Depicting the associating degrees between two concepts and their relationships are major works for constructing a multi-relationship fuzzy concept network. This paper indicates some drawbacks of the existing methods of calculating associating degrees between concepts, and proposes a new method for overcoming these drawbacks. We also use some examples to compare the proposed method with the existing methods for calculating the associating degrees between two concepts in a multi-relationship fuzzy concept networks.
Salton and Mcgill proposed information retrieval system based on the Boolean logic model [
In [
The rest of this study is organized as follows. Section 2 briefly reviews the concept of geometric mean, the fuzzy concept network [
In [
∏ i = 1 n a i n = a 1 × a 2 × ⋯ × a n n , (1)
where 1 ≤ i ≤ n . The geometric mean is well defined for sets of positive numbers, and is useful to deal with fuzzy aggregating problem and fuzzy decision-making problem.
Lucarella et al. have proposed the fuzzy concept networks for fuzzy information retrieval [
associated with a degree μ, where μ ∈ [ 0 , 1 ] , indicating the degree of strength of the relationship between two concepts or the degree of strength that a document contains a concept.
l = { 〈 c , r 〉 , u ( c , r ) | c ∈ C and r ∈ C } ,
where C represents the set of concepts, u is the membership function, u : C × C → [ 0 , 1 ] , which represents that the concept c and concept r are connected
by the link l, and their relevant is u(c, r), where u ( c , r ) ∈ [ 0,1 ] .
In the relevant value between concept c and concept r is u(c, r), and the relevant value between concept r and concept s is u(r, s). Then based on the transitivity of link relationship, we can obtain the relevant value between concept c and concept s by the following expression:
u ( c , s ) = min ( u ( c , r ) , u ( c , s ) ) .
Similarly, if u ( c 1 , c 2 ) , u ( c 2 , c 3 ) , ⋯ , u ( c n − 1 , c n ) are known, then based on the transitivity of relationship, we can obtain the relevant value between concept c1 and concept cn by the following expression:
u ( c 1 , c n ) = min ( u ( c 1 , c 2 ) , u ( c 2 , c 3 ) , ⋯ , u ( c n − 1 , c n ) ) .
Kracker proposed the multi-relationship fuzzy concept network [
1) Positive association: It relates concepts with a fuzzy similar meaning (e.g. person—individual) in some contexts.
2) Negative association: It relates concepts with fuzzy complementary relationship (e.g. male—female), fuzzy incompatible relationship (e.g. unemployed—freelance) or fuzzy antonymous relationship (e.g. small—large) in some contexts.
3) Generalization: A concept regarded as a fuzzy generalization of another concept if it includes that concept in an analytic or partitive sense (e.g. person—student).
4) Specialization: It is the inverse of fuzzy generalization.
Let C be a set of concepts in a multi-relationship fuzzy concept network. The fuzzy relationships between concepts are defined as follows [
1) Fuzzy positive associating P is a fuzzy relation, P : C × C → [ 0 , 1 ] , which is reflexive, symmetric, and max-*-transitive.
2) Fuzzy negative association N is a fuzzy relation, N : C × C → [ 0 , 1 ] , which is anti-reflexive, symmetric, and max-*-nontransitive.
3) Fuzzy generalization G is a fuzzy relation, G : C × C → [ 0 , 1 ] , which is anti-reflexive, anti-symmetric, and max-*-transitive.
4) Fuzzy specialization S is a fuzzy relation, S : C × C → [ 0 , 1 ] , which is anti-reflexive, anti-symmetric, and max-*-transitive.
A multi-relationship fuzzy concept network is denoted as MRFCN (E, L), where E is a set of nodes, and where represents a concept or a document as in
1) c i → ( 〈 μ P , P 〉 , 〈 μ N , N 〉 , 〈 μ G , G 〉 , 〈 μ S , S 〉 ) c j , means that the directed edge l connect ci to cj with a four-tuple ( 〈 μ P , P 〉 , 〈 μ N , N 〉 , 〈 μ G , G 〉 , 〈 μ S , S 〉 ) , where μ P ∈ [ 0 , 1 ] , μ N ∈ [ 0 , 1 ] , μ G ∈ [ 0 , 1 ] and μ S ∈ [ 0 , 1 ] .
2) c i → μ d j , means that document dj has concept ci with the degree of strength, where μ ∈ [ 0 , 1 ] .
Furthermore, Horng et al. proposed an algorithm with eight steps to construct multi-relationship fuzzy concept networks automatically [
In [
Case 1: If concept ci and concept cj contain different words, then they are not related.
Case 2: If concept ci and concept cj contain almost the same words, but the weighs of the words in concept ci are larger than those in concept cj, then concept ci is said to dominate concept cj and should be more general than concept cj.
Case 3: If concept ci and concept cj contain almost the same words, but the weights of the words in concept ci are smaller than those in concept cj, then concept ci is said to be dominated by con concept cj and should be more specific than concept cj.
Case 4: If most words contained in concept cj are also contained in concept ci, but many words contained in concept ci are not contained in concept cj, then concept ci concerns more aspects than concept cj and should be more general than concept cj.
Case 5: If most words contained in concept ci are also contained in concept cj, but many words contained in concept cj are not contained ci, then concept ci concerns fewer aspect than concept cj and should be more specific than concept cj.
Case 6: If concept ci and concept cj contain almost the same words, and the weight of the words are similar in both concepts, then these two concepts should be similar to each other and have a fuzzy positive associating relationship.
Young proposed a method for calculating the associating degree between concepts [
M ( c i ) = w i 1 / t 1 + w i 2 / t 2 + ⋯ + w i h / t h , (2)
where M : C → [ 0 , 1 ] W , wi1 is the weight of word tj in concept ci, and h is the number of words in the word set WS. Then, calculating the associating degree
between concepts denoted G(ci, cj) and equal to the degree of subsethood of M(ci) in M(cj). A method to calculate G(ci, cj) is shown as follows:
G ( c i , c j ) = { ( | M ( c i ) ∩ M ( c i ) | | M ( c i ) | ) = ( ∑ k = 1 h min ( w k i , w k j ) ∑ k = 1 h w k i ) , if M ( c j ) ≠ ϕ 1 , if M ( c i ) ≠ ϕ (3)
where wki is the weight of word tk in concept ci, wki is the weight of word tk in concept cj, WC(ci) is the number of words contained on concept ci, WC(cj) is the number of words contained in concepts cj, and h is the number of words in the word set WS. According to Subsection 2.2, we can understand that fuzzy specialization relationship is the inverse of the fuzzy generalization relationship. Thus
S ( c i , c j ) = G ( c i , c j ) . (4)
Moreover, based on Subsection 2.1, the degree of fuzzy positive association relationship between concept ci and concept cj, denoted as P(ci, cj), is calculated as follows:
P ( c i , c j ) = min ( G ( c i , c j ) , S ( c i , c j ) ) . (5)
However, Horng et al. [
Example 3.1: Assume that there are five words t1, t2, …, and t5 in the word set WS and assume that the corresponding fuzzy subset M(ci) and M(cj) of concept ci and concept cj in the word set WS are shown as follows:
M ( c i ) = 0.3 / t 1 + 0.3 / t 2 + 0.4 / t 3 + 0.4 / t 4 + 0.3 / t 5 ,
M ( c j ) = 0.8 / t 2 + 0.9 / t 3 .
According to Case 4 of the above six cases for deciding fuzzy relationship between concepts, concept ci should be more general than the concept cj because concept ci contains all the words contained in concept cj (i.e., the words t2, t3). However, Young’s method yield the same associating degrees G(ci, cj) and G(cj, ci) as follows:
G ( c i , c j ) = 0 . 3 + 0 . 4 0 . 3 + 0 . 3 + 0 . 4 + 0 . 4 + 0 . 3 = 0 . 4 1 ,
G ( c j , c i ) = 0 . 3 + 0 . 4 3 . 8 + 0 . 9 = 0 . 4 1 .
According to the above results, we cann’t know which concept is more general than the other one.
Therefore, Horng et al. [
G ( c i , c j ) = { ( ∑ k = 1 h min ( w k i , w k j ) ∑ k = 1 h w k i ) W C ( c i ) max ( W C ( c i ) , W C ( c j ) ) , if M ( c j ) ≠ ϕ 1 , if M ( c i ) ≠ ϕ (6)
where wki is the weight of word tk in concept ci, wkj is the weight of word tk in concept cj, WC(ci) is the number of words contained on concept ci, WC(cj) is the number of words contained in concepts cj, and h is the number of words in the word set WS. The proposed method can overcome the drawback of Young’s method.
However, we also found the formula (6) proposed by Horng et al. still has some drawbacks for dealing with associating degrees between concepts (i.e., the result is not fitting for one of the above six cases). In the following, we use some examples to illustrate these drawbacks.
Example 3.2: Assume there are four words t1, t2, t3 and t4 in the word set WS, and assume that the corresponding fuzzy subset M(ci) and M(cj) of concept ci and concept cj in the word set WS are shown as follows:
M ( c i ) = 0.3 / t 1 + 0.1 / t 2 + 0.2 / t 3 + 0.8 / t 4 ,
M ( c j ) = 0.4 / t 1 + 0.3 / t 2 + 0.8 / t 3 .
According to Case 2 of the above six cases for deciding fuzzy relationship between concepts, concept cj should be more general than concept ci because concept ci and concept cj contained almost the same words (i.e., the words t1, t2, and t3), but all the weights of the words in concept cj also contained in concept ci are larger than concept ci. Based on Horng et al.’s method, we calculate the G(ci, cj) and G(cj, ci), respectively, as follows:
G ( c i , c j ) = ( 0 . 3 + 0 . 1 + 0 . 2 0 . 3 + 0 . 1 + 0 . 2 + 0 . 8 ) 4 4 = 0 . 4 2 ,
G ( c j , c i ) = ( 0 . 3 + 0 . 1 + 0 . 2 0 . 4 + 0 . 3 + 0 . 8 ) 3 4 = 0 . 5 .
Since G(cj, ci) is larger than G(ci, cj), we can see that concept ci is more general than concept cj. However, the relationship between the two concepts ci and cj does not coincide with human intuition for violating Case 2.
Example 3.3: Assume that there are six words t 1 , t 2 , ⋯ , t 6 in the word set WS, and assume that the corresponding fuzzy subset M(ci) and M(cj) of concept ci and concept cj in the word set WS are shown as follows:
M ( c i ) = 0.6 / t 1 + 0.3 / t 2 + 0.4 / t 3 + 0.7 / t 4 + 0.6 / t 5 + 1 / t 6 ,
M ( c j ) = 0.8 / t 1 + 0.5 / t 2 + 0.7 / t 3 + 1 / t 4 + 0.8 / t 5 .
According to Case 3 of the above six cases for deciding fuzzy relationship between concepts, concept cj is general than ci because the two concepts ci and cj contained almost the same words (i.e., the word t1, t2, t3, t4 and t5), but all the weights of the words in concept cj also contained in concept ci are larger than concept ci. Based on Horng et al.’s method, we calculate the G(ci, cj) and G(cj, ci), respectively, as follows:
G ( c i , c j ) = ( 0 . 6 + 0 . 3 + 0 . 4 + 0 . 7 + 0 . 6 0 . 6 + 0 . 3 + 0 . 4 + 0 . 7 + 0 . 6 + 1 ) 6 6 = 0 . 6 7 6 7 ,
G ( c j , c i ) = ( 0 . 6 + 0 . 3 + 0 . 4 + 0.7 + 0.6 0 . 8 + 0 . 5 + 0 . 7 + 1 + 0 . 8 ) 5 6 = 0 . 6 8 4 2 .
Since G(cj, ci) is larger than G(ci, cj), we can see that concept ci is more general than concept cj. However, the relationship between the two concepts ci and cj is not fitting for human intuition because of violating Case 3.
Example 3.4: Assume that there are seven words t 1 , t 2 , ⋯ , t 7 in the word set WS, and assume that the corresponding fuzzy subset M(ci) and M(cj) of concept ci and concept cj in the word set WS are shown as follows:
M ( c i ) = 1 / t 1 + 0.8 / t 2 + 0.9 / t 3 ,
M ( c j ) = 0.2 / t 1 + 0.1 / t 2 + 0.2 / t 3 + 0.1 / t 4 + 0.2 / t 5 + 0.1 / t 6 + 0.1 / t 7 .
According to Case 4 of the above six cases for deciding fuzzy relationship between concepts, concept cj should be more general than the concept ci because concept cj contains all the words contained in concept ci, (i.e., the words t1, t2 and t3), but the words t4, t5, t6 and t7 contained in concept cj are not contained in concept ci. Based on Horng et al.’s method, we calculate the G(ci, cj) and G(cj, ci), respectively, as follows:
G ( c i , c j ) = ( 0 . 2 + 0 . 1 + 0 . 2 1 + 0 . 8 + 0 . 9 ) 3 7 = 0 . 4 8 5 ,
G ( c j , c i ) = ( 0 . 2 + 0 . 1 + 0 . 2 0 . 2 + 0 . 1 + 0 . 2 + 0 . 1 + 0 . 2 + 0 . 1 + 0 . 1 ) 4 4 = 0 . 5 .
Since G(ci, cj) is larger than G(cj, ci), we can see that concept ci is more general than concept cj. However, the relationship between the two concepts ci and cj does not coincide with human intuition because of violating Case 4.
According to the above discussion, we found that formula (5) proposed by Horng et al. has some drawbacks for calculating the degrees between concepts. In order to obtain more accurate associating degrees between concepts for automatically constructing multi-relationship fuzzy concept networks, to develop a new method for calculating associating degrees between concepts is necessary.
In this section, we present a new method for calculating associating degrees between concepts based on geometric mean operator. The new method for calculating associating degrees between concepts shown as follows:
G ( c i , c j ) = { ∑ k = 1 h w k i × w k j 2 × ∑ k = 1 k w k i + ( 0 . 5 ) 1 + 2 × R O U N D ( W C ( c i ) m a x ( W C ( c i ) , W C ( c j ) ) ) , i f M ( c j ) ≠ ϕ 1 , i f M ( c i ) ≠ ϕ (7)
where wki is the weight of word tk in concept ci, wkj is the weight of word tk in concept cj, WC(ci) is the number of words contained in concept ci, WC(cj) is the number of the words contained in concept cj, and h is the number of words in the word set WS. ROUND(.) is a round off function, e.g. ROUND(0.4) = 0 and ROUND(0.6) = 1. The main idea of the proposed method is to include the rate of words contained in concept. We have found that if we increase the importance of the rate of words contained in concept while the rate above 0.5, we can get the appropriate association degrees between concepts. Regarding the weight values of the word in concept, some references use the number of words in the document to calculate the weight [
In the following, we use the examples discussed in Section 3 to compare the proposed method with existing methods.
1) If we use formula (7) to deal with Example 3.1, we can calculate the two associating degrees G(ci, cj) and G(cj, ci), respectively, as follows:
G ( c i , c j ) = 0.3 × 0.8 + 0.4 × 0.9 2 × ( 0.3 + 0.3 + 0.4 + 0.4 + 0.3 ) + ( 0.5 ) 1 + 2 × ROUND ( 5 5 ) = 0.44556
G ( c j , c i ) = 0.3 × 0.8 + 0.4 × 0.9 2 × ( 0.8 + 0.9 ) + ( 0.5 ) 1 + 2 × ROUND ( 2 5 ) = 0.82056 .
Since G(cj, ci) is larger than G(ci, cj), we can see that concept ci is more general than concept cj, and it coincides with the intuition of the human being.
2) If we use formula (7) to deal with Example 3.2, we can calculate the two associating degrees G(ci, cj) and G(cj, ci), respectively, as follows:
G ( c i , c j ) = 0 . 3 × 0 . 4 + 0 . 1 × 0 . 3 × 0.2 × 0.8 2 × ( 0.3 + 0.1 + 0.2 + 0.8 ) + ( 0.5 ) 1 + 2 × ROUND ( 4 4 ) = 0 . 4 5 3 4 3 ,
G ( c j , c i ) = 0 . 3 × 0 . 4 + 0 . 1 × 0 . 3 × 0.2 × 0.8 2 × ( 0 . 4 + 0 . 3 + 0 . 8 ) + ( 0.5 ) 1 + 2 × ROUND ( 3 4 ) = 0 . 4 3 1 5 4 .
Since G(cj, ci) is larger than G(ci, cj), we can see that concept ci is more general than concept cj, and it coincides with the intuition of the human being for observing Case 2.
3) If we use formula (7) to deal with Example 3.3, we can calculate the two associating degrees G(ci, cj) and G(cj, ci), respectively, as follows:
G ( c i , c j ) = 0.6 × 0.8 + 0.3 × 0.5 + 0.4 × 0.7 + 0.7 × 0.1 + 0.6 × 0.8 2 × ( 0.6 + 0.3 + 0.4 + 0.7 + 0.6 + 1 ) + ( 0.5 ) 1 + 2 × ROUND ( 6 6 ) = 0 . 5 6 0 9 ,
G ( c j , c i ) = 0.6 × 0.8 + 0.3 × 0.5 + 0.4 × 0.7 + 0.7 × 1 + 0.6 × 0.8 2 × ( 0.8 + 0.5 + 0.7 + 1 + 0.8 ) + ( 0.5 ) 1 + 2 × ROUND ( 5 6 ) = 0.5380 .
Since G(ci, cj) is larger than G(cj, ci), we can see that concept cj is more general than concept ci, and it coincides with the intuition of the human being for observing Case 3.
4) If we use formula (7) to deal with Example 3.4, we can calculate the two associating degrees G(ci, cj) and G(cj, ci), respectively, as follows:
G ( c i , c j ) = 0.1 × 0.2 + 0.8 × 0.1 × 0.9 × 0.2 2 × ( 0.1 + 0.8 + 0.9 ) + ( 0.5 ) 1 + 2 × ROUND ( 3 7 ) = 0.71376 ,
G ( c j , c i ) = 0.3 × 0.4 + 0.1 × 0.3 × 0.2 × 0.8 2 × ( 0.2 + 0.1 + 0.2 + 0.1 + 0.2 + 0.1 + 0.1 ) + ( 0.5 ) 1 + 2 × ROUND ( 7 7 ) = 0.70216 .
Since G(ci, cj) is larger than G(cj, ci), we can see that concept cj is more general than concept ci, and it coincides with the intuition of the human being for observing Case 4.
From the previous discussions, we can obtain the proposed method is useful than the two existing methods proposed by Young and Horng et al. respectively for calculating the associating degrees between two concepts for deciding their relationship in a multi-relationship fuzzy concept network.
In this paper, we firstly pointed out some drawbacks of the existing methods for calculating the associating degree between two concepts, and presented a method based on geometric mean operator for overcoming these drawbacks. We used some examples to compare the proposed method with the existing methods. The proposed method is more useful than the existing methods to calculate the associating degrees between two concepts for constructing their relationship in a multi-relationship fuzzy concept networks for document retrieval.
This work was supported in part by the Ministry of Science and Technology, under Grant 104-2410-H-239-007.
Chen, S.-J. (2018) A Method for Calculating the Association Degrees between Concepts of Concept Networks. Journal of Computer and Communications, 6, 55-65. https://doi.org/10.4236/jcc.2018.65005