A Concept Similarity Method in Structural andSemantic LevelsSheng Yan, Li Yun, Luan Luan

Institute of Information Engineering, Yang Zhou UniversityYang Zhou, China, 225009

liyun@yzu.edu.cn

Abstract—Based on the formal concept analysis, combining therough set and semantic ontology, this paper proposes a formalconcept similarity measurement by improving the originalTversky similarity model. The method used the hierarchicalinformation of concept lattice in the lattice structure and its ownsemantic meaning to measure the similarity of from the structureinformation of the concepts and semantic informationrespectively, which reflect the true similarity of formal concept tosome extent.

Keywords formal concept analysis, hierarchical information,semantic, concept similarity

I. INTRODUCTIONAs the basic unit of human’s thinking and knowledge, the

concept has been deeply focused by philosophers, scientistsand become an important subject of artificial intelligence.Concept lattice is the core structure of Formal ConceptAnalysis (FCA)[1], which describes the concepts and theirrelationship. It is a highly simplified description of objectiveworld to a certain extent. The advantage of the simplification isits good mathematical nature. Prof Wille create the FCA basedon this simplification. However, it is precisely because of thissimplification makes the concept lattice can not simply beunderstood as part of the real world model, while is seen as anartificial data sets derived from a number of indications.

Based on the concept lattice, combined with rough settheory[2] and semantic ontology theory[3] , we improved thebasic Tversky[4] similarity model and put forward a method ofcomputing concept similarity. Measuring the concept similarityby structure and semantic can reflect the true similarity ofconcept from different levels.

II. BACKGOUNDS

A. Formal Concept AnalysisFormal Concept Analysis (FCA)[1] is an useful data analysis

technique based on number theory, whose content is formalcontext, formal concept and the relation among the formalconcepts. Formal Context is defined as a triple K= (G, M, I),where G is the set of objects, M is the set of attributes and I isthe binary relationship between G and M, that is I G M, gIm.In the formal context K, two mapping function f and g isdefined as fellow:

' ,A m M gRm g A (1)' ,B g G gRm m B (2)

They are called the Galois connection between G and M. Ifthe tuple(A, B) from P(G) P(M) satisfied two conditions:A=g(B) and B=f(A), or A=B’ and B=A’, then, we called (A, B)an formal concept from formal context K, denoted C=(A, B),which B and A is called the Intent and Extent of formalconcept C separately. Assumed that C1= (A1, B1) and C2= (A2,B2) are two formal concepts, the order relation“ ”is defined asC1 C2 B2 B1. C1 is the sub concept of C2, C2 is superconcept of C1. All formal concepts and their relations consist ofa concept lattice.

Formal concept lattice is a complete lattice, because anyconcepts have only and unique suprema and infima which aregiven by:

( , ) , ''t t t t

t T t T t T

A B A B(3)

( , ) '',t t t t

t T t T t T

A B A B(4)

They are also called Meet and Join operation.

B. Rough SetRough set theory[2] is characterized by that we don’t need to

give out the quantity of certain characteristics or attributes andfrom a given set of the problem description, confirm theapproximate domain by the indiscernibility classes directly andthen find out the inherent laws of problems.Def 1: Assume that U denotes a non-empty finite universe,

R denotes a set of equivalence relations on U, U/R is all of theequivalence classes on R, for any X U, the lowerapproximated set and upper approximate set of X on R can bedefined respectively as below:

}][|/{)( RxYRUYXR(5)

}!][|/{)( RxYRUYXR (6)( )R X is the maximum set of objects which belong to X on

the basis of known knowledge; ( )R X is the union set of theequivalence classes [X]R whose intersection with X is notempty and also is the minimum set of object which maybebelong to X. The boundary regions of set X are defined

Second International Symposium on Information Science and Engineering

978-0-7695-3991-1/09 $26.00 © 2009 IEEE

DOI 10.1109/ISISE.2009.133

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as )()()( XRXRXBN . Sets have uncertainty owing to the

existence of the boundary regions. The bigger the boundaryregions are, the lower their accuracy is, and the greater theirroughness is. When condition )()( XRXR is satisfied, set

X is called the accuracy set of R; otherwisewhen )()( XRXR , set X is called the rough set of R. The

rough set can be depicted by the upper and lowerapproximations of precise set.They are also called Meet and Join operation.

C. Tversky Similarity ModelTversky[4] put forward an concept similarity method based

on the set theory:

1 2

1 2 1 2 2 1

( )( , )( ) ( ) ( )

f B BS a bf B B f B B f B B (7)

There into, , 0, f is an function for measure thesimilarity between B1 and B2, the attribute set of B1 and B2 is aand b.After this, Rodriguez and Egenhofer[5] have proposed an

assessment of semantic similarity among entity classes indifferent ontologies based on the normalization of Tversky’ssimilarity model:

1 2

1 2 1 2 2 1

| |( , )

| | ( , ) | | (1 ( , )) | |B B

S a bB B a b B B a b B B (8)

Where the function |.| represents the cardinality of a setdepth(a), depth(b) represents the depth of a, b in ontologies. If

( ) ( )depth a depth b , then ( )( , )( ) ( )depth a

a bdepth a depth b

,

else ( )( , ) 1( ) ( )depth a

a bdepth a depth b

.

Zhao[3,6] proposed extension of the measure with theemployment of rough set theory:

1 2

1 2 1 2 2 1

| ( ) |( , )| ( ) | | | (1 ) | |

LA

LA LA LA LA LA

B BS a bB B B B B B (9)

Where intent( {( , ) | }),LAB x y y B is the setof all concepts of lattice.Wang[7]advanced a more complex method to compute the

similarity of formal concepts:

1 21 1 2 2

1 2 1 2 2 1

1 2

1 2 1 2 2 1

| ( ) |(( , ), ( , ))

1 1| ( ) | | | | |

2 2| ( ) |

(1 )1 1

| ( ) | | | | |2 2

LA

LA

LA LA LA LA LA

LA

LA LA LA LA LA

A AS A B A B

A A A A A A

B B

B B B B B B (10)This formula is very complex because it contains the part of

the intent and extent of concepts. The paper improved the basicTversky similarity model combined the Lattice and Rough settheory to obtain more accurate results.

D. Information EntropyThe main idea of information entropy is: There are many

events in system S= (E1, E2, … En), each probability of events

P=(P1, P2, …,Pn), then each events’ information content isI(E)=-logPi, the average information content of the wholesystem is:

1 1

( ) ( ) logn n

i i

H S PiI E Pi Pi(11)

The average information content is called as informationentropy. The paper compute the information content of eachattribute using entropy to obtain the semantic similarity.

III. CONCEPT SIMILARITY METHOD

A. Structure SimilarityConcept lattice is a complete lattice, which has lots of

available information, such as that the upper the node is, themore abstract it is and the less information it has while themore downer the node is, the more specificity it is and themore information it has. When we compared formal concept C1to other concepts C2 and C3, the level number of two conceptsC1 and C2 is the closer than the one of C1 and C3, then,obviously, C2 should be better than C3 because of closerhierarchy which the true similarity of C1 and C2 is bigger thanthe similarity of C1 and C3. Thus, the use of hierarchyinformation of formal concept is necessary in calculating theconcept similarity.

Secondly, considering the hierarchy structure of formalconcept and combining with rough set theory, using theapproximate set of intent instead of intent self to calculate thesimilarity helps to contain more specific information. Based ontwo points above, this paper improves the Tversky similaritymodel and adds information of hierarchy and approximates toobtain a more effective result.

Def 2 The structure similarity among concepts Formalconcepts C1 (A1, B1) and C2 (A2, B2) are two formal concepts inconcept lattice, the structure similarity is defined as follows:

1 2

1 21 2 1 2 1 2 1 2 2 1

| ( ) |( , )

| ( ) | ( , ) | | (1 ( , )) | |LA

LA LA LA LA LA

B BStructSim C C

B B C C B B C C B B (12)

1 2( ) intent ( {( , ) | }),

LAB B x y y B stands for

all formal concepts, | | represents the cardinality of a set, |B1LA-B2LA| shows the number of intents which belong to the lowerapproximate set of B1 but don’t belong to the lowerapproximate set of B2, |B2LA-B1LA| shows the number of intentswhich belong to the lower approximate set of B2 but don’tbelong to the lower approximate set of B1;

11 2

1 21 2

12 1

1 2

( ), ( ) ( )

( ) ( )( , )

( )1 , ( ) ( )

( ) ( )

depth Cdepth C depth C

depth C depth CC C

depth Cdepth C depth C

depth C depth C ,depth(C1) represents the level number of C1 in concept lattice,which the number of top node is noted as 0. The coefficientrepresents the hierarchical relationship among the concepts.The formal context (Figure 1) is from literature[8] to

illustrate how to compute the structure similarity of concepts.In the formal context, we used Godin[9] to construct the formallattice below (Figure 2):

621

Considering the formal concepts C1((Garfield, Socks),(mammal, cat)), C2((Grayfriar’s Bobby, snoopy), (dog,mammal)), C3((Grayfriar’s Bobby), (real, dog, mammal)), thestructure similarity between C1 and C2 , C1 and C3 according toformula (12):

1 2

1 1( , )

1 1 21 *1 (1 )*12 2

StructSim C C

1 3

1 5( , )

2 2 131 *1 (1 )*25 5

StructSim C C

The similarity between C1 and C2 is bigger than the onebetween C1 and C3.

B. Semantic SimilarityIn order to measure the semantic similarity, we adopt

WordNet[10] as the basic copus to extract the IS-A relationsamong the words and then add the probabilty of each word toobtain a weighted IS-A relations. At last, we get a semanticsimilarity table by computing the information content of eachword.

WordNet is a broad coverage of English vocabularysemantic network. The paper mainly uses the IS-A relation ofwords and SynSet to obtain a weighted IS-A hierarchicalrelationship.

Def 3 The weighted IS-A relationships[11]: Given an englishcorpus , the weighted IS-A relationships H( ) can be generatedby each word’s probability and their IS-A relationship. Theprobability of each word is defined as:

( )

( )( ) ( )

i sub n

freq np n p i

M (13)( )freq n represents the number of word n in the whole

corpus; M represents the number of all words in the whole

corpus; ( )( )

i sub n

p irepresents the sum of probability of words

which are the children nodes of word n. If word n is in bottomand it has no child, then the relative value is 0. At the sametime, the top node is defined as Top and its ( ) 1p Top .

Figure 3 A weighted IS-A hierarchical relationship

From the IS-A relationships, we can get the informationcontent of each word according to the information contentformula I(E)=-logPi, for example:

( ) log 0.000011599=4.935579I cat ,( ) log 0.000011658=4.933376I dog ,( ) log 0.00012528 3.902118I mammal ,From the weighted IS-A hierarchical relationship, we can

see that the word is upper, its information content is less and ifthe parent node of two words has more information content,they share more information and more similar. The paper adoptthe similarity of information content advanced in theliterature[11] to improve the semantic similarity of formalconcepts.

Def 4 Information content similarity, ics(n1, n2) [12] Givenan English corpus and a weighted IS-A hierarchicalrelationshipH( ), any two words n1, n2 in corpus, if n1= n2 or n1belongs to the Synset of n2, then ics(n1, n2)=1;

else 1 2

1 2

2 ( ')( , )

( ) ( )I n

ics n nI n I n

. n’ is the maximum common

parent node of n1 and n2,1 2( , )

( ') max{ ( )}n S n n

I n I n , 1 2( , )S n n is

all parent node of n1 and n2.According to this definition, the similarity of any two

words can be obtained. For example:2 ( ) 2*3.902118( , ) 0.790786( ) ( ) 4.935579 4.933376I mammalics cat dog

I cat I dogBecause the intent of a formal concept reflects its nature,

we can just consider the intent to measure the similarity offormal concepts.

Def 5 Semantic similarity among concepts[11]:The semanticsimilarity of two formal concepts C1(A1, B1) and C2(A2, B2) isdefined as:

1 21 2

1 2

( , )( , )

max{| |, | |}M B B

SemanticSim C CB B (14)

Figure 2 The relative concept lattice

Figure 1 The formal context

622

1 1 2 2

1 2 1 2,

( , ) max{ ( , )}b B b B

M B B ics b b , b1 and b2 can only

occur once in every combination.For example, the semantic similarity between formal

concept C1((Garfield, Socks), (mammal, cat)) andC2((Grayfriar’s Bobby, snoopy), (dog, mammal)) is:

1 2

(( , ),( , ))( , ) 0.895393max{| ( , ) |,| ( , ) |}M mammal cat dog mammal

SemanticSim C Cmammal cat dog mammal

While the semantic similarity between formal conceptC1((Garfield, Socks), (mammal, cat)) and C3((Grayfriar’sBobby), (real, dog, mammal)) is:

1 3(( , ),( , , ))

( , ) 0.596929max{| ( , ) |,| ( , , ) |}M mammal cat real dog mammal

SemanticSimC Cmammal cat real dog mammal

C. The formula of similarity among conceptsAfter obtaining the structure and semantic similarity of two

formal concepts, we can combine with the two similarities tocompute the final similarity.

Def 6 Sim(C1, C2) The final similarity between formalconcepts C1(A1, B1) and C2(A2, B2) is:

1 2 1 2 1 2( , ) ( , ) (1 ) ( , )Sim C C StructSim C C SemanticSim C C (15)is a weighted coefficient and adjust the importance of

structure and semantics similarity, whose value is 0~1.

For example, we compute the similarity betweenC1((Garfield, Socks), (mammal, cat)) and C2((Grayfriar’sBobby, snoopy), (dog, mammal)) is:

1 2

1( , ) 0.5 * 0.5 * 0.895393 0.6976972

Sim C C

While the similarity between C1((Garfield, Socks),(mammal, cat)) and C3((Grayfriar’s Bobby), (real, dog,mammal)) is:

1 3

5( , ) 0.5 * 0.5 * 0.596929 0.490772

13Sim C C

It shows that the similarity of formal concepts C1 and C2 ishigher than the one of C1 and C3, which is in line with theactual situation.

IV. CONCLUSIONS AND FUTURE WORKThe paper proposes a concept similarity method which

improved the basic Tversky model and measures the similarityamong the concepts in structural and semantic levels and triedbest to find the true similarity of different concepts. If the

method is applied to ontology engineering, such as ontologymapping, ontology merging, we can measure the similaritybetween ontology concepts; And in information retrieval, it canbe used for matching keywords as a basic tool to measuresimilarity. The future work is to verify effects and accuracy.

ACKNOWLEDGMENTThis research was supported in part by the Chinese

National Natural Science Foundation under grant No.60673060, Natural Science Foundation of Jiangsu Provinceunder contract BK2008206 and Natural Science Foundation ofEducation Department of Jiangsu Province under contract08KJB520012.

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