A. Zenebe, A F. Norcio/ Fuzzy Sets and Systems 160(2009)76-94 Membership degree of movie to genres for User 7 Movie (lj) Gi (vector for jth movie) 0.683 1.000 0.000 0.000 4 0.438 0.000 1000 0.000 Based on the heuristics illustrated above, the possibility for item Ij to take different values of X varies and the membership function should meet the following four criteria: (1)assigning higher degree of membership to major values than minor values; (2) assigning O to values that are not associated with the item; (3)degrees of membership should be normalized to the range of [0, 1]; and(4)the same value of X at similar rank positions between different items should have varying degrees of membership values if the number of values of X associated with the items are different. We represent this type of heuristic with a Gaussian-like membership function, as shown in(1). Ax,(Ij)=rk/2v where N=Ilil is the number of values of X associated with Ij and rk(I<rk <ILiD is the rank position of value xk and a> I is a parameter used as a threshold to control the difference between consecutive values of X in/j. Moreover, is the only parameter that needs to be determined For example, using Eq(1)with a set to 1. 2(after various experimental trials), the movie 'King Kong(2005 0.211),(Thriller,.168)). Furthermore, for User 7, Table 1 shows the representation of some of the rated movies represented in terms of genres: for Lil= 5 and xj=[(Action, 1),(Adventure, 0.366), (Drama, 0. 272),(Fanta Ross et al. [23] stated that a membership function(MF)should match"the intuitively plausible semantic description of imprecise properties of the objects in X. "The reason we make the first important genre completely satisfied is movies are advertised exclusively as having a specific genre such as dram a and comedy; as well users have similar perception to movies. The soundness of the representation and inference method are further investigated in Section 5 using the movie dataset The reciprocal function defined as 1/k for k= l to N is also investigated. The problem with the reciprocal function is it does not consider the total number of different genres that exist in movies leading to the same degree of membership value for the same genre at same rank position with different number of genres. A uniform distribution membership of genres in a movie, assuming a movie with multiple genres has equal degree of genre presence for all occurring genres represented by l or 0, is the baseline crisp set representation. The heuristic that leads to the membership function in(1) is developed based on the analysis of the movie dataset literature on movies [10] and preliminary experimental trials conducted on a. This heuristic, for instance, assumes that two genres will not have equal degree of presence in a movie. That is, two or more genres cannot exist in"equal degree of presence"or"in equal amount"in a movie. For example, let us say a movie X has drama and comedy. The movie cannot have dramatic and comedy nature with exact equal amount, i.e. with exact degree of membership 0.5 each. This is also true in the movies database used in this research. This assumption is logical because a movie cannot have exactly the same"content"of two or more genres. If the amount of content is close then a needs to be tuned Moreover, in future research, studying various membership functions is needed in order to get an optimal fuzzy set based recommender system. The basis of our study is, provided that a membership function that satisfies the four criteria identified above. A is used, the recommendations by FTM will be better than crisp-set based approach. The example, one can use movie actresses/actors as the second attribute. The actors in a movie can be represented in a vco. a representation scheme can be extended to recommender systems based on a combination of multiple attributes. Fe A=a1, a2,..., ak) for K actors. The role or importance of an actor or actress ak in a movie mi can be represented by degree of membership associated with the fuzzy variable 'degree of role or importance. That is, A j=l(ak, Ha ( i) for k= I to k], where pa(lj) can be determined heuristically or using machine learning
A. Zenebe, A.F. Norcio / Fuzzy Sets and Systems 160 (2009) 76–94 81 Table 1 Membership degree of movie to genres for User 7 Movie (Ij ) Gj (vector for j th movie) Rating xj1 xj2 xj3 … xj15 xj16 56 4 0.683 1.000 0.438 … 0.000 0.000 79 5 1.000 0.000 0.000 … 0.467 0.000 89 3 0.683 0.000 0.000 … 0.000 1.000 … … … … … …… … 254 2 0.438 0.000 1.000 … 0.000 0.000 Based on the heuristics illustrated above, the possibility for item Ij to take different values of X varies and the membership function should meet the following four criteria: (1) assigning higher degree of membership to major values than minor values; (2) assigning 0 to values that are not associated with the item; (3) degrees of membership should be normalized to the range of [0, 1]; and (4) the same value of X at similar rank positions between different items should have varying degrees of membership values if the number of values of X associated with the items are different. We represent this type of heuristic with a Gaussian-like membership function, as shown in (1). xk (Ij ) = rk/2 √∗|Lj|(rk−1) , (1) where N = |Lj | is the number of values of X associated with Ij and rk (1rk |Lj |) is the rank position of value xk, and > 1 is a parameter used as a threshold to control the difference between consecutive values of X in Ij . Moreover, is the only parameter that needs to be determined. For example, using Eq. (1) with set to 1.2 (after various experimental trials), the movie ‘King Kong (2005)’ is represented in terms of genres: for |Lj | = 5 and xj = {(Action, 1), (Adventure, 0.366), (Drama, 0.272), (Fantasy, 0.211), (Thriller, 0.168)}. Furthermore, for User 7, Table 1 shows the representation of some of the rated movies. Ross et al. [23] stated that a membership function (MF) should match “the intuitively plausible semantic description of imprecise properties of the objects in X.” The reason we make the first important genre completely satisfied is movies are advertised exclusively as having a specific genre such as drama and comedy; as well users have similar perception to movies. The soundness of the representation and inference method are further investigated in Section 5 using the movie dataset. The reciprocal function defined as 1/k for k = 1 to N is also investigated. The problem with the reciprocal function is it does not consider the total number of different genres that exist in movies leading to the same degree of membership value for the same genre at same rank position with different number of genres. A uniform distribution membership of genres in a movie, assuming a movie with multiple genres has equal degree of genre presence for all occurring genres represented by 1 or 0, is the baseline crisp set representation. The heuristic that leads to the membership function in (1) is developed based on the analysis of the movie dataset, literature on movies [10] and preliminary experimental trials conducted on . This heuristic, for instance, assumes that two genres will not have equal degree of presence in a movie. That is, two or more genres cannot exist in “equal degree of presence” or “in equal amount” in a movie. For example, let us say a movie X has drama and comedy. The movie cannot have dramatic and comedy nature with exact equal ‘amount’, i.e. with exact degree of membership 0.5 each. This is also true in the movies database used in this research. This assumption is logical because a movie cannot have exactly the same “content” of two or more genres. If the amount of content is close then needs to be tuned. Moreover, in future research, studying various membership functions is needed in order to get an optimal fuzzy set based recommender system. The basis of our study is, provided that a membership function that satisfies the four criteria identified above. A is used, the recommendations by FTM will be better than crisp-set based approach. The representation scheme can be extended to recommender systems based on a combination of multiple attributes. For example, one can use movie actresses/actors as the second attribute. The actors in a movie can be represented in a vector A = {a1, a2,...,ak} for K actors. The role or importance of an actor or actress ak in a movie mi can be represented by degree of membership associated with the fuzzy variable ‘degree of role or importance’. That is, Aj = {(ak, ak (Ij )), for k = 1 to K}, where ak (Ij ) can be determined heuristically or using machine learning.
A Zenebe, A F. Norcio/Fuzzy Sets and Systems 160(2009)76-94 3. 2. User feedback representation using fuzzy set .. User rating is the most widely used feedback in recommender systems. It is a proxy variable used for measuring ser degrees of interest in an item. User ratings are represented and interpreted as binary values-those liked or disliked. In five-scale ratings, those above 3 are considered as liked. However, user rating is intrinsically imprecise as a may give different ratings to the same item at different times and situations due to the difficulty to make a distinction between rating 4 and 5, and 1 and 2. Moreover, the same rating, say 4 on a scale of 5, given by two users does not necessarily imply equal degrees of interest in an item. For pessimistic users, a rating of 4 may mean strongly liked but for optimistic raters it may mean somewhat liked Is the difference between ratings 3 and 4 the same as the difference between 4 and 5? These all contribute to fuzziness that arises from the human thinking processes instead of randomness associated with the ratings. Therefore, user interest based on user rating is treated as a fuzzy variable and its uncertainty is represented using a possibility distribution function. Let the fuzzy variable degrees of interest in an item(DI)consisting of strongly liked (SL), liked(L), indifferent 1), disliked(D), and strongly disliked (SD) fuzzy values, and associated with user rating(R)expressed in continuum from minimum value(Min) to maximum value(Max). Then, the proposition'a user has strongly liked an item Ihas the possibility distribution function IR()= HsL(R =r), for r between Min and Max Under this interpretation or semantic of the fuzzy variable Dl, the user rating is represented and inferred using a possibility distribution function by treating the rating as a fuzzy number [22]. For instance a rating 4 on 5 scale which refers to strongly liked is represented in terms of its possibility distribution values =IASL(R=4)/5, HL(R 4)/4, .., HsD(R= 4)/1. For instance, for User 7 in Table 1, the possibility distribution of User 7s rating of High on movie 56 can be expressed as: IASL (R= 4)=0.50, H(R=4)=1,HI(R = 4)=0.50, AD(R=4)= 0.25, ASD(R=4)=0.0 F; and possibility distribution of User 7s rating of Very High on movie 79 can be expressed as:{sL(R=5)=1,(R=5)=0.55,(R=5)=0.20,pD(R=5)=0.0,sD(R=5)=0.0 without losing generality, a half triangular fuzzy number, which is the simplest model of uncertain quantity, is used to represent the degree of positive experience a user has in relation to an item. The half triangular fuzzy number membership function, for user rating r on Ii E [Min, Max] and for a fuzzy set A on DI, is defined as HA(i=(-Min)/(Max- Min) As a result, a set of items liked by a user, denoted by E, is defined as: (i: HA(i)>0.5, i.e., li: r>(Min+ Max)/2) 3.3. Inference engine and algorithm Based on the representation scheme defined for items and user feedback, the inference engine consisting of the recommendation score aggregation methods and the similarity measures is defined below. 3.3. 1. Fuzzy set theoretic similarity measures One of the most important issues in recommender systems research is computing similarity between users, and between items(products, events, services, etc. ) This in turns highly depends on the appropriateness and reliability of the methods of representation. The set-theoretic, proximity-based and logic-based are the three classes of measures of similarity [11]. In fuzzy set and possibility framework, similarity of users or items is computed based on the membership unctions of the fuzzy sets associated to the users or item features. Based on the work of Cross and Sudkamp [11], those similarity measures that are relevant for item recommendation application are adapted. For items 1; and lk that are defined as ((xi, Hx (D)),i=l,., N and ((i, Hx(Ik)),i=l,., N, a similarity measure between lj and lk is denoted by S(Ik, Ij), and the different similarity measures are defined n(x,(k),px2() SI(k, ID)= (Fx1(k),Px(1)) S2(k,l)= ∑Hx2()*x1() (△(2(4)2)√(∑(ax()2)
82 A. Zenebe, A.F. Norcio / Fuzzy Sets and Systems 160 (2009) 76–94 3.2. User feedback representation using fuzzy set User rating is the most widely used feedback in recommender systems. It is a proxy variable used for measuring user degrees of interest in an item. User ratings are represented and interpreted as binary values-those liked or disliked. In five-scale ratings, those above 3 are considered as liked. However, user rating is intrinsically imprecise as a user may give different ratings to the same item at different times and situations due to the difficulty to make a distinction between rating 4 and 5, and 1 and 2. Moreover, the same rating, say 4 on a scale of 5, given by two users does not necessarily imply equal degrees of interest in an item. For pessimistic users, a rating of 4 may mean strongly liked but for optimistic raters it may mean somewhat liked. Is the difference between ratings 3 and 4 the same as the difference between 4 and 5? These all contribute to fuzziness that arises from the human thinking processes instead of randomness associated with the ratings. Therefore, user interest based on user rating is treated as a fuzzy variable and its uncertainty is represented using a possibility distribution function. Let the fuzzy variable degrees of interest in an item (DI) consisting of strongly liked (SL), liked (L), indifferent (I ), disliked (D), and strongly disliked (SD) fuzzy values, and associated with user rating (R) expressed in continuum from minimum value (Min) to maximum value (Max). Then, the proposition ‘a user has strongly liked an item I’ has the possibility distribution function R(I ) = SL(R = r), for r between Min and Max. Under this interpretation or semantic of the fuzzy variable DI, the user rating is represented and inferred using a possibility distribution function by treating the rating as a fuzzy number [22]. For instance a rating 4 on 5 scale which refers to strongly liked is represented in terms of its possibility distribution values = {SL(R = 4)/5, L(R = 4)/4,..., SD(R = 4)/1}. For instance, for User 7 in Table 1, the possibility distribution of User 7’s rating of High on movie 56 can be expressed as: {SL(R = 4) = 0.50, L(R = 4) = 1, I(R = 4) = 0.50, D(R = 4) = 0.25, SD(R = 4) = 0.0}; and possibility distribution of User 7’s rating of Very High on movie 79 can be expressed as: {SL(R = 5) = 1, L(R = 5) = 0.55, I(R = 5) = 0.20, D(R = 5) = 0.0, SD(R = 5) = 0.0}. Without losing generality, a half triangular fuzzy number, which is the simplest model of uncertain quantity, is used to represent the degree of positive experience a user has in relation to an item. The half triangular fuzzy number membership function, for user rating r on Ii ∈ [Min, Max] and for a fuzzy set A on DI, is defined as: A(Ii) = (r − Min)/(Max − Min). (2) As a result, a set of items liked by a user, denoted by E, is defined as: {Ii: A(Ii) > 0.5, i.e., Ii:r>(Min+Max)/2}. 3.3. Inference engine and algorithm Based on the representation scheme defined for items and user feedback, the inference engine consisting of the recommendation score aggregation methods and the similarity measures is defined below. 3.3.1. Fuzzy set theoretic similarity measures One of the most important issues in recommender systems research is computing similarity between users, and between items (products, events, services, etc.). This in turns highly depends on the appropriateness and reliability of the methods of representation. The set-theoretic, proximity-based and logic-based are the three classes of measures of similarity [11]. In fuzzy set and possibility framework, similarity of users or items is computed based on the membership functions of the fuzzy sets associated to the users or item features. Based on the work of Cross and Sudkamp [11], those similarity measures that are relevant for item recommendation application are adapted. For items Ij and Ik that are defined as {(xi, xi (Ij )), i = 1,...,N} and {(xi, xi (Ik)), i = 1,...,N}, a similarity measure between Ij and Ik is denoted by S(Ik, Ij ), and the different similarity measures are defined as S1(Ik, Ij ) = i min(xi (Ik), xi (Ij )) i max(xi (Ik), xi (Ij )), (3) S2(Ik, Ij ) = i xi (Ik) ∗ xi (Ij ) (( i (xi (Ik))2)) (( i (xi (Ij )2)) , (4)