Applications38(2011)14609-14623 directly interrelated equations. In the same way, various real re- Table\ 0 ye uom on to the use m he user sults are provided(obtained with MovieLens)grouped into"case Meas of study"subsections where the integrities and defects of each of Name Measures descriptions the alternatives mentioned can be compared. The main subsections in which this section is structured al ime of the user on the ite preliminary definitions, similarity measures, obtaining a users k neighborhoods, prediction of the value of an item, obtaining the Mean a Prediction quality recommendations, quality of the recommendation: precision and (u Cover age on the user or the Rs recall, quality of the novelty: novelty-precision and novelty-recall erage of the rs and quality of the trust: trust-precision and trust-recall Recommendation precision on the user Recommendation quality Recommendation recall on the user Recommendation recall of the rs In this subsection we specify the definitions, parameters, mea- Novelty quality sures and sets used in the equations, as well as the values of the ning example. In order to simplify the equations of the other Novelty recall of the Rs subsections, we do not specify here the different learning and test wu Trust precision on the user Trust quality groups which must be used in the framework operation. Trust precision of the rs 2.1.1. Formalization Trust recall of the rs Given an rs with a database of l users and m items rated range min,., max), where the absence of ratings will be repre- ented by the symbol·. (1,1),(2,·),③3.·),(4,2),(5.4,(6.1),(7,· We define the basic sets: (N: natural numbers, R: real numbers) (8.·),(9,·),(10,-(11,),(12,·,(13.4),(14.1 U={u∈N1≤u≤L},set users ={i∈N1≤i≤M}, set of items (1,4),·,(3,·),4.3),5·),(6,·,7,·), efine vote v of user u on item i as rui=v (8.·),9,5).(10.4)11.·),(12.·),(13,-,(14,·) 'e define the average of the valid votes of user u as Tu We define the cardinality of a set c as its number of B=01…,(2)(3,,4.(5-,(6(7,3 valid elements (8.3),(9,4),(10.5),(11,-,(12-),(13,5),(14.· Below we present the tables of parameters(Table 1), measures 2.2.1. Introduction (Table 2)and sets(table 3)used in the formalizations made in the 6. The proposed framework will allow to compare future similar- stantiate the behavior of well-known similarity measures and 2.1.2. Running exampl propose the one that gives the best results, so that it can act as a reference for future comparisons. The user to user similarity mea- U={u∈N1≤u≤5},I={∈N1≤i≤14}, sures most commonly-used in RS are: Pearson Correlation, cosine, V={U∈N1≤v≤5Vv=·} Constrained Pearsons Correlation and spearman rank correlation. (1.5),(2,·),(3.·,(4.3),(5,·,(6.4),(7,1) tween two users x and y: sim(x, y) based on their ratings of items (8-),(9,-),(10.4,、(11),(12.,2),(13,4,(14. hat both users have rated(9). Axy={∈lrx≠·Aryi≠· 2.2. 2 Running example Parameters In order to make the example easier to follow we will use a sim- Square Difference(MSD) of two users x and y We represent the votes issued in table format (Table 4): We obtain the table of similarities between users (Table 5), tak # Ratings received ing into account that MSD(x, y)= MSD(, x). The maximum similar- Relevant item threshold #Trust users #Trust items Calculation example: MSD(U1, U2)=i((5-1)+(3-2)+directly interrelated equations. In the same way, various real re￾sults are provided (obtained with MovieLens) grouped into ‘‘case of study’’ subsections where the integrities and defects of each of the alternatives mentioned can be compared. The main subsections in which this section is structured are: preliminary definitions, similarity measures, obtaining a user’s k￾neighborhoods, prediction of the value of an item, obtaining the accuracy, standardization process, obtaining the coverage, top N recommendations, quality of the recommendation: precision and recall, quality of the novelty: novelty-precision and novelty-recall and quality of the trust: trust-precision and trust-recall. 2.1. Preliminary definitions In this subsection we specify the definitions, parameters, mea￾sures and sets used in the equations, as well as the values of the running example. In order to simplify the equations of the other subsections, we do not specify here the different learning and test groups which must be used in the framework operation. 2.1.1. Formalization Given an RS with a database of L users and M items rated in the range {min,...,max}, where the absence of ratings will be repre￾sented by the symbol . We define the basic sets: (N: natural numbers, R: real numbers) U ¼ fu 2 Nj1 6 u 6 Lg; set of users ð1Þ I ¼ fi 2 Nj1 6 i 6 Mg; set of items ð2Þ V ¼ fv 2 Nj min 6 v 6 maxg [ fg; set of possible votes ð3Þ Ru ¼ fði; vÞji 2 I; v 2 Vg; ratings of user u ð4Þ We define vote v of user u on item i as ru;i ¼ v ð5Þ We define the average of the valid votes of user u as ru ð6Þ We define the cardinality of a set C as its number of valid elements #C ¼ #fx 2 Cjx – g ð7Þ #Ru ¼ #fi 2 Ijru;i – g ð8Þ Below we present the tables of parameters (Table 1), measures (Table 2) and sets (Table 3) used in the formalizations made in the paper. 2.1.2. Running example U ¼ fu 2 Nj1 6 u 6 5g; I ¼ fi 2 Nj1 6 i 6 14g; V ¼ fv 2 Nj1 6 v 6 5 _ v ¼ g R1 ¼ h1; 5i;h2; i;h3; i;h4; 3i;h5; i;h6; 4i;h7; 1i; h8; i;h9; i;h10; 4i;h11; i;h12; 2i;h13; 4i;h14; i R2 ¼ h1; 1i;h2; i;h3; i;h4; 2i;h5; 4i;h6; 1i;h7; i h8; i;h9; i;h10; i;h11; i;h12; i;h13; 4i;h14; 1i R3 ¼ h1; 5i;h2; 2i;h3; i;h4; 4i;h5; i;h6; i;h7; i; h8; 3i;h9; 5i;h10; 4i;h11; i;h12; i;h13; 4i;h14; i ; R4 ¼ h1; 4i;h2; i;h3; i;h4; 3i;h5; i;h6; i;h7; i; h8; i;h9; 5i;h10; 4ih11; i;h12; i;h13; i;h14; i R5 ¼ h1; i;h2; i;h3; i;h4; i;h5; i;h6; i;h7; 3i; h8; 3i;h9; 4i;h10; 5i;h11; i;h12; i;h13; 5i;h14; i 2.2. Similarity measures 2.2.1. Introduction The proposed framework will allow to compare future similar￾ity measures and methods, in the meantime, it is advisable to sub￾stantiate the behavior of well-known similarity measures and propose the one that gives the best results, so that it can act as a reference for future comparisons. The user to user similarity mea￾sures most commonly-used in RS are: Pearson Correlation, cosine, Constrained Pearson’s Correlation and Spearman rank correlation. The similarity approaches usually compute the similarity be￾tween two users x and y: sim(x,y) based on their ratings of items that both users have rated (9). Ax;y ¼ fi 2 Ijrx;i – ^ ry;i – g: ð9Þ 2.2.2. Running example In order to make the example easier to follow we will use a sim￾ilarity measure that is very easy to calculate manually: the Mean Square Difference (MSD) of two users x and y. MSDðx; yÞ ¼ 1 #Ax;y X i2Ax;y ðrx;i ry;iÞ 2 : We represent the votes issued in table format (Table 4): We obtain the table of similarities between users (Table 5), tak￾ing into account that MSD(x,y) = MSD(y,x). The maximum similar￾ity is reached at value 0. Calculation example: MSDðU1; U2Þ ¼ 1 4 ½ð5 1Þ 2 þ ð3 2Þ 2 þ ð4 1Þ 2 þ ð4 4Þ 2  ¼ 6:5. Table 1 Parameters. Name Parameters descriptions L #Users M #Items min #min rating value max #max rating value K #Neighborhoods N #Recommendations b #Range to the current time c #Ratings received h Relevant item threshold q #Trust users h #Trust items Table 2 Measures. Name Measures descriptions Usage ru, i Rating of the user on the item General use tu, i Rating time of the user on the item pu, i Prediction to the user on the item mu Mean absolute error on the user Prediction quality m Mean absolute error of the RS cu Coverage on the user c Coverage of the RS tu Recommendation precision on the user Recommendation quality t Recommendation precision of the RS xu Recommendation recall on the user x Recommendation recall of the RS nu Novelty precision on the user Novelty quality n Novelty precision of the RS lu Novelty recall on the user l Novelty recall of the RS wu Trust precision on the user Trust quality w Trust precision of the RS zu Trust recall on the user z Trust recall of the RS J. Bobadilla et al. / Expert Systems with Applications 38 (2011) 14609–14623 14611