M.Y.H. Al-Shamri, K.K. Bharadwaj/ Expert Systems with Applications 35(2008)1386-1399 1393 same fuzzy set. In the following, we define an appropriate fuzzy distance function that satisfies all the above four conditions g0. Definition 3. Let a and b be the membership vectors correspond to two crisp values a, and b for a given feature 0 with I fuzzy sets. The fuzzy distance between a and b is defined as Fig. 3. Membership functions for genre interestingness measure. fd(a, b=d(a, b)xd(a, b) where d(a, b) is simply the difference operator, a and b are The distance between two fuzzy sets or points is discussed vectors of size L, and d(a, b )is any vector distance metric extensively in the literature(Dumitrescu, Lazzerini, Jair 2000: Klir& Yuan, 1995). For our model, a vector of 21 In our experiments, the Euclidean distance function is features represents the user. A local fuzzy distance should used for d(a, b) be found for each feature. Hence, for each pair of users, we have 21 local fuzzy distances. The global fuzzy distance b)=∑(a-b) is obtained by two methods(Gadi, Daoudi, Matusiak 999). The first method uses a fuzzy IF-THEN rule of where a is the membership value of the feature a in its jth Cae F(x, is close to y1)and (x2 is close to y2) .and(x21 is fuzzy set. To illustrate the effectiveness of the proposed fuzzy se to y21)THEN (x is similar to y). distance function, let us consider the following example In this case, the global fuzzy distance is given by Example 2. Suppose we have to compute fuzzy distance fd(x,y)=min(fd(x1, yi), fd(x2,y2),., fd(x2l, y21)) .(16) between two users having the age The second method considers each fuzzy distance between two features as an opinion. The global fuzzy distance is a Case i: 35 and 40, global opinion from all. In fuzzy logic, this requires an Case ii: 45 and 60, aggregation operator. The aggregation operator may Case iii: 18 and 23 the average of the 21 local fuzzy distances According to Fig. 2, 21 (17) Case i: a=(0, 1, 0), b=(0, 1, 0). Therefore For our model, formula(16)does not work well because it considers only the feature with the minimum distance and d(a, b)=V2x(0-0)+(1-1)=0, ignores the remaining features. According to fuzzy sets and distance concepts, we need a local fuzzy distance metric, fd(xi, yi), that fulfils the following conditions (A)a zero value for the same feature values Case ii: a=(0, 1, 0), b=(0, 0, 1). Therefore (B)A zero value for different feature values in the same d(a, b) fuzzy set having the same membership values (1-0)2 C) Minimize the distance between any two feature values d(60, 45)=60-45=15 belonging to the same fuzzy set having near member ship values. fd(60,45)=√2×15( Far users) (D) Maximize the distance between any two feature val es belonging to two different fuzzy sets Case ii: a=(1,0, 0), b=(0.8, 0.2, 0). Therefore The condition (A)is the basic requirement for any dis- d(a, b) (08-1)2+(02-0)2=0283, ance function. To clarify the condition(B), assume that d (23, 18)=23-18=5, we have two users with age 40 and 35. Both users are mid dle-aged with membership values of one(Fig. 2). The crisp fd(23,18)=0.283×5( Near users) distance between them is 5, whereas they are similar users from the fuzzy sets point of view. To make the distance Results in Example 2 show that formula(18)satisfies all between the two users zero, we need another term gives the four conditions of the required local fuzzy distance zero value for this and similar cases. What makes these metric. Accordingly, for the fuzzy approach, the fuzzy dis two users similar is their equal membership values in the tance function between two users can be aggregated usingThe distance between two fuzzy sets or points is discussed extensively in the literature (Dumitrescu, Lazzerini, & Jain, 2000; Klir & Yuan, 1995). For our model, a vector of 21 features represents the user. A local fuzzy distance should be found for each feature. Hence, for each pair of users, we have 21 local fuzzy distances. The global fuzzy distance is obtained by two methods (Gadi, Daoudi, & Matusiak, 1999). The first method uses a fuzzy IF-THEN rule of the form: IF (x1 is close to y1) and (x2 is close to y2)... and (x21 is close to y21) THEN (x is similar to y). In this case, the global fuzzy distance is given by fdðx; yÞ ¼ minffdðx1; y1Þ; fdðx2; y2Þ; ... ; fdðx21; y21Þg: ð16Þ The second method considers each fuzzy distance between two features as an opinion. The global fuzzy distance is a global opinion from all. In fuzzy logic, this requires an aggregation operator. The aggregation operator may be the average of the 21 local fuzzy distances: fdðx; yÞ ¼ 1 21 X 21 i¼1 fdðxi; yi Þ: ð17Þ For our model, formula (16) does not work well because it considers only the feature with the minimum distance and ignores the remaining features. According to fuzzy sets and distance concepts, we need a local fuzzy distance metric, fd(xi,yi), that fulfils the following conditions: (A) A zero value for the same feature values. (B) A zero value for different feature values in the same fuzzy set having the same membership values. (C) Minimize the distance between any two feature values belonging to the same fuzzy set having near member￾ship values. (D) Maximize the distance between any two feature val￾ues belonging to two different fuzzy sets. The condition (A) is the basic requirement for any dis￾tance function. To clarify the condition (B), assume that we have two users with age 40 and 35. Both users are mid￾dle-aged with membership values of one (Fig. 2). The crisp distance between them is 5, whereas they are similar users from the fuzzy sets point of view. To make the distance between the two users zero, we need another term gives zero value for this and similar cases. What makes these two users similar is their equal membership values in the same fuzzy set. In the following, we define an appropriate fuzzy distance function that satisfies all the above four conditions. Definition 3. Let a and b be the membership vectors correspond to two crisp values a, and b for a given feature with l fuzzy sets. The fuzzy distance between a and b is defined as fdða; bÞ ¼ dða; bÞ dða; bÞ; ð18Þ where d(a,b) is simply the difference operator, a and b are vectors of size l, and d(a,b) is any vector distance metric. In our experiments, the Euclidean distance function is used for d(a,b): dða; bÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X l j¼1 ðaj bjÞ 2 vuut ; ð19Þ where aj is the membership value of the feature a in its jth fuzzy set. To illustrate the effectiveness of the proposed fuzzy distance function, let us consider the following example. Example 2. Suppose we have to compute fuzzy distance between two users having the age: Case i: 35 and 40, Case ii: 45 and 60, Case iii: 18 and 23. According to Fig. 2, Case i: a = h0, 1, 0i, b = h0, 1, 0i. Therefore dða; bÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð0 0Þ 2 þ ð1 1Þ 2 q ¼ 0; dð35; 40Þ ¼ 40 35 ¼ 5; fdð35; 40Þ ¼ 0 5 ¼ 0 ðSimilar usersÞ: Case ii: a = h0, 1, 0i, b = h0, 0, 1i. Therefore dða; bÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 0Þ 2 þ ð0 1Þ 2 q ¼ ffiffiffi 2 p ; dð60; 45Þ ¼ 60 45 ¼ 15; fdð60; 45Þ ¼ ffiffiffi 2 p 15 ðFar usersÞ: Case iii: a = h1, 0, 0i, b = h0.8, 0.2, 0i. Therefore dða; bÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:8 1Þ 2 þ ð0:2 0Þ 2 q ¼ 0:283; dð23; 18Þ ¼ 23 18 ¼ 5; fdð23; 18Þ ¼ 0:283 5 ðNear usersÞ: Results in Example 2 show that formula (18) satisfies all the four conditions of the required local fuzzy distance metric. Accordingly, for the fuzzy approach, the fuzzy dis￾tance function between two users can be aggregated using 0 0.5 1 1.5 012345 Genre interestingness measure Membership value Fig. 3. Membership functions for genre interestingness measure. M.Y.H. Al-Shamri, K.K. Bharadwaj / Expert Systems with Applications 35 (2008) 1386–1399 1393