lem is characterized by a single decision maker facing a set ... alternatives from best to worst (Individual De- ... through a given web site) to a given decision sup-.
Collaborative Filtering Methods based on Fuzzy Preference Relations Patrice Perny and Jean-Daniel Zucker LIP6 - Paris 6 University 4 Place Jussieu 75252 Paris Cedex 05 - France
Abstract
Decision Making (GDM)). These two problems have received much attention in the past decades and Decision Theory has provided many sophisticated preference models and decision rules to face various situations (see e.g. [10], [4], [15], [14], [17], [5], [16]). However, the recent multiplication of distributed information systems shared by multiple individuals and the growing success of the internet network is opening new spaces for decision support activities. As an illustration of such a new activity, we are interested in the conception of decision support systems based on the implicit sharing of preferences and experiences between di erent individuals facing similar decision problems (i.e. decision problems with similar sets of alternatives). We introduce in this paper a collaborativecentered approach to decision support. "Collaborative Decision Making" (CDM) is basically concerned with a new category of decision problems where any individual seeks recommendation for his personal choices, the other individual being only considered as possible advisors. Despite the multiplicity of possible advisors, the problem addressed in CDM is not a matter of group decision making or negotiation between individuals. Indeed, in CDM, the problem is to nd a recommendation that best t the preferences of a single decision maker, rather than to seek a compromise solution satisfying all the users. The recommendation provided to the decision maker is based on other users experiences without any explicit communication or cooperation. The individuals that have contributed to the recommendation are
This paper introduces a new approach for decision support. It is characterized by a collaborative decision making process relying on the implicit sharing of preferences and experience between di erent individuals facing similar decision problems. A recommendation principle is described, based on fuzzy ltering methods de ned from individual fuzzy preference relations and fuzzy similarity relations between users. This approach is illustrated in the context of movie recommendation tasks on the internet. Keywords: multicriteria analysis, collaborative decision making, fuzzy preference relations, fuzzy similarity relations, fuzzy ltering.
1 Introduction The usual practice of decision support is oriented to deal with two typical problems. The rst problem is characterized by a single decision maker facing a set of feasible alternatives (evaluated according to one or possibly several criteria). The aim of the analysis is then to identify the best alternatives for the decision maker, or to rank the alternatives from best to worst (Individual Decision Making (IDM)). The second problem involves several decision makers, experts or judges, facing the same decision problem with several possible alternatives. The aim of the analysis is there to nd a compromise solution between possibly con icting individual preferences (Group 1
Perny, P. & J.-D. Zucker (1999). Collaborative Filtering Methods based on Fuzzy Preference Relations. EUROFUSE-SIC'99, Budapest.
collaborative ltering methods for the selection of objects. In the third section we brie y present an internet-based movie-recommendation system implementing both methods.
not even aware of their role of advisors. On the other hand, despite the fact that each individual is making a decision, CDM is not either an individual decision making problem. Indeed, there are several decision problems (one per individual) to be addressed simultaneously. Moreover, each individual has an in uence on the recommendation provided to other individuals. Therefore the set of decision problems as a whole cannot only be viewed as a collection of independent decision problems. In this paper we assume that several independent individuals have been connected (e.g. through a given web site) to a given decision support system and have expressed their preferences (e.g. using grades of satisfaction) about some universal objects (e.g. movies, CDs, books, ...). The problem we address is to provide each individual with relevant recommendation regarding objects that he has not yet graded. The approach we propose is to base the building of recommendations on fuzzy selections generated by collaborative ltering methods. The basic idea of collaborative ltering (see [6], [7], [13], [1]) is to investigate similarities of judgement and preferences between individuals so as to recommend an object o to the individual i when o is well graded by a signi cant amount of individuals sharing with u almost the same value system. It is clear that collaborative ltering requires a critical number of individuals to be reached. To deal with the case where the number of individuals is not enough, a content-oriented ltering based on a more conventional multicriteria analysis of similarities between objects ought to be preferred. We propose here an hybrid approach integrating this two ltering mechanisms so that each user is provided with a fuzzy subset of possibly interesting objects graded by a level of con dence attached to the recommendation. The paper is organized as follows. In the rst section, we present a fuzzy relational model linking individuals to objects. This model enables the comparison of objects, individuals and the representation of fuzzy preferences. In the second section, we introduce both content-oriented and
2 The construction of a fuzzy relational system 2.1 The initial information A collaborative decision problem is characterized by a set of individuals I , a set of objects O and a matrix of grades assigned to some objects by some individuals. More precisely, assuming that any individual i 2 I has graded a set of objects G(i) O, we want to assess the value of every object in O n G(i) for every individual i, so as to provide him, with a relevant selection of objects. We denote x 2 X the grade given to o by individual i, for all i 2 I and all o 2 G(i). The gradation scale X is assumed to be an absolute scale, bounded and completely ordered. Within X we distinguish 3 particular grades, namely the maximal grade x , the minimal grade x and a neutral grade x0 dividing the scale X into two parts: the part [x0 ; x ] corresponds to positive evaluations whereas the part [x ; x0 ] corresponds to negative evaluations. Moreover, as mentioned in the introduction, we want to complete the collaborative approach with a recommendation principle based on the multiattribute analysis of objects. For this reason, we also assume that a database is available, in which every object o 2 O is described by a tuple (d1 (o); : : : ; d (o)) of values representing the image of o within a multiattribute space Y = Y1 : : : Y . We present now the construction of a fuzzy relational system allowing the similarity between objects, the similarity between individuals and the preferences of individuals over objects to be represented. io
n
n
2
2.2 The construction of fuzzy relations
where u+ is a non-decreasing function valued in the unit interval and such that u+ (x0 ) = 0 and u+ (x ) = 1. The negative side of preferences is represented by a fuzzy relation P ; de ned by: ( ; o 2 G(i) ; P (i; o) = u0 (min(x ; x0 )) ifotherwise where u; is a non-decreasing function valued in the unit interval and such that u; (x0 ) = 0 u; (x) = 1. Functions u+ and u; are used to de ne the membership of every object to the fuzzy subsets of positive and negative examples. Such fuzzy subsets are de ned for each individual and characterize their personal preference pro le. Notice that, by construction, minfP + (i; o); P ; (i; o)g = 0 for all i 2 I and all o 2 G(i). Therefore, positive and negative examples attached to a given individual are disjoint fuzzy sets of objects.
2.2.1 Fuzzy similarity between objects
For each attribute Y , we construct a onedimensional similarity relation de ned on O. There is no space here to give a comprehensive presentation of the various techniques that can be used for the construction of . The construction depends on the nature of the attribute. As an illustration, consider the 2 following cases: Attributes valued on an ordered numerical scale: (o; o0 ) = (jd (o) ; d (o0 )j) where is a non-increasing function valued into [0; 1] and such that (0) = 1. Attributes valued on a nominal scale: ( d (o) = d (o0 ) (o; o0 ) = 10 ifotherwise : j
j
io
j
j
j
j
j
j
j
Then the overall fuzzy similarity over objects is de ned on O O by:
2.2.3 Fuzzy in uence Relations
(o; o0 ) = ( (o; o0 ); : : : ; (o; o0 )) 1
n
Following the approach adopted for the comparison of objects, the similarity between individuals could also be based on a multiattribute pro le representing the individual. Notice however that in most cases, the collaborative decision support system, to be really useful, must allow the users to be anonymous. Thus, the only available information to be used in the production of relevant recommendations is the set of grades given to objects by each user. They re ect the value system of each user and allow the preference pro le of users to be compared. For every pair of individuals (i; j ), the comparison of their preference pro le can be made by investigating the grades of the objects belonging to G(i; j ) = G(i) \ G(j ). More precisely, within the set of objects, the preference pro le of each individual i 2 I is characterized by two fuzzy sets, namely: X + X ; + ;
where is a weighted compromise operator, e.g. a weighted quasi-linear mean : ! X ; 1 (x1 ; : : : ; x ) = ! (x ) n
n
i
i
i=1
where is a continuous strictly monotonic function on the unit interval and w ; j = 1; : : : ; n are factors weighting the relative importance of each attributes. j
2.2.2 Fuzzy preference relations Here, preference relations are not used to compare objects but to express fuzzy positive or negative opinions of individuals over objects. Two types of fuzzy preference relations are constructed on I O: The positive side of preferences is represented by a fuzzy relation P + de ned by: ( + o 2 G(i) + P (i; o) = u0 (max(x ; x0 )) ifotherwise
G
(i) =
o2G(i)
P
(i; o)=o
G
(i) =
o2G(i)
P
(i; o)=o
Such fuzzy sets are useful to evaluate the similarity between two individuals (i; j ), and the extend to which an individual i can in uence an individual j . Thus, from the comparison of the sets
io
3
(G+ (i); G; (i)) with (G+ (j ); G; (j )) we derive 2 fuzzy in uence relations over individuals. positive in uence: we say that i has a positive in uence over j when I + (i; j ) holds:
weakening slightly the condition. For instance, we can substitute the min operator used in equations (1) and (2) by a compromise operator. The relations C and D are symmetric and C is re exive. They could be used to construct fuzzy clusters of individuals having similar preference pro les. Let us show now how these fuzzy relations are used in ltering methods.
I + (i; j ) [8 o 2 G(i; j ); ((P + (i; o) P + (j; o)) ^ (P ; (i; o) P ;(j; o))] negative in uence: we say that i has a negative in uence over j when I ; (i; j ) holds: I ; (i; j ) [8 o 2 G(i; j ); ((P + (i; o) P ; (j; o)) ^ (P ; (i; o) P +(j; o))] Notice that from relations I + and I ; , it is easy
3 Fuzzy ltering methods In the rst subsection we present a contentoriented ltering method based on the fuzzy similarity relation over objects, as de ned in 2.2.1. In the second subsection, we show how the fuzzy relation I + can be used in fuzzy collaborative ltering methods. A similar presentation could be done with relation I ; , and possibly with C and D. However, there is no space here to present all these options. Finally, we present a hybrid ltering method integrating the two approaches.
to identify the pairs of individuals having concordant preference pro les, and the pairs having discordant pro les. Such notions can be represented by relations C and D de ned as follows:
C (i; j ) (I +(i; j ) ^ I + (j; i)) D(i; j ) (I ;(i; j ) ^ I ; (j; i)) Because P + and P ; are fuzzy relations, the above logical equations must be interpreted in a [0, 1]-valued logic. This yields to the following fuzzy relations:
I + (i; j ) I ; (i; j ) C (i; j ) D(i; j )
3.1 Content-oriented ltering
This rst method is derived from multicriteria methods (see [12] and [11]) as well as fuzzyneighbors algorithms (see [2] and [8]). It can be applied for each individual independently because it is totally non-collaborative. The basic idea of content-oriented ltering is to provide every individual i with objects similar to those he uses to like. Following this principle, we ought to recommend an object o to individual i if the following condition holds:
T (p++(o); p;; (o)) (1) = min T (p+; (o); p;+ (o)) (2) f 2 ( )g = T (I + (i; j ); I + (j; i)) (3) ; ; = T (I (i; j ); I (j; i)): (4) =
min
f 2 o
o
g
G(i;j )
G i;j
ij
ij
ij
ij
where T is a t-norm, I is the fuzzy implication de ned as the quasi-inverse of T , and p (o) = I (P (i; o); P (j; o)) for all u, v in f+; ;g.
F +(i; o) [:(o 2 G(i)) ^ (9o0 2 N (o); ( (o0 ; o) ^ P + (i; o0 )))] where N (o) is the set of the k most similar objects from o (i.e. objects o0 with the k greatest values (o0 ; o)). Similarly, we ought to discard an object o from
uv ij
u
k
v
The relations I + and I ; are re exive, but not symmetric and not transitive. They will be used in the next section to export preferences from an individual to another one. Notice that the presence of a universal quanti er in the logical equations de ning in uence relations may have drastic e ects in some applications (especially if the number of individuals involved in the process is small). This diculty can be easily overcome by
k
the list of objects to be recommended to individual i if the following condition holds:
F ;(i; o) [:(o 2 G(i)) ^ (9o0 2 N (o); ( (o0 ; o) ^ P ; (i; o0 )))] k
4
version of the condition is given by:
The numerical translation of such equations gives: _ F + (i; o) = T ( (o0 ; o); P + (i; o0 )) 2 k( ) _ ; F (i; o) = T ( (o0 ; o); P ; (i; o0 )) 0
o
o0
N
2
CF ;(i; o) [:(o 2 G(i)) ^ (9j 2 N (i); (I + (j; i) ^ P ; (j; o)))] k
o
The numerical translation of these equations is: _ CF +(i; o) = T (I +(j; i); P + (j; o)) 2 k( ) _ CF ;(i; o) = T (I +(j; i); P ; (j; o))
k (o)
N
for all o 2 O n G(i), where T is a t-norm and _ the t-conorm associated to T . The choice of the t-norm depends on the application. For instance, choosing a no-idempotent t-norm allows reinforcement e ects when several positively graded objects are in N (o).
j
j
N
2
i
k (i)
N
for all o 2 O n G(i), where T is a t-norm and _ the t-conorm associated to T . Here also, the nal con dence level attached to a recommendation o for individual i is obtained by balancing the positive and negative arguments. We set:
k
The overall con dence level attached to a recommendation o for individual i is obtained by balancing the positive and negative arguments represented by F + (i; o) and F ; (i; o). We choose to de ne the nal con dence level of a content-based recommendation by:
CF (i; o) = maxfCF + (i; o) ; CF ;(i; o); 0g where is the attenuation coecient introduced above.
F (i; o) = maxfF + (i; o) ; F ; (i; o); 0g where is an attenuation coecient chosen in the unit interval so as to limit the importance of negative arguments in the ltering process.
3.3 Hybrid ltering The general recommendation system we are proposing can be seen as an hybrid ltering process aggregating the outputs of the contentoriented ltering process with those obtained by collaborative ltering. The nal aggregation is simply de ned, for all o 2 O n G(i), by the following equation:
3.2 Collaborative ltering
The index I + (j; i) introduced above measures the strength of the arguments justifying the transfer of preferences from j to i. Thus, when I + (j; i) is close to 1 and P + (j; o) is close to 1 for an object o 2 G(j )\(O nG(i)), we could infer that o must be recommended to i. More generally, for each individual i, we could inspect the fuzzy neighborhood of individuals having a positive in uence on him, and derive recommendations from their opinions. Following this principle, we decide to recommend an object o to individual i if the following condition holds:
FF (i; o) = :CF (i; o) + (1 ; )F (i; o) where 2 [0; 1] represents the weight of the other individuals in the recommendation. We set = 0 at the beginning of the decision support process, and then is increased as the number of individuals involved in the process increases. The fuzzy set: X FF (i; o)=o
CF + (i; o) [:(o 2 G(i)) ^ (9j 2 N (i); (I + (j; i) ^ P + (j; o)))] where N (i) is the set of the k most in uential individual for i (i.e. individuals j having the k greatest values P + (j; i)). Similarly, the negative k
o
k
2 n
O G(i)
represents the fuzzy selection of objects to be recommended to individual i, for every i 2 I . 5
4 Designing a system for movierecommendations
or not) the set of recommendation produced by "Film conseil" will naturally integrate this modi cation. For example, if two movies m and m0 have been recommended following similar computations, and if m is later graded negatively by the user, m0 will automatically be discarded from the top recommended movies.
To experiment our approach to collaborative decision making, we have developed a movierecommendation system called "Film Conseil", accessible from the web (http://132.227.69.2). There exists today several commercial products that propose collaborative ltering for movies or music: re y [3], Movie Critic [9], etc. The movie-recommendation task has one main practical advantage: it is relatively easy to attract users to give holistic judgments about movies and thus their preferences.
5 Conclusion and perspectives In this paper a new approach for decision support has been introduced, characterized by a collaborative decision making process relying on the implicit sharing of preferences and experience between di erent individuals facing similar decision problems. The recommendation principle proposed is two-fold. On the one hand, it is based on fuzzy ltering methods that only use individual fuzzy preference relations; on the other hand it relies on fuzzy in uence relations between users allowing an implicit cooperation between users. This approach has been experimented on a movie recommendation task. The content-oriented recommendation principle has proved to be particularly useful when only a few user had given their preferences. As the number of users is increasing the collaborative principles are increasingly used. What makes the interest of such a collaborative approach is to support the decision process on objects for which nothing may be known but the preferences of several users.
In the system "Film Conseil", the objects considered are movies. Each movie m of the database that counts about 13000 di erent movies is described by a tuple of attribute values. The attributes used by the system are the ID of the movie (a key to index images), the title, the Genre (western, comedy, drama, ...), the origin (USA, India, England, France, ...), the duration, the year produced, the lm maker and the rst four main actors. Within the site "Film conseil", each individual is represented by his preference pro le, i.e. the set of grades he has assigned to movies. The fuzzy similarity between movies and the fuzzy in uence relations linking users are computed periodically and stored in an independent database. Then, "Film conseil" uses the di erent ltering approaches introduced in the previous section so as to provide the user with an ordered list of recommendations. A level of con dence in the recommendation is associated to each recommended movie and is precisely used to rank them.
The rst contribution of this paper is to propose a framework for using fuzzy preferences relations to design collaborative decision support systems. The second contribution is to integrate two types of recommendation that play a complementary role to propose the most relevant recommendation. One of the current limitation of this work concerns the ability to take into account feedback from individuals regarding recommendations that have been made to them. Suppose for example that a movie recommended to a user is later graded as irrelevant. Although the method, by construction, guarantees that the system will not make such a recommendation again,
The user may also request an explanation for these recommendations. This explanation is generated automatically by the system, as a userfriendly translation of the numerical computations performed to derive the recommendation. It should be noted that, when the user grades new movies (even recommended by the system 6
[10] J. Von Neumann, O. Morgenstern (1947), Theory of games and economic behavior, Princeton University Press. [11] P. Perny (1998), \Multicriteria Filtering Methods based on concordance and discordance principles", Annals of Operations Research, 80, 137{165. [12] P. Perny and B. Roy (1992), "The use of fuzzy Outranking Relations in Preference Modelling", Fuzzy sets and Systems, 49, 33{ 53. [13] P. Resnick, N. Iacovou, M. Suchak, P. Bergstrom, J. Riedl (1994), GroupLens: An Open Architecture for Collaborative Filtering of Netnews. In: CSCW '94. (Eds: Furuta, Richard; Neuwirth, Christine) ACM Press, New York, 175{186. [14] B. Roy (1996), Multicriteria Methodology for Decision Aiding, Kluwer Academic Publishers, Dordrecht. [15] L.J. Savage (1972), The Foundations of Statistics. Dover, New York. [16] R. Slowinski (1998), Handbook of Fuzzy Sets and Possibility Theory, Operations Research and Statistics, Kluwer Academic Publishers. [17] Ph. Vincke (1992), Multicriteria Decision Aid, Wiley.
it means that the procedure per se ought to be adapted. For this reason we investigate two directions. The rst one consists in using a machine learning algorithm to learn a description of each user's preferences. Such a description may then be used to lter the recommendations provided by the two other principles. A second direction is to learn, from the point of view of each individual, which among the other individuals are good advisors and bad advisors. In this case the feedback provided by the user is used to learn a degree of con dence in other individuals advises.
References [1] ACM (1997), Recommender Systems, special section of communications of the ACM 40, 3, 56{89. [2] M. Bereau and B. Dubuisson. A fuzzy extended k-nearest neighbor rule. Fuzzy Sets and Systems, (44):17{32, 1991. [3] www. re y.com [4] P. Fishburn (1970), Utility theory for decision making, Wiley, New York, 1976. [5] J. Fodor, M. Roubens (1994), Fuzzy Preference Modelling and Multicriteria Decision support, Kluwer Academic Publishers. [6] P.W. Foltz, S.Dumais (1992), Personalized Information Delivery: An analysis of Information-Filtering Methods, Commun. ACM 35(12, December), 51{60. [7] D. Goldberg, D. Nichols, B.M. Oki, D. Terry (1992): Using Collaborative Filtering to Weave an Information Tapestry. Commun. ACM 35(12, December), 61-70. [8] L. Henriet, P. Perny (1996), "Methode multicriteres non-compensatoires pour la classi cation oue d'objets", proceedings of the meeting \Logique Floue et Applications", 9{ 16. [9] http://www.moviecritic.com/ 7
