The Effect of Sparsity on Collaborative Filtering Metrics

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1 The Effect of Sparsty on Collaboratve Flterng Metrcs Jesus Bobadlla & Francsco Serradlla Unversdad Poltecnca de Madrd Crta. De Valenca, Km 7, 8031 Madrd, Span Abstract Ths paper presents a detaled study of the behavor of three dfferent content-based collaboratve flterng metrcs (correlaton, cosne and mean squared dfference) when they are processed on several rato matrces wth dfferent levels of sparsty. The total number of experments carred out s 648, n whch the followng parameters are vared: metrc used, number of k-neghborhoods, sparsty level and type of result (mean absolute error, percentage of ncorrect predctons, percentage of correct predctons and capacty to generate predctons). The results are llustrated n two and three-dmensonal representatve graphs. The conclusons of the paper emphasze the superorty of the correlaton metrc over the cosne metrc, and the unusually good results of the mean squared dfference metrc when used on matrces wth hgh sparsty levels, leadng us to nterestng future studes. Keywords: recommender systems, sparsty, collaboratve flterng, metrc 1 Introducton At present, Recommender Systems (RS), are broadly used to mplement Web.0 servces (Janner 007) as mentoned by Knghts and Ln (007), based on Collaboratve Flterng (CF). RS make predctons about the preferences of each user based on the preferences of a set of smlar users. Ths way, a trp to Canary Islands could be recommended to an ndvdual who has rated dfferent destnatons n the Carbbean very hghly, based on the postve ratngs about the holday destnaton of Canary Islands of an mportant number of ndvduals who also rated destnatons n the Carbbean very hghly. There are a large number of applcatons based on RS (Jnghua 007, Baragla 004, and Fesenmaer 00), some of whch are centered on the move recommendaton area (Konstan 004, Antonopoulus 006, L 005). The qualty of the results offered by a RS greatly depends on the qualty of the results provded by ts CF (Adomavcus 005, Herlocker 004) phase;.e. t s Copyrght 009, Australan Computer Socety, Inc. Ths paper appeared at the 0th Australasan Database Conference (ADC 009), Wellngton, New Zealand. Conferences n Research and Practce n Informaton Technology (CRPIT), Vol. 9. A. Bouguettaya, X. Ln, Eds. Reproducton for academc, not-for proft purposes permtted provded ths text s ncluded. essental to be capable of adequately selectng the group of users most smlar to a gven ndvdual. The smlarty among users can be computed n three dfferent ways: content-based methods, model-based methods and hybrd approaches. Content-based methods (Breese 1998, Kong 005) use smlarty metrcs (Herlocker 004) whch operate drectly on the ndvdual user s ratos (n the trp recommender example, that s each value voted for each travel destnaton). Model-based methods (Breese 1998) use user ratos to create a computable model (Bayesan classfer (Cho 007), neural network (Ingoo 003), fuzzy system [16], etc.) and from ths model they predct the clusters of smlar users. At present, for reasons of predctablty and effcency, commercal RS (Lnden 003) are mplemented usng content-based CF metrcs. Modelbased CF can usually be found n non-commercal research phases. The majorty of CF research ams to ncrease the accuracy and coverage (Gagls 006, L 005, Fuyuk 006, and Manolopoulus 007); nevertheless, t s advsable to mprove certan other factors: effectveness of recommendatons, searchng for good tems, credblty of recommendatons, precson and recall measures, etc.). Memory-based methods work on two-dmensonal matrces of U users who have rated a number of tems I. We can consder a RS runnng n an e-travel agency, where, over the years, thousands of travelers have rated hundreds of destnatons, for example. An mportant problem n obtanng effectve predctons usng RS s the fact that most of the users only rate a very small proporton of the tems; ths s known as the sparsty problem. When the matrx s very sparse, t means there are many users who have rated very few tems and ths leads to two man negatve effects: The set of smlar users (k-neghborhoods) (Herlocker 00) does not sutably match the preferences of the recommended user (there are not enough common rated tems to establsh a relable smlarty result between two users). It s not easy to recommend tems to the user, as you are not lkely to fnd enough k- neghborhoods who had rated the same tems postvely.

2 Consequently, the accuracy and the majorty of the man effectveness measures of the CF predctons drop when they are appled to extremely sparse matrces, leadng to the users losng confdence n the RS servce as a whole. The sparsty problem has tradtonally been tackled usng user profle nformaton to renforce the smlarty measure. The CF technques called demographc flterng (Pazzan 1999) use all the possble addtonal nformaton to establsh the smlarty among users such as gender, age, educaton, area code, etc. Another approach n order to reduce the sparsty problem s the use of a dmensonalty reducton technque such as Sngular Value Decomposton (SVD) (Sarwar 000). The demographc flterng approach has two mportant restrctons: More often than not there s no demographc nformaton (or not enough demographc nformaton) n the RS database. Establshng smlartes based on demographc nformaton s very rsky and can easly lead to ncorrect recommendatons. The dmensonalty reducton approach removes unrepresentatve users or tems. At present some research works use statstcal technques such as Prncple Component Analyss (PCA) (Goldbergh 001) and nformaton retreval technques such as Latent Semantc Indexng (LSI) (Deerwester 1990, and Hofmann 003). The man problem wth the reducton approach s the nherent loss of nformaton n the reducton process. Alternatvely, other approaches exst wth whch to deal wth the sparsty problem, such as the use of trust nferences (Papagels 005), attracton-weghted nformaton flterng (Bruyn 004) and topographc organzaton of user preferences patterns (Polccova 004). Content-Based Metrcs Content-based methods work on a table of U users who have rated a number of tems I. The predcton of a nonrated tem for a user u s computed as an aggregate of the ratngs of the K most smlar users (k-neghborhoods) for the same tem, where denotes the set of k- neghborhoods. The most common aggregaton approaches are the average (1) and the weghted sum (). r 1 u, rk, (1) K r kk u, sm( u, k) kk r () k, 1 sm( u, k) (3) kk The smlarty approaches usually compute the smlarty between two users x and y: sm(x,y) based on ther ratngs of tems that both users have rated (4). I rx, and ry, (4) The most popular smlarty metrcs are Pearson correlaton (5) and cosne (6), although we wll complete the experments n ths paper by addng the least known Mean Squared Dfference (MSD) metrc (7). r x, rx ry, ry r rx r r x, y, sm( x, y) (5) y rx, ry, sm( x, y) (6) r x r, y, I 1 sm( x, y) (7) I r x, y r y, 1 The research work shown n ths paper s based on comparatve experments usng Pearson (5), cosne (6) and MSD (7) metrcs, the average aggregaton approach (1), and the Mean Absolute Error (8). MSD has been selected due to ts unque behavor, whch s very dfferent to the correlaton and cosne metrcs, manly when t s used n sparse rato matrces. 3 Desgn of Experments In order to dscover the behavor of each of the three metrcs analyzed, we used the MoveLens database 1 as a base, whch has been a reference for many years n research carred out n the area of CF. The database contans 943 users, 168 tems and 100,000 ratngs, wth a mnmum of 0 tems rated per user. The tems represent cnema flms and the ratng ranges vary from 1 to 5 stars. In all the experments carred out, for each flm that each user has rated, the average value of the ratos gven by ther k-neghborhoods for that flm has been calculated and the predcton has been compared wth the value rated by the user, thus obtanng the calculaton of the mean absolute error (MAE) [8]. Where μ acts as a normalzng factor, usually computed as: 1 Our acknowledgements to the GroupLens Research Group

3 E I user 1 1 K kk r I k, r user, The prevous process was carred out for each of the followng k-neghborhoods values: 15, 30, 60, 90, 10, 150, 180, 10 and 40, coverng from 1.6% to 5.4% of the total number of users. In order to obtan comparable results based on dfferent sparsty levels, we have made several reductons on the orgnal database contanng 100,000 ratngs; each reducton process has removed a fxed number of database ratos usng a random functon. In ths way, we have obtaned fve addtonal databases: 80,000 ratngs, 60,000 ratngs, 40,000 ratngs, 0,000 ratngs and 10,000 ratngs. The MoveLens orgnal database presents a /(943*168) percentage of sparsty, the 80,000 database presents a 100-8*10 6 /(943*168) percentage of sparsty, and so on. Therefore, the sparsty range covered n the experments s: 93.7%, 94.96%, 96.%, 97.48%, 98.74% and 99.37%. In turn, all of these calculatons have been carred out 3 tmes (one for each metrc ncluded n the study). The total number of experments carred out s 648 (9 k-neghborhoods * 6 sparsty levels * 3 metrcs * 4 types of results). The experments have been grouped n such a way that the followng can be determned: Accuracy. Number of recommendatons made. Number of perfect predctons. Number of bad predctons. We consder a perfect predcton to be each stuaton n whch the predcton of the number of stars recommended for one user n one flm matches the value rated by that user for that flm. We consder a bad predcton to be each stuaton n whch the predcton of the number of stars recommended for one user n one flm s dfferent by more than stars from the value rated by that user for that flm. We consder a recommendaton made to be each stuaton n whch a user has rated an tem and at least one of the user s k-neghborhoods has also rated t, n such a way that a predcton could be made and an MAE error obtaned. 4 Results The results secton has been dvded nto sx subsectons: the frst one shows comparatves of the three metrcs studed, processed usng dfferent levels of sparsty; n ths case no detaled nformaton on k-neghborhoods s ncluded as each result (each graph) has been obtaned by calculatng the average of the ndvdual results of all nne (15 to 40) k-neghborhoods. The remanng three subsectons refer to each one of the three metrcs, respectvely, and they contan all the (8) detaled nformaton obtaned when processng all the possble varatons: range of k-neghborhoods / range of sparsty levels. The last subsectons llustrate the detals obtaned by comparng the correlaton metrc wth the cosne and the MSD metrcs Comparson of CF Metrcs usng dfferent sparsty levels The frst results presented here refer to accuracy, processed usng the Mean Absolute Error. The x-axs represents the dfferent percentages of sparsty. Fgure 1 shows better results wth the correlaton metrc than wth the cosne metrc. In fact, there s an mprovement n the correlaton results, n contrast to the cosne results, where the error ncreases as the sparsty percentage ncreases. The Mean Squared Dfference metrc shows much better results than the cosne and correlaton metrcs; nevertheless t s necessary to adjust ths good result wth the very poor behavor obtaned n Fgure. The MAE values ndcate the mean absolute dfference between predctons and real rated values, consequently, a value of 0.5 on the MAE axs means a half-star error n the values predcted from the MoveLens database. Fgure 1. Mean Absolute Error Fgure shows the percentage of recommendatons that each metrc s able to produce. These percentages are obtaned by dvdng each number of predctons obtaned by computng the dfferent levels of sparsty by the number of ratos of each database (10,000, 80,000, etc.), for example, usng the 10,000-rato database (99.37% of sparsty) the cosne metrc was able to compute an average of 8688 predctons, thus the last damond poston n Fgure has the value 86.88%. The correlaton metrc once agan shows better behavor than the cosne metrc as t s able to obtan a larger number of recommendatons; nevertheless, as expected, the amount of predctons decreases as the sparsty level ncreases (t s more dffcult to obtan

4 recommendatons when the quantty of nformaton avalable to make predctons decreases). The rsng secton of the cosne functon n Fgure can be explaned by the erroneous behavor of the cosne metrc when the sparsty of the vectors s too hgh, whch s the case when the sparsty of the database s hgh. The MSD metrc provdes very poor results as t obtans a low quantty of predctons. Ths s ths metrc s Achlles heel and s the aspect that should be mproved n any metrc derved from t. Fgure 3 shows that the correlaton metrc s able to acheve a greater number of predcton hts than the cosne metrc. Whereas the cosne hts drop n lne wth the sparsty level, the correlaton metrc even manages to mprove ts results when the percentage of sparsty s hgh. A very mportant objectve of CF metrcs s to avod ncorrect recommendatons, to prevent users from losng confdence n the system. Fgure 4 presents the percentage of ncorrect recommendatons (more than two stars of dfference between predctons and real ratos). As we can see, the cosne metrc does not respond well to an ncrease n sparsty, whereas the correlaton metrc responds well. The MSD metrc does not produce a hgh quantty of predctons (Fgure ), but t appears to acheve a good number of hts wth ts recommendatons (Fgures 3 and 4), partcularly when the sparsty levels are hgh. Fgure 4. Percentage of bad predctons Fgure. Percentage of recommendatons made It s mportant to realze that the cosne metrc mproves the MAE and the percentage of perfect predcton results compared to the correlaton metrc when the sparsty percentage s not very hgh (database of 100,000 ratngs). The MSD exhbts excellent behavor when the sparsty levels are hgh; nevertheless, t s mportant to realze that the overspecalzaton effect (recommendng tems that are too well-known) can be easly produced. Fgure 3. Percentage of perfect predctons 4. Correlaton Metrc Ths secton shows the detaled results obtaned from the correlaton metrc experments. As n the prevous secton, the aspects of study are: MAE accuracy, percentage of recommendatons made, percentage of perfect predctons and percentage of bad predctons. Each result s presented as a three-dmensonal graph where the x-axs represents the number of k- neghborhoods computed n each experment and the z- axs represents the percentage of sparsty (.e. the 100,000, 80,000, databases used). Fgure 5a) shows an even declne of the MAE when the sparsty percentage ncreases (as we saw n Fgure 1). In ths case we can observe that correlaton works better when the number of k-neghborhoods s not low. Fgure 5b) shows the poorest results when the number of k-neghborhoods s low (less than 60). The evoluton presented n Fgure would mprove by selectng more than 60 k-neghborhoods. The same s true when the correlaton metrc obtans perfect predctons (Fgure 5c) and bad predctons (Fgure 5d). As a result of ths, we can hghlght the good results obtaned by ths metrc, especally when the number of neghborhoods s not low and the sparsty level s hgh. Fgure 5c) shows how the rsng correlaton slope presented n Fgure 3 can be enhanced by selectng a number of k-neghborhoods hgher than 10.

5 Fgure 5. Results of the correlaton metrc: a) Mean Absolute Error, b) percentage of recommendatons made, c) percentage of perfect predctons, d) percentage of bad predctons 4.3 Cosne Metrc Although t was prevously establshed that the correlaton metrc presented better behavor than the cosne metrc, t s relevant to pont out the detals of the cosne, whch s much less regular than the Pearson metrc. In general, t can be sad that the cosne metrc works better when the sparsty level s low and the number of k- neghborhoods s hgh. Ths fact can be observed n Fgure 6, where the best results are gven n the quadrant: k-neghborhoods from 150 to 40 and sparsty from 93.7 to 96.. Fgure 6a) shows the best results when the values are smaller (from 0.65 to 0.75); the same stuaton s presented n Fgure 6d) (from 0.5 to 1.5). Fgures 6b) and 6c) gve the best results when the values are larger (from 80 to 100 and from 68 to 78, respectvely). By studyng Fgures 6a) to 6d) (cosne) we can observe that the slopes of the surfaces are hgher than those correspondng to Fgures 5a) to 5d) (correlaton), both on the sparsty axs and the k-neghborhoods axs (when k>60); ths means that the nfluence of both parameters s hgher n the cosne metrc. 4.4 Mean Squared Dfferences Metrc The results obtaned by applyng the MSD metrc are sgnfcantly dfferent to those obtaned by the cosne and correlaton metrcs. The mean absolute error (Fgure 7a) presents very low (good) values for all the k- neghborhoods and the percentage of sparsty ranges. We can observe that the best results (lowest errors) are obtaned by selectng the lowest k-neghborhood values and, partcularly, when the sparsty percentage s hgh. There can be no doubt that the weak pont of the MSD metrc s ts poor capacty to generate a large

6 Fgure 6. Results of the cosne metrc: a) Mean Absolute Error, b) percentage of recommendatons made, c) percentage of perfect predctons, d) percentage of bad predctons number of predctons. As can be seen n Fgure 7b), the percentage of recommendatons obtaned usng the MSD metrc s lower than that obtaned usng the cosne metrc and even more so n the case of the correlaton metrcs. In ths aspect, t can be observed that the functon s slope s much more sgnfcant n the k- neghborhoods axs than n the percentage of sparsty axs; therefore, ths aspect can be mproved by choosng a hgh k-neghborhood value as opposed to workng wth hgh sparsty databases. The qualty of the recommendatons (measured as hgh levels of perfect predctons combned wth low levels of bad predctons) s very good when usng the MSD metrc, n comparson to the cosne and correlaton metrcs; ths s manly due to the low percentage of bad recommendatons. By observng Fgures 7c) and 7d) t can be determned that the best results (more perfect predctons and fewer bad predctons) are obtaned at the hghest values of sparsty. The poorest results occur when the hghest k-neghborhood values are combned wth the lowest percentages of sparsty levels. In short, when usng the MSD metrc wth low values of sparsty, t s necessary to choose a sutable k- neghborhood value to obtan a balance between qualty (Fgures 7a,c,d) and capacty to recommend (Fgure 7b); the hghest values of the k-neghborhood parameter offer us a better capacty for recommendaton, whle the lowest values of the k-neghborhood parameter lead to an mprovement n the qualty. The most nterestng observaton n Fgure 7 s that all the objectves (low error, hgh capacty to recommend, hgh percentage of perfect predctons and low percentage of bad predctons) are mproved at the same tme when the sparsty value ncreases. Ths characterstc confers a specal mportance to the MSD metrc to be used n very sparse RS databases and t opens a way to creatng new specalzed MSD-based metrcs.

7 Fgure 6. Results of the cosne metrc: a) Mean Absolute Error, b) percentage of recommendatons made, c) percentage of perfect predctons, d) percentage of bad predctons 5 Conclusons The sparsty levels of RS databases have an mportant nfluence on the results of content-based collaboratve flterng metrcs. The mpact of the sparsty nfluence depends on the k-neghborhood value selected, the man objectve we want to maxmze (MAE, capacty to recommend, etc) and, logcally, on the metrc used. When the sparsty level ncreases: The Pearson correlaton metrc mproves ts MAE and has a negatve effect on the capacty to recommend. In addton, ts percentage of good predctons shows a slght ncrease. The cosne metrc has a negatve effect on all the aspects studed (MAE, capacty to recommend, correct predctons, ncorrect predctons); ths negatve behavor can be reduced by selectng hgh k-neghborhood values. The Mean Squared Dfference (MSD) greatly mproves all the results except for the capacty to recommend. The correlaton metrc obtans much better results than the cosne metrc when workng wth sparse RS databases, especally when the k-neghborhood value s not hgh (preferably 60 and 90). By usng databases wth a hgh degree of sparsty, the MSD metrc obtans better results than the correlaton metrc n all the aspects studed except for the capacty to generate a large number of predctons. The MSD metrc presents unusually good behavor when appled to sparse RS rato matrces. However, t should be used wth cauton due ts very poor capacty to generate recommendatons and ts hgh probablty of sufferng from the effects of overspecalzaton; nevertheless, the MSD metrc offers a serous alternatve to the standard metrcs when t s used n sparse rato matrces and can be selected as a reference n desgnng new content-based CF metrcs capable of satsfactorly tacklng the RS sparsty problem.

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