Available online at w w w .researchinform ation.co.uk
T h e J o u rn a l o f G r e y S ystem V o lu m e 27 N o.2, 2015
A fuzzy similarity measure for collaborative filtering using nonadditive grey relational analysis Yi-Chung Hu1*, Yu-Jing Chiu1, Yu-Lin Liao1’2, Qingchang Li3’4 1. Department o f Business Administration, Chung Yuan Christian University, Chung Li District, Taoyuan City, Taiwan 2. Department o f Business Administration, Chien Hsin University o f Science and Technology, Chung Li District, Taoyuan City, Taiwan 3. School o f Management, Xiamen University, Xiamen, P.R. China 4. School o f Business Administration, Jimei University, Xiamen, P.R. China
Abstract Previously, we proposed a novel similarity measure for neighborhood methods using the grey relational analysis (GRA). Nevertheless, the traditional GRA have drawn our attention to its assumption on noninteraction among criteria involved. Since criteria are not always independent, the adequacy o f additivity should be taken into account. This motivates us to use the nonadditive GRA to design a novel similarity measure for neighborhood methods in collaborative filtering on the basis o f the previous study. The neighborhood methods are then used to recommend items that users with similar preferences liked in the past. The applicability o f the proposed fuzzy similarity-based method to movie recommendation is examined. Experimental results on movie recommendation demonstrate that the generalization ability o f the multi-criteria neighborhood method using the proposed fuzzy similarity measure performs well in comparison to that using the additive similarity measure on the basis o f GRA.
Key words: Grey Relational Analysis; Neighborhood Method; Collaborative Filtering; Fuzzy Integral; Movie Recommendation
I. Introduction The grey relational analysis (GRA) proposed by Deng [l21 can be used to effectively measure the relationships among patterns by dealing with data series produced by the systems [l3' 22'24]. The GRA has been widely applied to various multi-criteria decision problems such as [14-19, 22-25, 30-31]. In view of the nature of practical decision problems often characterized by multiple criteria [9>10], and the possible improvement of recommendation accuracy for multi-criteria recommenders[1,81, Hu [25] used the GRA to develop a new similarity measure for neighborhood methods in collaborative filtering. Undoubtedly, the grey relational
C o lla b o ra tiv e F ilte rin g
* Corresponding Author: Yi-Chung Hu, Department o f Business Administration, Chung Yuan Christian University, Chung Li District, Taoyuan City, Taiwan; Email:
[email protected]
M e a s u re
93
Yi-Chung Hu et al/ The Journal o f Grey System 2015 (27) grade (GRG), which is used to represent the grade of relationship between two users with respect to a certain item, plays a critical role. Traditional GRG is computed by using the weighted average method, assuming that criteria do not interact [26l However, since the criteria are not always independent, the assumption of additivity is not always reasonable [28). This motivates us to use a nonadditive technique, namely the nonadditive GRA fl4,16' 3I], to design a novel similarity measure for collaborative filtering. The Choquet fuzzy integral, which is a generalization of the weighted average, is incorporated into the GRA since the Choquet fuzzy integral does not assume the independence of one element from another f28' 29]. By using the nonadditive GRA, this paper aims to design a fuzzy similarity measure revealing that the greater the strength of the relationship of one user with another one, the greater similarity of the former to the latter. To examine the recommendation performance, the proposed fuzzy similarities are applied to movie recommendation. Movie recommendation has been an important issue for online sites such as the popular Netflix and Yahoo! Movies. The uniqueness of Netflix and Yahoo! Movies indicates that personalization recommender systems have become an important component in personalization applications. Collaborative filtering, whose objective is to estimate how well a user will like an item that he/she has not rated [2], is appropriate for movie recommendation. The reason is that movie recommendation has relied on interest ratings on movies in the past from users. To evaluate the recommendation performance of the proposed fuzzy similarity measure for collaborative filtering, a set of user-submitted movie ratings from the popular Yahoo! Movies (http://movies.yahoo.com) is collected for several hundred randomly chosen movies. On the Yahoo! Movies platform, users are asked to provide information on four criteria (i.e., story, acting, direction, and visuals) for each movie. From this viewpoint, it is reasonable to treat movie recommendation as a multi-criteria decision problem. Neighborhood methods that measure the similarity between users have played an important role in collaborative filtering [5l Neighborhood methods are in fact the most prevalent approach for collaborative filtering [8]. Compared to the additive measures using the additive GRA introduced in [25], this paper contributes right to propose the fuzzy similarity measures using the nonadditive GRA for neighborhood methods in multi-criteria collaborative filtering. The remainder of the paper is organized as follows. Section 2 describes the proposed similarity measures using the nonadditive GRA. A genetic algorithm (GA) for constructing the recommendation model using the proposed fuzzy similarity measures is briefly introduced in Section 3. Section 4 applies the proposed recommendation approach to movie recommendation on the Yahoo! Movies platform. Section 5 contains a discussion and conclusions.
2. Neighborhood Methods Using Nonadditive GRA
C o lla b o ra tiv e F ilte rin g M e a s u re
94
Collaborative filtering aims to predict the overall rating of a particular item (e.g., s) for an active user (e.g., u) based on the overall ratings assigned to 5 by those users who are similar to u. Of course, u has not previously evaluated item 5. An estimate of the overall rating that user u would give to item s can be based on the similarity, denoted by sim(u, if), between users u and u' who rated item s. Further introduction of commonly-used similarity measures such as cosine-based, Pearson correlation coefficient (PCC), Spearman-rank-correlation-based
Yi-Chung Hu et al/ The Journal o f Grey System 2015 (27) (SRC-based) measures can be referred to [25]. Below, Subsection 2.1 introduces the additive similarity measure proposed in [25], Since the Choquet fuzzy integral in terms of a fuzzy measure can be then used to aggregate the GRCs for the nonadditive GRA, Subsection 2.2 introduces the fuzzy measure. The formulation of the proposed fuzzy similarity measure is presented in Subsection 2.3. Finally, the neighborhood method is briefly introduced in Subsection 2.4. 2.1 Additive Similarity Measure Using Traditional GRA
x“ = ( r “ ,..., r ") for user u with respect to
Without losing generality, let
item 5 be the reference sequence, and all the other
x“ = ( r “s ,..., rn“ ) for user
u' with respect to item s be the comparative sequences (u * u'), where r “ and r“ s (1 < i< n ) denote the partial rating for criterion i that users u and u' would give to item s, respectively. The grey relational coefficient (GRC) %k
of u' with
respect to s on criterion k indicates the degree of relationship between u' and u on criterion k (1 1, then-1 < A < 0; if J ' ” | jxi
jUj =1, then A = 0. For instance, if n = 2, then /j(X)
1 holds. It is obvious that A > 0 when
+ ju2+ 2fJ\H2=
+ /r2< 1.
2.3. The Proposed Fuzzy Similarity Measure Let f “ be a nonnegative real-valued measurable function defined on X such that f “ : A->[0, 1], The element in X with min{ f ‘‘ (xj)\j = 1, 2,..., n) is renumbered as one, where f “ (x}) =
denotes the performance or
observation value of Xj. That is, x/s are renumbered by rearranging fix]) into the descending order, that is, f “ (xt) < f “ (x2)
