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)