 g fx6 ; x10 ; x11 ; x12 g fx8 ; x9 g fx9 g fx10 ; x11 g fx11 g fx10 ; x11 ; x12 g fx13 ; x14 ; x15 ; x16 ; x17 ; x18 ; x19 ; x20 ; x21 g fx15 ; x16 ; x17 ; x18 ; x19 ; x20 ; x21 g fx17 ; x18 ; x19 g fx18 g fx19 g fx20 ; x21 g

m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 m13 m14 m15

fm1 g fm1 ; m2 g fm1 ; m2 ; m3 ; m9 g fm1 ; m2 ; m4 ; m9 ; m14 g fm1 ; m2 ; m3 ; m5 ; m9 g fm1 ; m2 ; m6 ; m7 ; m8 g fm1 ; m7 g fm1 ; m2 ; m7 ; m8 g fm1 ; m9 g fm1 ; m10 g fm1 ; m7 ; m10 ; m11 ; m14 g fm1 ; m7 ; m10 ; m12 ; m14 g fm1 ; m2 ; m6 ; m7 ; m8 ; m10 ; m13 g fm1 ; m14 g fm1 ; m2 ; m6 ; m7 ; m8 ; m9 ; m15 g

Proposition 1. For any ðAi ; Bi Þ 2 LðX; M; IÞ; i ¼ 1; 2; 3; then S^LA ð; Þ satisfies 1. S^LA ððA1 ; B1 Þ; ðA1 ; B1 ÞÞ ¼ 1. 2. S^LA ððA1 ; B1 Þ; ðA2 ; B2 ÞÞ ¼ S^LA ððA2 ; B2 Þ; ðA1 ; B1 ÞÞ. 3. if ðA1 ; B1 Þ 6 ðA2 ; B2 Þ 6 ðA3 ; B3 Þ (or ðA1 ; B1 Þ 6 ðA2 ; B2 Þ 6 ðA3 ; B3 Þ), S^LA ððA1 ; then S^LA ððA1 ; B1 Þ; ðA3 ; B3 ÞÞ 6 S^LA ððA1 ; B1 Þ; ðA2 ; B2 ÞÞ, ^ B1 Þ; ðA3 ; B3 ÞÞ 6 SLA ððA2 ; B2 Þ; ðA3 ; B3 ÞÞ.

Proof. The conclusions 1–3 can be directly obtained from the definition of S^LA ð; Þ and properties of rough set [11]. Compared with other similarity measures, the proposed measure is more convenient to calculate. By the Lemmas 1 and 2, the similarity of two concepts is directly calculated from the objects classes and attributes classes, which can avoid looking through all concepts of formal context and constructing Hasse diagram. h Example 2. The partial classes of objects and attributes of information Table 1 are calculated as Table 2, respectively. There are 15 objects and attributes classes, respectively. Moreover, there are 27 concepts (Fig. 1), 12 join-irreducible concepts, 12 meet irreducible generated from the information Table 1. Combining Tables 1 and 2, we can check all joinirreducible elements (except special nodes with empty element in extent) and meet-irreducible elements are contained in P and Q, respectively. Thus, it is suitable to regard P and Q as candidate sets of low approximation to define similarity measure. As an example, we calculate the similarity measure between concepts C 1 (labeled by 7) and C 2 (labeled by 18) in Hasse diagram (Fig. 1). Assume x ¼ 12 then, C 1 :¼ ðA1 ; B1 Þ ¼ ðfx15 ; x16 ; x17 ; x18 ; x19 ; x20 ; x21 g; fm1 ; m7 ; m10 gÞ

Table 3 The characteristics of the databases Database

Source

jXj

jMj

jCj

jPj

jQ j

Zoo Postoperative Flare Tic-tac-toe

UCI UCI UCI UCI

101 87 1389 985

27 26 49 29

340 2280 28,742 59,505

59 77 525 985

25 26 47 29

Where j X j represents the number of objects, j M j represents the number of attributes, j C j represents the number of concepts, j P j represents the number of objects classes, j Q j represents the number of attributes classes.

Example 3. The characteristics of four databases (http://mlearn.ics.uci.edu/databases) are discussed in Table 3. Obviously, calculating similarity of concepts based on object classes and attributes classes is more simper than constructing Hasse diagram [17] and looking through all concepts [22,23]. In particular, we only consider attributes classes as candidate set (i.e., x ¼ 0), the proposed method is more conveniently to calculate the similarity of concepts.

5. Conclusion In this paper, we introduce a new similarity measure based on rough set theory, which is a development of Tversky’s similarity model. In the proposed method, we use an important fact that objects classes and attributes classes contain all meet-irreducible elements and join-irreducible elements, respectively. The method is both featural and structural, and is easy to understand. The similarity of two concepts can be directly calculated from objects and attributes classes and is more conveniently to calculate the concepts similarity of the large context.

C 2 :¼ ðA2 ; B2 Þ ¼ ðfx9 ; x11 ; x20 ; x21 g; fm1 ; m2 ; m6 ; m7 ; m8 gÞ ðA1 \ A2 Þ^LA ¼ fx20 ; x21 g; A^1LA B^1LA B^2LA

ðB1 \ B2 Þ^LA ¼ fm1 ; m7 g

¼ fx15 ; x16 ; x17 ; x18 ; x19 ; x20 ; x21 g;

A^2LA

¼ fx9 ; x11 ; x20 ; x21 g

¼ fm1 ; m7 ; m10 g ¼ fm1 ; m2 ; m6 ; m7 ; m8 g

S^LA ððA1 ; B1 Þ; ðA2 ; B2 ÞÞ

! 1 2 2 ¼ 0:432 ¼ þ 2 2 þ 52 þ 22 2 þ 12 þ 32

In ontology mapping, the similarity measure threshold be 0.43, then C 1 and C 2 can be mapped onto each other. From the above example, the similarity of two concepts can be directly calculated from objects and attributes classes, and both featural and structural information are considered. Specially, the proposed method is more conveniently to calculate the concept similarity of the large context.

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