Game-theoretic rough sets for recommender ... - Semantic Scholar

Knowledge-Based Systems 72 (2014) 96–107

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Game-theoretic rough sets for recommender systems Nouman Azam, JingTao Yao ⇑ Department of Computer Science, University of Regina, Regina, SK S4S 0A2, Canada

a r t i c l e

i n f o

Article history: Received 27 May 2014 Received in revised form 29 August 2014 Accepted 30 August 2014 Available online 20 September 2014 Keywords: Game-theoretic rough sets Probabilistic rough sets Game theory Recommender systems Rough sets

a b s t r a c t Recommender systems guide their users in decisions related to personal tastes and choices. The rough set theory can be considered as a useful tool for predicting recommendations in recommender systems. We examine two properties of recommendations with rough sets. The first property refers to accuracy or appropriateness of recommendations and the second property highlights the generality or coverage of recommendations. Making highly accurate recommendations for majority of the users is a major hindrance in achieving high quality performance for recommender systems. In the probabilistic rough set models, these two properties are controlled by thresholds ða; bÞ. One of the research issues is to determine effective values of these thresholds based on the two considered properties. We apply the gametheoretic rough set (GTRS) model to obtain suitable values of these thresholds by implementing a game for determining a trade-off and balanced solution between accuracy and generality. Experimental results on movielen dataset suggest that the GTRS improves the two properties of recommendations leading to better overall performance compared to the Pawlak rough set model. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction The internet or Web users are commonly confronted with situations where there are many potentially useful choices for selecting an item of interest. Making a suitable decision, such as a purchase decision, generally involves an urge to narrow down the search to a few but important choices. Web-based recommender systems (WRS) as a special kind of Web-based support systems, were introduced with the intent of handling relevant information in order to provide useful and customized recommendations to their users [2,30]. There are many techniques and approaches proposed in the literature that are used to develop different recommender systems. The most common approaches are content based, collaborative based, knowledge based and demographic based recommender systems [6]. In content based approach, the recommendations are based on contents or properties of items that are of interest to a user [32]. The collaborative based approach provides recommendations based on users having similar interests to the user in question [34]. The knowledge based approach makes use of the data pertaining to the user’s needs and preferences to recommend a suitable option [7]. The demographic based approach aims to group or cluster the users based on personal attributes and make recommendations based on the demographic group a user belongs ⇑ Corresponding author. Tel.: +1 306 585 4071. E-mail address: [email protected] (J.T. Yao). http://dx.doi.org/10.1016/j.knosys.2014.08.030 0950-7051/Ó 2014 Elsevier B.V. All rights reserved.

to [28]. Despite of some differences, all of these approaches require some sort of intelligent mechanisms to make effective recommendations. The collaborative based approach is comparatively more popular and successful way for building recommender systems [34]. In addition, the demographic based approach is sometimes treated and considered as an extension of the collaborative based approach [2]. For these reasons, we focus on collaborative and demographic based recommendations in this research. Rough set theory, emerged in the early 1980s, is an important and useful mathematical approach to handle vague and imperfect knowledge [25,27]. It can effectively process uncertain, incomplete and insufficient information to make useful inferences and reasonings [26]. The conventional Pawlak model in rough set theory is of qualitative nature in the sense that it does not allow any errors in the positive and negative regions [26,41,44]. Researchers argued that the qualitative absoluteness or intolerance to errors can lead to problems and limitations in practical applications [44,46,47]. Quantitative generalizations and models of rough sets were introduced that generally resort to some measures and thresholds to express error tolerance [46]. The probabilistic rough set models represent one class of these quantitative models and include the decision-theoretic rough set model [40,45], the variable precision rough set model [47,48], the Bayesian rough set model [10,31], the information-theoretic rough set model [9] and the game-theoretic rough set model [12,39]. An important realization in these models is that a pair of thresholds ða; bÞ is used to define the rough set approximations and the resulting three regions [11,41,43]. The

N. Azam, J.T. Yao / Knowledge-Based Systems 72 (2014) 96–107

determination and interpretation of thresholds are two important issues in the probabilistic rough sets [43]. Some notable attempts in this regard can be found in references [4,9,12,16,19–21]. Three-way or ternary decisions are obtained with probabilistic rough set models when the positive, negative and boundary regions are interpreted as regions of acceptance, rejection and deferment decisions [42]. The configuration of thresholds play a crucial role in obtaining suitable decision regions [4]. Setting the thresholds in order to decrease the number of deferment decisions improves the generality but results in many incorrect acceptance and incorrect rejection decisions. On the other hand, adjusting the thresholds to obtain more accurate decisions leads to many deferment decisions. How to obtain thresholds in order to balance the properties of accuracy and generality is a major obstacle in obtaining an effective probabilistic model. We employ the gametheoretic rough set (GTRS) model for such a purpose. The rough set theory has recently gained some attention in the recommender systems research [5,8,15,17,24,33]. It has been used with different intends in designing, developing and implementing recommender systems, such as, to obtain rules for determining the competence of players in a game [8], to visualize user preferences in menu selection [17], to reduce attributes for determining useful keywords in web page recommendations [5] and to deal with the problem of missing ratings (also termed as the problem of sparsity) [15]. The work presented in this research is different from existing work in three ways. Firstly, we look at multiple aspects of recommendation decisions simultaneously. Secondly, we go beyond the basic capabilities of rough sets by merging and combining it with the field of game theory in a GTRS model. Thirdly, we focus on ternary decision making aspect of rough sets for recommendations. The GTRS based thresholds can be used to obtain the three rough set regions which are helpful in applications for obtaining useful rules for decision support and reducing data processing time. In a recent article, the GTRS based three-way decisions were applied and analyzed in the medical field [37]. This article extends the GTRS to WRS for obtaining decision recommendations. Specifically, we focus on determining a tradeoff solution between the properties of accuracy and generality of rough set based recommendations. The GTRS provides benefits in at least two aspects. Firstly, the determination of thresholds based on a tradeoff solution between multiple criteria can comparatively lead to cost effective and moderate threshold levels [4,37]. Secondly, unlike other models where the users or experts are required to provide parameters or the notions of costs, risks or uncertainty are used to determine the thresholds, the GTRS obtains the thresholds based on the data itself [37]. The remaining of this article is organized as follows. Section 2 explains an architecture of WRS that incorporate a GTRS component for obtaining intelligent decision recommendations. Section 3 elaborates rough sets based recommendations and explains how the properties of accuracy and generality affect these recommendations. Section 4 describes the general GTRS model. In Section 5, a GTRS based approach for threshold determination is presented. Finally, Section 6 contains experimental results with the proposed approach on movielens dataset.

2. Architecture of Web-based recommender systems The Web has been increasingly used as a platform for supporting, developing and deploying recommender systems [6]. In some sense, the WRS may be viewed as a branch or sub-class of Web-based support systems that provide assistance to their users in the form of recommendations. A WRS may consist of different components with many functionalities ranging from supporting end user activities and interaction through interface

97

to maintaining and manipulating the knowledge within the system. Following our previous knowledge and understanding of the architecture of Web-based support systems [36,38], we consider the architecture of Web-based recommender systems as comprising of three fundamental layers, i.e. the interface, management and data layers. The Web and internet make up the interface layer. A client interaction with the system that is deployed on the server side either completely or partially is made possible through the support provided by the Web and internet. The Web browsers play a major role in presenting the interface to the clients. The interface enables the users to enter any relevant information that the system may need. In addition, it also allows the users to receive useful information about items or services that may be of potential interest to them in the form of effective recommendations. An effective and carefully designed Web interface plays a critical role for the success of any WRS. Special attention has to be given for its clarity, completeness and consistency. The management layer serves as a middleware in the three layer architecture. The information from the top and bottom layers are processed at this layer before being presented to an intended upper or lower layer. Some of the components that may be required to make up this layer include, a Database Management System, Knowledge Discovery/Data Mining and Control Facilities. The Database Management System is responsible for retrieving relevant information about the users and items from the data layer and make them available to other system components. The Knowledge Discovery component is responsible for discovering and mining important information based on users and items features. It is this component that ultimately determines the overall success of the system in the long term. Intelligent techniques such as logic, inference and reasoning about the data may be incorporated in this component to analyze data and make recommendations. We investigate the use of GTRS as a tool for making intelligent recommendations in this component. Finally, the Control Facility component may be needed to ensure that the system is being used in a right way. Access rights, permissions and confidential information should be properly handled by this component. The third layer in the three layer architecture is the data layer which contains data necessary for the operation of the system. This layer contains data about the users of the system, the items and their respective features and the user choices, preferences or ratings for different items. The data related to users and ratings may be captured explicitly in the form of queries or may be implicitly gathered based on the users interaction with the system. The Knowledge Discovery component (at the management layer) can retrieve and access this information for analysis purpose. This layer may also have a knowledge base component that may contain the results of data analysis, such as rules or patterns or certain parameter values that are used in making recommendations. 3. Rough sets based recommendations We consider recommendations with the probabilistic rough set model [41]. For the sake of completeness, we briefly review the main results of the Pawlak rough set model and the probabilistic rough set model. The Pawlak rough set model approximates a concept or a set by a pair of lower and upper approximations. For a set C, the lower and upper approximations are defined as [25,26],

aprðCÞ ¼ fx 2 U j ½x # Cg;

ð1Þ

aprðCÞ ¼ fx 2 U j ½x \ C – £g;

ð2Þ

where U is the set of objects called universe and ½x is an equivalence class (containing object x) and is based on an equivalence relation

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E # U  U. The concept C is any subset of the universe, i.e., C # U. According to the lower and upper approximations, the positive, negative and boundary regions of a concept C are defined as [26],

POSðCÞ ¼ aprðCÞ;

ð3Þ c

NEGðCÞ ¼ ðaprðCÞÞ ;

ð4Þ

BNDðCÞ ¼ aprðCÞ  aprðCÞ:

ð5Þ

The conditions in Eqs. (1) and (2), i.e., ½x # C and ½x \ C – £, are highlighting the qualitative relationships between an equivalence class ½x and a concept C. Specifically, the condition ½x # C reflects whether ½x is fully contained in C, i.e., ½x has a complete overlap with C and the condition ½x \ C – £ highlights whether ½x has some overlap with C. The degree of an overlap is not considered. The probabilistic rough set model considers the degree of an overlap between ½x and C in the form of conditional probability PðCj½xÞ, which is defined as,

PðCj½xÞ ¼

jC \ ½xj : j½xj

ð6Þ

The rough set approximations are defined by using thresholds ða; bÞ on conditional probability as [41,43],

aprða;bÞ ðCÞ ¼ fx 2 U j PðCj½xÞ P ag;

ð7Þ

aprða;bÞ ðCÞ ¼ fx 2 U j PðCj½xÞ > bg;

ð8Þ

where PðCj½xÞ is the conditional probability of an object x to be in C given that the same object is in ½x. It is assumed that 0 6 b < a 6 1 [9]. The three probabilistic regions are defined as,

POSða;bÞ ðCÞ ¼ aprða;bÞ ðCÞ ¼ fx 2 UjPðCj½xÞ P ag;

ð9Þ

NEGða;bÞ ðCÞ ¼ ðaprða;bÞ ðCÞÞc ¼ fx 2 UjPðCj½xÞ 6 bg;

ð10Þ

BNDða;bÞ ðCÞ ¼ aprða;bÞ ðCÞ  aprða;bÞ ðCÞ ¼ fx 2 Ujb < PðCj½xÞ < ag:

ð11Þ

One can define the Pawlak lower and upper approximations using the probabilistic rough set model by considering the equivalent form of conditions in Eqs. (1) and (2) in terms of conditional probability. The condition ½x # C is equivalent to PðCj½xÞ ¼ 1 and the condition ½x \ C – £ is equivalent to PðCj½xÞ > 0. This leads to the following definitions,

aprð1;0Þ ðCÞ ¼ fx 2 U j PðCj½xÞ P 1g;

ð12Þ

aprð1;0Þ ðCÞ ¼ fx 2 U j PðCj½xÞ > 0g;

ð13Þ

where PðCj½xÞ P 1 is used instead of PðCj½xÞ ¼ 0 for having consistency in format. Based on Eqs. (12) and (13), the Pawlak positive, negative and boundary regions based are defined as,

POSð1;0Þ ðCÞ ¼ fx 2 UjPðCj½xÞ P 1g;

ð14Þ

NEGð1;0Þ ðCÞ ¼ fx 2 UjPðCj½xÞ 6 0g;

ð15Þ

BNDð1;0Þ ðCÞ ¼ fx 2 Uj0 < PðCj½xÞ < 1g:

ð16Þ

This means that the Pawlak model is a special case of the probabilistic rough set model with ða; bÞ ¼ ð1; 0Þ. Eqs. (14)–(16) suggest that the Pawlak model requires PðCj½xÞ to be strictly one or zero for considering an object x to be included in either positive or negative regions. The probabilistic approach relaxes these conditions by introducing thresholds ða; bÞ. Particularly, an object x is considered to be in C, if its probabilistic relationship with C is at or above level a, i.e., PðCj½xÞ P a. A decision of acceptance for x is being made in this case. The same object is considered not to be in C if its probabilistic relationship with C is at or below level b, i.e., PðCj½xÞ 6 b. A decision of rejection is being made in this case. The inclusion or exclusion from C is indeterminate, if the probabilistic relationship

of the object with C is between the two thresholds, i.e., b < PðCj½xÞ < a. A decision of deferment or delay is being made in this case. A key observation in Eqs. (9)–(11) is that the decisions of objects belonging to any of the three regions are affected and determined by the choice of thresholds ða; bÞ. Considering making recommendation decisions with the probabilistic rough set model, different recommendations may be possible for the same object when different threshold values are being used. This means that a particular item recommended with a certain threshold settings may be deferred or even not recommended with another threshold settings. How different threshold levels will affect the overall quality of recommendation decisions is a critical issue in this context. We look at the properties of accuracy and generality to evaluate the quality of recommendations based on different threshold values. 3.1. Evaluating recommendations based on accuracy and generality The property of accuracy measures how close a recommender system predictions are to the actual user preferences. Generally speaking, from the early recommender systems to date, the majority of the published work focused on different ways of measuring this property to evaluate recommender systems [14]. Mcnee et al. argued that accuracy is not the only indicator of measuring or evaluating the performance of recommender systems [22]. Herlocker et al. discussed and pointed toward additional evaluation properties beyond the accuracy including the property of generality or coverage [13]. The property of generality may be interpreted and defined in different ways. We consider it as the relative number of users for whom recommendations are being made. The two properties provide different and complimentary aspects for evaluating recommendations using rough sets. The relationship between decision thresholds and the properties of accuracy and generality can be seen visually by considering Fig. 1(a)–(i). The Fig. 1 correspond to the Pawlak model. The Fig. 1 correspond to probabilistic two-way model and the Fig. 1(g)–(i), correspond to probabilistic three-way model. The ovals in these figures represent the concept of interest that we wish to approximate. The small rectangles are the equivalence classes. The Pawlak positive and negative regions are based on complete inclusion or exclusion of an equivalence class to be considered in the positive or negative regions. This is shown in Fig. 1(a). The probabilistic two-way model is based on inclusion of every equivalence class in either positive or negative region which is shown in Fig. 1(d). The probabilistic three-way model use the thresholds ða; bÞ to determine the inclusion of an equivalence class in any region. The Fig. 1 highlight the property of generality of the three models. The generality in this case corresponds to the area in which useful decisions in the form of acceptance or rejection (recommend or not recommend) are made. This region is obtained by taking the union of positive and negative regions. The Fig. 1 highlight the accuracy of the three models. It is noted that although the Pawlak positive and negative regions are completely accurate (no black region in Fig. 1(c)), however we may not be able to make useful decisions for majority of the objects (see Fig. 1(b)). On the other hand, the probabilistic two-way decision model allow to make useful decisions for all objects (Fig. 1)(d), i.e., all the objects are in decision region (Fig. 1(e)), however, it leads to many errors as shown by the black colour in Fig. 1(f). The probabilistic three-way model aims at making a suitable tradeoff between the two properties leading to effective levels of accuracy and generality as shown in Fig. 1(h) and (i). We further highlight the implications of accuracy and generality for recommender systems by considering an example of collaborative recommendations. Table 1 represents the ratings of users on

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99

Fig. 1. Accuracy versus generality.

different movies. A positive sign, i.e., + indicates that the user liked the movie and a negative sign, i.e.,  indicates that the user did not liked the movie. Each row in the table represents the ratings of a particular user corresponding to four movies. Suppose we are interesting in predicting the users ratings on Movie 4 using rough set analysis. One can not only use previous ratings on Movie 4 to predict the rating of a new user on Movie 4. This however is possible when we have additional features of the users or movies such as demographic information or user ratings on other movies. We make use of the rationale of collaborative recommendations according to which users having similar taste or preferences over the seen movies are likely to have similar preferences for the unseen movies. This means that the similarities between users on ratings of Movies 1–3 can be used to predict the users ratings on Movie 4. This calls for some sort of similarity measure between the users. It is important to point out that different methods are suggested in the literature for measuring the similarities between the users in collaborative recommendations. Su and Khoshgoftaar categorized these methods into three groups, namely, the correlation based similarity, vector based similarity and conditional probability based similarity [34]. In this article, we determine the similarity of users based on the notion of equivalence classes. A user is similar to another user if both of them are in the same equivalence class. We further explain this later in this section.

An information table is required to commence with the application of rough sets. Table 1 is essentially an information table since the set of objects (perceived as users) are described by a set of attributes (user ratings for movies). Hence, we are able to directly apply rough set analysis on Table 1. Moreover, Movies 1–3 are conditional attributes and Movie 4 is decision attribute in this example. This means that based on data of Movies 1–3, we want to the predict the decisions (which in this case is the ratings) for Movie 4. One may choose different subsets of conditional attributes for the same task. A rough set reduct in Table 1 may be investigated for this purpose. Let X i represents an equivalence class which is the set of users having the same rating pattern for Movies 1–3. Table 2 shows the equivalence classes that are formed based on the data in Table 1. The equivalence classes in this case are interpreted as the groups of users having identical taste or preferences on the considered movies. The concept of interest in this case is to determine the positive ratings on Movie 4, i.e., Movie 4 = +. We are unable to exactly specify this concept based on the equivalence classes X 1 ; . . . ; X 8 . For instance, we are unable to tell whether or not X 3 belongs to this concept since 4 users in X 3 like the movie and 1 user dislike it. Therefore, we approximate this concept in the probabilistic rough set model. The association of each equivalence class X i with the

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Table 1 User ratings for four movies.

U1 U2 U3 U4 U5 U6 U7 U8 U9 U 10 U 11 U 12 U 13 U 14 U 15 U 16 U 17 U 18 U 19 U 20 U 21 U 22 U 23 U 24 U 25 U 26

Movie 1

Movie 2

Movie 3

Movie 4

+ + +  +     +  + +  + +  +  +    +  +

+ +  + + + +  +  +      +    +     

+  + +  +  + + + +    + +    +  + +   

+ + + + +  + + + + + +   + + +         

S X 2 Þ \ CÞ ðX 8 \ C c Þj S S jX 1 X 2 X 8 j jfU 1 ; U 2 ; U 5 ; U 14 ; U 19 ; U 25 gj 6 ¼ ¼ ¼ 1:0; jfU 1 ; U 2 ; U 5 ; U 14 ; U 19 ; U 25 gj 6

Accuracyða; bÞ ¼

jMovie 4 ¼ þ jX i j

T

Xij

:

S

S X 2 X 8 Þj jUj fU 1 ; U 2 ; U 5 ; U 14 ; U 19 ; U 25 g 6 ¼ 0:2307: ¼ ¼ jU 1 ; U 2 ; . . . ; U 26 j 26 jðX 1

ð17Þ

3.2. Accuracy and generality of the Pawlak and probabilistic two-way model Let us look at the properties of accuracy and generality in the Pawlak rough set model. Based on certain thresholds ða; bÞ, these measures are defined as [4],

S

ð21Þ

S S ... X 5 Þ \ CÞ ððX 6 \ X 7 \ X 8 Þ \ C c Þj S S jX 1 X 2 ...: X 8 j jfU 1 ;...;U 5 ;U 7 ;U 9 ;U 10 ;U 11 ;U 13 ;...;U 19 ;U 22 ;...;U 26 gj ¼ jfU 1 ;U 2 ;...;U 26 gj 21 ¼ ¼ 0:8077; 26 ð22Þ

Accuracyða;bÞ ¼

jððX 1

Generalityða; bÞ ¼ ¼

S

jðX 1

X2

S

S

X2

S

... jUj

S

X 8 Þj

¼

fU 1 ; U 2 ; . . . ; U 26 g jU 1 ; U 2 ; . . . ; U 26 j

26 ¼ 1:0: 26

S Total number of classified objects by POSða;bÞ ðCÞ and NEGða;bÞ ðCÞ jPOSða;bÞ ðCÞ NEGða;bÞ ðCÞj ;¼ ; Number of objects in U jUj

where C c is the set complement of C, containing all objects in U that are not in C. The accuracy measures the relative number of correct classification decisions (or recommendation decisions) for objects compared to the total classification decisions. The generality

ð23Þ

This means that we are able to make recommendations for all the users, i.e., generality equals to 100%. This however is possible with an accuracy level of 80.77%. In general, by decreasing and lowering the expectation of being highly accurate in all the cases we are able to make recommendations for more users. For instance, if we lower our expectation of acceptance for recommendation by lowering the value of a from 1.0 to 0.8, then X 3 will also be included in the positive region. This means that for the objects in X 3 , i.e., fU 3 ; U 10 ; U 15 ; U 16 ; U 20 g, we predict posi-

S Number of correctly classified objects by POSða;bÞ ðCÞ and NEGða;bÞ ðCÞ jðPOSða;bÞ ðCÞ \ CÞ ðNEGða;bÞ ðCÞ \ C c Þj S ;¼ j; Total number of classified objects by POSða;bÞ and NEGða;bÞ jPOSða;bÞ ðCÞ NEGða;bÞ ðCÞ

Generalityða; bÞ ¼

ð20Þ

These results mean that with the Pawlak model, we are able to make recommendations that are 100% accurate, however these recommendation are possible for only 23.07% of the users. Arguably, one may want to make recommendations for more users. This is possible in the probabilistic two-way model. Let us considering a special case of probabilistic two-way model defined by a ¼ b ¼ 0:5. The three regions are determined as S POSð0:5;0:5Þ ðCÞ ¼ fX 1 ; X 2 ; X 3 ; X 4 ; X 5 g, BNDð0:5;0:5Þ ðCÞ ¼ ;, and NEGð0:5;0:5Þ ðCÞ ¼ fX 6 ; X 7 ; X 8 g. The accuracy and generality in this case are calculated as,

The conditional probabilities of equivalence classes X 1 ; . . . ; X 8 based on Eq. (17) are calculated as 1.0, 1.0, 0.8, 0.75, 0.67, 0.33, 0.2 and 0.0, respectively. These conditional probabilities represent the level of agreement between similar users to positively rate the movie. Generally speaking, the users having similar opinions or preferences on certain items, does not necessarily imply that they will always be in perfect agreement on other items of interest. The probability of an equivalence class X i is determined as PðX i Þ ¼ jX i j=jUj which means that the probability of X 1 is jX 1 j=jUj ¼ 1=26 ¼ 0:037. The probabilities of other equivalence classes X 2 ; . . . ; X 8 are similarly calculated as 0.077, 0.192, 0.154, 0.115, 0.115, 0.192 and 0.115, respectively.

Accuracyða; bÞ ¼

jððX 1

Generalityða; bÞ ¼

concept, i.e., PðCjX i Þ or conditional probability needs to be determined for this purpose which is given by,

PðCjX i Þ ¼ PðMovie 4 ¼ þjX i Þ ¼

measures the number of objects for whom classification decisions can be made compared to all objects. The three regions of the Pawlak model are determined as S S POSð1;0Þ ðCÞ ¼ fX 1 ; X 2 g, BNDð1;0Þ ðCÞ ¼ fX 3 ; X 4 ; X 5 ; X 6 ; X 7 g, and NEGð1;0Þ ðCÞ ¼ fX 8 g. The properties of accuracy and generality are calculated as,

ð18Þ

ð19Þ

tive ratings. However, from Table 1 it is noted that 4 out of 5 of these recommendations are correct, i.e., U 20 has a – rating for Movie 4. How much to increase the level of generality at the cost of decrease in the level of accuracy will require some kind of tradeoff

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analysis between the two properties. The thresholds ða; bÞ control this tradeoff. Determining effective values for the thresholds would lead to a moderate, cost effective and efficient levels for accuracy and generality. We consider the GTRS model for this purpose. 4. The game-theoretic rough set model A major contribution of GTRS is that it provides a mechanism for determining thresholds by realizing a tradeoff between multiple cooperative of competitive criteria in the probabilistic rough set model [12,39]. Although the GTRS works for multiple criteria but we review a typical two-player GTRS based game which is of interest and relevance here. Considering the thresholds ða; bÞ and a set of criteria C ¼ fc1 ; c2 g which are being employed to analyze rough sets. Let the evaluations of these criteria based on ða; bÞ be denoted as c1 ða; bÞ and c2 ða; bÞ, respectively. It is sensible to select thresholds which optimize the following,

arg max c1 ða; bÞ; ða;bÞ

arg max c2 ða; bÞ:

ð24Þ

ða;bÞ

This however may not be always possible in reality. The configuration of thresholds to improve or optimize evaluations based on one criteria may come at a cost of decrease in some other criteria. The GTRS philosophy is to consider such criteria in a game environment in order to reach an effective and near optimal solution. A game is formally defined as a tuple fP; S; ug [18], where:  P is a finite set of n players,  S ¼ S1  . . .  Sn , where Si is a finite set of strategies available to player i. Each vector s ¼ ðs1 ; s2 ; . . . ; sn Þ 2 S is called a strategy profile where player i plays strategy si , and  u ¼ ðu1 ; . . . ; un Þ where ui : S#R is a real-valued utility or payoff function for player i. The solution concept of Nash equilibrium is generally used to determine possible game solutions in GTRS. Let si ¼ ðs1 ; s2 ; . . . ; si1 ; siþ1 ; . . . ; sn Þ be a strategy profile without player i strategy. This means that ðs1 ; s2 ; . . . ; sn Þ ¼ ðsi ; si Þ. The strategy profile ðs1 ; s2 ; . . . ; sn Þ is a Nash equilibrium, if for all players i; si is the best response to si . This is equivalently represented in mathematical form as [18],

8i; 8s0i 2 Si ;

ui ðsi ; si Þ P ui ðs0i ; si Þ;

where ðs0i – si Þ:

ð25Þ

Eq. (25) intuitively means that a strategy profile such that no player would want to change his strategy, provided he has the knowledge of other players strategies. In other words, none of the players has any incentive or benefit to change their respective strategies, given Table 2 Equivalence classes based on the data in Table 1. X1 X3 X5 X7

X2 X4 X6 X8

¼ fU 1 g ¼ fU 3 ; U 10 ; U 15 ; U 16 ; U 20 g ¼ fU 7 ; U 17 ; U 21 g ¼ fU 12 ; U 13 ; U 18 ; U 24 ; U 26 g

¼ fU 2 ; U 5 g ¼ fU 4 ; U 6 ; U 9 ; U 11 g ¼ fU 8 ; U 22 ; U 23 g ¼ fU 14 ; U 19 ; U 25 g

the other players chosen strategies. It may be viewed as a kind of a balance point between the players. The players in GTRS are considered as different criteria highlighting various aspects of rough set based decision making, such as, accuracy or applicability of decision rules [4]. Suitable measures are selected and used to evaluate these criteria. The strategies are realized in terms of different or varying levels of the properties, such as the risks, costs or uncertainty associated with different regions which can lead to different threshold levels. The strategies can alternatively be formulated as direct modification of thresholds [4]. Each criterion may be affected in a different way based on different strategies within the game. The ultimate goal of the game is to find an acceptable solution based on the considered criteria which can be used to determine effective thresholds. Table 3 represents a typical two-player GTRS based game. The players in this game are represented as criteria c1 and c2 , respectively. Each cell of the table contains a pair of utility functions which is calculated based on the respective strategy profile. Each cell of the table corresponds to a strategy profile and contains a pair of utility functions based on that strategy profile. For instance, the top right cell corresponds to a strategy profile ðs1 ; s1 Þ which contains utility functions uc1 ðs1 ; s1 Þ and uc2 ðs1 ; s1 Þ. The game solution, such as the Nash equilibrium [35], is utilized to determine a possible strategy profile and the associated thresholds. The determined thresholds are then used in the probabilistic rough set model to obtain the three regions and the implied three-way decisions. Table 4 shows the meaning of fundamental game elements and components in the GTRS model (see Table 5). 5. Analyzing accuracy versus generality tradeoff with GTRS We now investigate the use of game-theoretic rough set model for analyzing the properties of accuracy and generality of rough sets based classification. Earlier in Section 3, the relationship between probabilistic thresholds and the two considered properties was demonstrated with an example. An important observation was that the configuration of probabilistic thresholds control the tradeoff between the accuracy and generality. We aim to find a mechanism that effectively adjusts the threshold values based on the two properties using the GTRS model. 5.1. Formulating tradeoff between accuracy and generality as a game From description of GTRS in Section 4, it is evident that we need to identify at least three components of a game in order to formulate and analyze problems with GTRS. This includes information Table 4 The meaning of different symbols in a two-player GTRS game. Game symbols

Meaning in the GTRS

P Si sm 2 S 1 ; s n 2 S 2 S ðsm ; sn Þ 2 S uc1 ðsm ; sn Þ; uc2 ðsm ; sn Þ

The set of criteria considered as game players; fc1 ; c2 g The set of strategies for player ci ; fs1 ; s2 ; . . .g A particular strategy sm of player c1 and sn of player c2 Joint strategy set. S ¼ S1  S2 A strategy profile where c1 plays sm and c2 plays sn Utility based on criteria c1 and c2 respectively

Table 5 Payoff table for the game.

Table 3 Payoff table for a two-player GTRS based game.

G

c2

c1

s1 s2 ...

s1

s2

...

uc1 ðs1 ; s1 Þ; uc2 ðs1 ; s1 Þ uc1 ðs2 ; s1 Þ; uc2 ðs2 ; s1 Þ ...

uc1 ðs1 ; s2 Þ; uc2 ðs1 ; s2 Þ uc1 ðs2 ; s2 Þ; uc2 ðs2 ; s2 Þ ...

... ... ...

A

s1 ¼ a # s2 ¼ b " s3 ¼ a # b "

s1 ¼ a#

s2 ¼ b "

s3 ¼ a# b"

uA ðs1 ; s1 Þ; uG ðs1 ; s1 Þ uA ðs2 ; s1 Þ; uG ðs2 ; s1 Þ uA ðs3 ; s1 Þ; uG ðs3 ; s1 Þ

uA ðs1 ; s2 Þ; uG ðs1 ; s2 Þ uA ðs2 ; s2 Þ; uG ðs2 ; s2 Þ uA ðs3 ; s2 Þ; uG ðs3 ; s2 Þ

uA ðs1 ; s3 Þ; uG ðs1 ; s3 Þ uA ðs2 ; s3 Þ; uG ðs2 ; s3 Þ uA ðs3 ; s3 Þ; uG ðs3 ; s3 Þ

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about multiple criteria which are players in a game, the strategies or available actions for each player and the utilities or payoff functions of the players. We begin by identifying these components. The players should reflect the overall purpose and intention of the game. The objective in this game is to achieve and improve the overall quality of recommendations. Since we are interested in the properties of accuracy and generality, we consider them as game players. The player set in this case is P ¼ fc1 ; c2 g with player c1 representing accuracy and is denoted as A and player c2 representing generality and is denoted as G. The game is to obtain effective levels of thresholds by determining a tradeoff solution between the players. The players are affected by considering different ða; bÞ thresholds. The strategies which represent possible moves or options available to the players are therefore formulated in terms of different modifications or changes in thresholds. Each player will aim at selecting a strategy that configures the thresholds in order to maximize its benefits or utilities. How to formulate strategies in terms of thresholds was recently being investigated in [3]. The study identified four different approaches to formulate strategies, including the two ends approach, the middle start approach, the random approach and the range approach [3]. Not much has been reported with regards to the relative performance of these approaches. We consider the two ends approach in this study as this approach provides useful results in the previous studies [4,12]. The approach configures the thresholds from initial values of thresholds ða; bÞ ¼ ð1; 0Þ. This facilitates the comparison with the Pawlak model which is off particular interest in this study. Three types of strategies are considered, namely, s1 ¼ a # (decrease of a), s2 ¼ b " (increase of b) and s3 ¼ a # b " (decrease of a and increase of b). The payoff functions are used to measure the consequences of choosing a certain strategy. They should reflect possible benefits or performance gains of a particular player in selecting a strategy. As discussed above, the players A and G representing accuracy and generality are affected by considering different threshold values which are considered as possible strategies. Therefore, the utilities of these two players are determined as values of accuracy and generality defined in Eqs. (18) and (19). From player A’s perspective, an accuracy value of 1.0 means a maximum possible gain or payoff and an accuracy of 0.0 represents a minimum possible gain. In the same way, from player G’s perspective, a generality value of 1.0 means a maximum possible gain and a generality value of 0.0 represents a minimum gain. For a particular strategy profile, say ðsm ; sn Þ that configures and leads to thresholds ða; bÞ, the associated certainty or utility of the players are represented by,

uA ðsm ; sn Þ ¼ Accuracyða; bÞ;

ð26Þ

uG ðsm ; sn Þ ¼ Generalityða; bÞ;

ð27Þ

where uA and uG are the payoff functions of players A and G, respectively. 5.2. Competition between accuracy and generality We form a game as a competition among the properties of accuracy and generality. The payoff table shown as Table 6 is constructed for this purpose. Each row of the payoff table represents the strategies of player A and each column represents the strategies of player G. Every cell in the table corresponds to a certain strategy profile of the form ðsm ; sn Þ. The payoffs corresponding to the strategy profile ðsm ; sn Þ are given by uA ðsm ; sn Þ and uG ðsm ; sn Þ for players A and G, respectively. A particular player would prefer a strategy over another strategy if it provides more payoff during the game. A strategy profile ðsm ; sn Þ would be the Nash equilibrium or the game solution if the following conditions are being satisfied based on Eq. (25),

Table 6 The payoff table for the example game. G

A

s1 ¼ a # s2 ¼ b " s3 ¼ a # b "

For player A : For player G :

s1 ¼ a #

s2 ¼ b "

s3 ¼ a # b "

(0.83, 0.69) (0.85, 0.77) (0.83, 0.89)

(0.85, 0.77) (0.86, 0.54) (0.83, 0.89)

(0.83, 0.89) (0.83, 0.89) (0.8077, 0.1.0)

8s0m 2 S1 ; uA ðsm ; sn Þ P uA ðs0m ; sn Þ; with ðs0m – sm Þ; ð28Þ 8s0n 2 S2 ; uG ðsm ; sn Þ P uG ðsm ; s0n Þ; with ðs0n – sn Þ: ð29Þ

This means that none of the players are benefitted by switching to a different strategy other than the one specified by the profile ðsm ; sn Þ. We now examine the issue of determining the threshold changes based on a certain strategy during the game. Four types of fundamental changes in the two thresholds are noted in Table 6. These changes are defined as,

a ¼ single player suggests to decrease a; a ¼ both the players suggest to decrease a;

ð31Þ

bþ ¼ single player suggests to increase b; bþþ ¼ both the players suggest to increase b:

ð32Þ ð33Þ

ð30Þ

The above definitions enable us to associate and calculate a threshold pair with each strategy profile. For instance, a threshold pair based on a strategy profile ðs1 ; s1 Þ ¼ ða# ; a# Þ is determined as ða ; bÞ, as both the players suggest to decrease threshold a (see Eq. (31)). In the same way, the strategy profile ðs3 ; s3 Þ ¼ ða# b" ; a# b" Þ is determined as ða ; bþþ Þ based on Eqs. (31) and (33). In the next section, we examine how to determine the values for the variables in Eqs. (30) and (33) based on a repetitive game mechanism. 5.3. Repetitive threshold learning with GTRS Modifying the thresholds continuously to improve the utility levels of the players will lead to a learning mechanism. The learning principle employed in such a mechanism is based on the relationship between modifications in thresholds and their impact on the utilities of players. We exploit this relationship in order to define the variables ða ; a ; bþ ; bþþ Þ. This will help in continuously improving the thresholds to reach their effective values. A repeated or iterative game is considered to achieve this. Algorithm 1. GTRS based threshold learning algorithm Input: A data set in the form of an information table. Initial values of a ; a ; bþ and bþþ . Output: Thresholds ða; bÞ. 1: Initialize a ¼ 1:0; b ¼ 0:0. 2: Repeat 3: Calculate the utilities of players based on Eq. (18) and (19). 4: Populate the payoff table with calculated values. 5: Calculate equilibrium in a payoff table using Eqs. (28) and (29). 6: Determine the selected strategies and the corresponding thresholds ða0 ; b0 Þ. 7: Calculate a ; a ; bþ and bþþ based on Eqs. (34)–(37). 8: ða; bÞ ¼ ða0 ; b0 Þ 9: Until PðBNDða;bÞ ðCÞÞ ¼ 0 or PðPOSða;bÞ ðCÞÞ > PðCÞor Accuracyða; bÞ < Generalityða; bÞ or a < 0:5 or b P 0:5

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Let ða; bÞ represent the initial threshold values for a particular iteration of the game. The game will determine equilibrium using Eqs. (28) and (29) which will be used to obtain the output thresholds, say ða0 ; b0 Þ. Based on the thresholds ða; bÞ and ða0 ; b0 Þ, we define the four variables that appeared in Eqs. (30)–(33) as,

a ¼ a  ða  ðGeneralityða0 ; b0 Þ  Generalityða; bÞÞÞ; a ¼ a  cða  ðGeneralityða0 ; b0 Þ  Generalityða; bÞÞÞ; bþ ¼ b  ðb  ðGeneralityða0 ; b0 Þ  Generalityða; bÞÞÞ; bþþ ¼ b  cðb  ðGeneralityða0 ; b0 Þ  Generalityða; bÞÞÞ:

ð34Þ ð35Þ ð36Þ ð37Þ

The threshold values for the next iteration are updated to ða ; b0 Þ. The constant c in Eqs. (34) and (37) is introduced to reflect the desired level of change in thresholds and should be greater than 1. A lower value of c allows to fine tune the thresholds but involves more computations [4]. On the other hand, a higher value of c results in lesser computations however, fine tuning is not possible. The iterative process stops when either the boundary region becomes empty, or the positive region size exceeds the prior probability of the concept C, or Generalityða; bÞ exceeds Accuracyða; bÞ or when a < 0:50 or b P 0:5. These conditions are mathematically expressed as, 0

PðBNDða;bÞ ðCÞÞ ¼ 0;

ð38Þ

PðPOSða;bÞ ðCÞÞ > PðCÞ;

ð39Þ

Accuracyða; bÞ < Generalityða; bÞ:

ð40Þ

a < 0:5 or b P 0:5:

ð41Þ

The above learning procedure is explained as Algorithm 1. Given a particular data set, the algorithm will return ða; bÞ values for classifying objects.

6. Experimental results and discussion 6.1. Threshold calculation and recommendations with GTRS We elaborate the role of GTRS for threshold determination by considering the example discussed in Section 3. Considering a game as discussed above in Section 5 between accuracy and generality having three strategies each. For the sake of simplicity, we consider a non-repeated (one time) game where an increase or decrease of 25% in thresholds are considered. The game is being played with an initial thresholds of ða; bÞ ¼ ð1; 0Þ. A particular strategy, say s1 is now interpreted as 25% decrease in a which leads to a ¼ 0:75. Thresholds corresponding to a strategy profile are calculated according to two rules. If both the players suggest to increase or decrease a threshold, a new value for the threshold will be determined as the sum of two changes. In case a single player suggests a change in the thresholds, the threshold value will be determined as an increase or decrease suggested by that player. For instance, thresholds values corresponding to a strategy profile say ðs1 ; s2 Þ = (25%decrease in a, 25% increase in b) will be determined as ða; bÞ ¼ ð0:75; 0:25Þ. Finally, the utility of the two players are determined by using Eqs. (26) and (27). Table 6 represents the payoff table corresponding to this game based on the data in Table 1. The cell containing bold values, i.e., (0.83,0.89) with its corresponding strategy profile ðs2 ; s3 Þ is the Nash equilibrium or game solution. This means that none of the two players achieve a higher payoff, given the other players chosen action. The thresholds based on this strategy profile is given by ða; bÞ ¼ ð0:75; 0:5Þ. The GTRS based results for predicting user ratings on Movie 4 are interpreted as follows. We are able to make 83% correct predictions for 89% of the users when we reduce and set the levels for acceptance and rejection of recommendations

for the movie at 0.75 and 0.5, receptively. Comparing these results with the Pawlak model, the GTRS is able to make recommendations for an additional 66% of the users at a cost of 17% decrease in accuracy. Please be noted that the accuracy and generality of the Pawlak model are 100% and 23.07%, respectively, as discussed in Section 3. Now let us examine how GTRS can be used to perform recommendations on Movie 4 for an unknown user based on the data in Table 1. The system will look for a user group or an equivalence class X i that has the same rating pattern as that of the user in question. The conditional probability of the respective equivalence class, i.e., PðMovie 4 ¼ þjX i Þ is used to make a recommendation decision as follows,

Recommend : Not recommend :

PðMovie 4 ¼ þjX i Þ P a; PðMovie 4 ¼ þjX i Þ 6 b;

Delay recommendation : b < PðMovie 4 ¼ þjX i Þ < a;

ð42Þ

where ða; bÞ ¼ ð0:75; 0:5Þ according to the GTRS based results. The above rules can be used to determine whether or not to recommend a movie while ensuring that these recommendations are based on 89% of the users having an accuracy level of 83%. It may be noted that in contrast to the conventional approaches, the GTRS also provide a third decision type, i.e., delay recommendation. This provides some flexibility in cases where the available information is insufficient for a recommendation. Whereas conventional approaches are generally based on two-way classification, the chances of misclassifications are more in situations that lack sufficient information to reach a certain conclusion. 6.2. Experimental setup and data We investigate the usage of GTRS in movie recommendations. The movielen dataset is typically considered for movie recommendation application [1,23]. The version 1M of the movielen, which contains about 1 million user ratings, is considered in this paper. The dataset consists of three different tables, namely, the user table, the ratings table and the movie table. The user table contains demographic information about the 6040 users including their ages, genders and occupations. The movie table contains information about 3952 movies including their titles and genres. The ratings table contains 1 million user ratings on a 5-star scale. Each user has at least 20 ratings in this table. For the sake of ease in computation, we reduced and considered the first 58,000 ratings from the ratings table (the number was selected randomly). These ratings correspond to 400 users which is about 15% of the total users. In addition, we converted the 5-star scale to a binary scale (‘‘like’’, ‘‘dislike’’) as discussed in reference [29]. Two sets of experiments were conducted based on this conversion. In the first setup, the ratings of 4 or 5 indicate ‘‘like’’ while ratings of 1–3 indicate ‘‘dislike’’ and in the second setup only rating 5 indicate ‘‘like’’ while the remaining indicate ‘‘dislike’’. The second setup is similar to the problem of predicting the rating 5 for the movies. We call the first setup as Task 1 and the second setup as Task 2. We also reduced the number of movies to the top 10 most frequently rated movies by the users (irrespective of their ratings). In case of missing ratings, we used a value of 0. The prediction on each movie was considered based on the user ratings for the other nine movies. For testing the results, 10-fold cross validation was used in all experiments. 6.3. Collaborative recommendations Table 7 presents the results on the training data of Task 1. The Pawlak model achieves 100% accurate predictions for predicting

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Table 7 Train results for data with Task 1. Prediction for movie

Accuracy

Weigthed acc.

Generality

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0.9140 0.9525 0.9829 0.9712 0.9605 0.9739 0.9792 0.9615 0.9766 0.9687

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.8950 0.9162 0.9724 0.9498 0.9418 0.9394 0.9732 0.9298 0.9587 0.9605

0.8240 0.8439 0.9292 0.9019 0.8899 0.8958 0.9636 0.8756 0.9316 0.9321

0.9540 0.9198 0.9782 0.9546 0.9594 0.9271 0.9875 0.9314 0.9625 0.9825

0.6481 0.6878 0.8584 0.8037 0.7798 0.7916 0.9271 0.7512 0.8633 0.8641

Average

0.9641

1.0

0.9437

0.8988

0.9557

0.7975

user ratings on the movies. However, these predictions are possible for a certain portion of the users, for instance, in case of Movie 1, these predictions are possible for 64.81% of the users and for movie 2 they are applicable to 68.78% of the users. Arguably, one would like to extend these predictions to cover more users. The GTRS makes it possible by allowing a slight decrease in accuracy. For instance, in case of Movie 1, the GTRS provides an increase in the generality by 30.59% at a cost of 8.6% decrease in accuracy. An important issue while considering and comparing the performances of three-way decision models is that they require additional information for some objects that belong to the boundary region. In the absence of such information, one may not be very confident and will have remaining doubts about their performances. We look at the performances of the two models in a worst case scenario where we have no access to additional information and we are asked to make certain decisions about the deferred cases. Suppose random decisions are being made with a 50% chance of being correct and another 50% of being incorrect. Under these conditions, the GTRS for Movie 1 provides 91.40% correct decisions for 95.4% of the objects and 50% correct decisions for the remaining 4.6% of the objects. The average or weighted accuracy in this case is ð91:40  95:4Þ þ ð50  4:6Þ ¼ 89:50%. In the same way, the Pawlak model for movie 1 provides 100% correct decisions for 64.81% of the objects and 50.0% correct decision for the remaining 35.19% of the objects leading to an average accuracy of ð100  64:81Þ þ ð50  35:19Þ ¼ 82:40%. These accuracies are shown in the fourth and fifth columns of Table 7 under the title of Weighted acc. Another interesting aspect of the two models is their requirements for additional information. For the objects in boundary regions, it is generally assumed that additional information is required to make a certain decision about them [9]. Since, in most if not all cases, the GTRS leads to reduced size of the boundary region compared to the Pawlak model, it means that the GTRS will require comparatively lesser additional information for the undecided objects. For instance, for movie 1, the GTRS requires additional information for 4.6% of the objects while the Pawlak model needs additional information for 35.19% of the objects. Let us look at the average performance of the two models over the prediction of 10 popular movies. The Pawlak model provides 100% accurate predictions for 79.75% of the users. The bold point form in the Table 7 and hereafter represents the best results. The GTRS provides 96.41% accurate predictions for 95.57% of the users. This means that when we decrease the accuracy level by 4.14%, we are able to extend the recommendation decisions to 15.82% of the users. Moreover, the GTRS provides better weighted accuracy of 94.37% compared to 89.88% with the Pawlak model. The results on testing data of Task 1 are presented in Table 8. The GTRS provides comparatively better accuracy for 7 out of 10

movies, i.e., for Movies 1, 3, 4, 5, 6, 7 and 8. Additionally, it always result in superior generality and weighted accuracy. The average results over the 10 movies further confirms the superiority of GTRS. An average accuracy of 61.83% is obtained with GTRS compared to 60.41% with its counterpart. The average generality of GTRS is 94.5% compared to 82.8%, an average increase of 11.7%. The weighted accuracy is 61.15% which is 2.18% more than the Pawlak. Let us now look at the results for Task 2. Table 9 presents the training results which are similar to the training results in Table 7. There are some differences in the testing results which are presented in Table 10 compared to the testing results in Table 8.

Table 8 Test results for data on Task 1. Prediction for movie

Accuracy

Weighted acc.

Generality

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0.4898 0.6425 0.5873 0.5950 0.5802 0.6348 0.6680 0.6407 0.6194 0.7252

0.4448 0.6426 0.5484 0.5749 0.5659 0.6126 0.6598 0.6303 0.6269 0.7344

0.4909 0.6279 0.5848 0.5906 0.5733 0.6215 0.6653 0.6281 0.6126 0.7196

0.4622 0.6051 0.5423 0.5618 0.5533 0.5917 0.6538 0.6017 0.6120 0.7134

0.8965 0.8977 0.9713 0.9537 0.9143 0.9016 0.9838 0.9102 0.9428 0.9750

0.6847 0.7371 0.8730 0.8256 0.8083 0.8145 0.9627 0.7807 0.8827 0.9102

Average

0.6183

0.6041

0.6115

0.5897

0.9347

0.8279

Table 9 Train results for data on Task 2. Prediction for movie

Accuracy

Weighted acc.

Generality

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0.9375 0.9818 0.9807 0.9795 0.9763 0.9799 0.9865 0.9756 0.9828 0.9782

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.9234 0.9503 0.9590 0.9601 0.9685 0.9570 0.9841 0.9527 0.9707 0.9782

0.8764 0.9124 0.9199 0.9125 0.9415 0.9215 0.9786 0.9246 0.9504 0.9565

0.9678 0.9346 0.9548 0.9596 0.9836 0.9522 0.9951 0.9519 0.9750 1.0

0.7529 0.8248 0.8397 0.8251 0.8829 0.8430 0.9572 0.8493 0.9008 0.9129

Average

0.9759

1.0

0.9604

0.9294

0.9675

0.8589

Table 10 Test results for data on Task 2. Prediction for movie

Accuracy

Weighted acc.

Generality

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0.6488 0.7613 0.7372 0.6958 0.7092 0.7684 0.7596 0.7676 0.7535 0.8201

0.5962 0.7382 0.7147 0.6631 0.6877 0.7521 0.7554 0.7444 0.7436 0.8131

0.6403 0.7431 0.7260 0.6851 0.7029 0.7544 0.7564 0.7529 0.7478 0.8169

0.5754 0.7005 0.6850 0.6377 0.6692 0.7195 0.7484 0.7115 0.7272 0.7948

0.9428 0.9303 0.9526 0.9451 0.9701 0.9477 0.9875 0.9452 0.9775 0.9900

0.7835 0.8418 0.8617 0.8444 0.9017 0.8706 0.9726 0.8655 0.9327 0.9415

Average

0.7421

0.7208

0.7326

0.6969

0.9589

0.8816

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Overall we note an increase in the accuracy of the two models. Compared to the testing results for Task 1, these results are more encouraging for GTRS. The GTRS outperforms the Pawlak model in all aspects. The average results further substantiate this. An average accuracy of 74.21% is determined which is 2.13% higher than the Pawlak. The generality is 95.89% compared to 88.16% and the weighted accuracy is 73.26% versus 69.69%.

6.4. Demographic recommendations In order to gain further insights into the relative performance of the two models, we considered the demographic based approach to recommendations. The demographic information of the users were extracted from the user table of the movielen dataset. Information tables (similar to Table 1) are created based on the extracted user information. This allows for the application of rough set based techniques for demographic based recommendations. Tables 11 and 12 present the training and testing results for the Task 1, i.e., the binary scale of rating where the ratings of 4 or 5 indicate ‘‘like’’ and 1–3 indicate ‘‘dislike’’. The trend in the training results are very similar to the results obtained with collaborative based approach in Tables 7 and 9. The testing results are comparatively different in this case. The Pawlak model provides better testing accuracy. This however is not always true in case of weighted accuracy, for instance, in case of Movies 2, 6 and 10. The generality results are consistent with the previous results. The GTRS significantly improves the generality by 20–25%. Although the average accuracy of GTRS which is lesser than the Pawlak (i.e., 71.53%) by 10.67%, the difference in average weighted accuracy is only 2.17% (i.e., 56.61% compared to 58.78%). The

Table 13 Train results for data on Task 2.

Table 11 Train results for data on Task 1. Prediction for movie

Accuracy

Weighted acc.

Generality

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0.9024 0.9098 0.8977 0.9007 0.9034 0.9106 0.8859 0.9016 0.9190 0.9116

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.7428 0.7058 0.7592 0.7648 0.7489 0.7216 0.7323 0.7528 0.7738 0.7653

0.6494 0.6448 0.6875 0.7168 0.6942 0.6588 0.6630 0.6830 0.7139 0.6851

0.6033 0.5021 0.6518 0.6609 0.6169 0.5397 0.6019 0.6294 0.6535 0.6446

0.2988 0.2897 0.3750 0.4336 0.3884 0.3176 0.3261 0.3661 0.4278 0.3702

Average

0.9043

1.0

0.7467

0.6797

0.6104

0.3593

Table 12 Test results for data on Task 1. Prediction for movie

average generality with GTRS is 62.98% showing an average improvement of 21.94%. Tables 13 and 14 summarize the training and testing results for the Task 2, i.e., the binary scale of rating where the ratings of 5 indicate ‘‘like’’ and 1–4 indicate ‘‘dislike’’. The testing results in this case are better compared to Task 1. The accuracy difference between the two models in majority of the cases is around 2–4%. The weighted accuracy of GTRS is superior in 9 out of 10 cases. Although the average accuracy of GTRS is lesser (79.59% compared to 83.58%), the weighted accuracy is better (65.48% versus 61.31%). The average generality of GTRS is 52.8%, which is 19.32% more than the Pawlak. Fig. 2 summarizes the average results for predicting the 10 movies using the collaborative based approach. Each set of 6 bars represents the results corresponding to either the train or test data for the two tasks. The bars with the pattern of rectangles and those with the pattern of grid represent the accuracy obtained with GTRS and the Pawlak models, respectively. The bars with the pattern of single sloped lines and the bars with the pattern of group of three sloped lines represent the weighted accuracy of GTRS and the Pawlak models, respectively. The white and black bars shows the generality of GTRS and Pawlak, respectively. It is noted that Pawlak model always provide 100% accuracy for the training data. The GTRS also achieves high accuracy on the training data above 95%. The GTRS improves the accuracy on the testing data. Comparison of the weighted accuracies suggest that the GTRS provides better weighted accuracy compared to Pawlak irrespective of training or testing data. It is also noted that GTRS always provides better generality. This means that we need lesser additional information

Prediction for movie

Accuracy

Weighted acc.

Generality

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0.8718 0.9048 0.9113 0.9303 0.9083 0.9196 0.9261 0.9269 0.9359 0.9457

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.7006 0.7039 0.7063 0.6954 0.7208 0.7157 0.7202 0.7234 0.7050 0.7189

0.6198 0.6420 0.6275 0.6311 0.6186 0.6435 0.6552 0.6409 0.6418 0.6654

0.5396 0.5036 0.5015 0.4540 0.5408 0.5140 0.5167 0.5232 0.4703 0.4911

0.2395 0.2840 0.2549 0.2622 0.2373 0.2869 0.3105 0.2819 0.2836 0.3308

Average

0.9181

1.0

0.7110

0.6386

0.5055

0.2772

Table 14 Test results for data on Task 2.

Accuracy

Weighted acc.

Generality

Prediction for movie

Accuracy

Weighted acc.

Generality

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

GTRS

Pawlak

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0.4330 0.7064 0.5545 0.5236 0.5754 0.6775 0.5849 0.6578 0.6221 0.7506

0.5485 0.7429 0.7078 0.6437 0.6915 0.7585 0.7914 0.7654 0.6996 0.8041

0.4573 0.6081 0.5373 0.5167 0.5470 0.5976 0.5529 0.5995 0.5808 0.6640

0.5179 0.5812 0.5912 0.5693 0.5821 0.5957 0.6112 0.6102 0.5945 0.6247

0.6368 0.5238 0.6847 0.7082 0.6234 0.5500 0.6234 0.6308 0.6620 0.6545

0.3690 0.3341 0.4390 0.4825 0.4289 0.3702 0.3816 0.4153 0.4736 0.4102

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0.5985 0.7854 0.7723 0.7988 0.8125 0.8377 0.7759 0.8715 0.8398 0.8663

0.7141 0.8240 0.8422 0.8160 0.8345 0.8453 0.8328 0.8759 0.8639 0.9096

0.5588 0.6448 0.6423 0.6435 0.6738 0.6803 0.6452 0.6692 0.6673 0.6933

0.5670 0.6103 0.6092 0.6036 0.5980 0.6154 0.6166 0.6284 0.6257 0.6568

0.5974 0.5073 0.5225 0.4801 0.5562 0.5339 0.5261 0.5362 0.4923 0.5276

0.3129 0.3403 0.3192 0.3279 0.2931 0.3341 0.3505 0.3417 0.3453 0.3827

Average

0.6086

0.7153

0.5661

0.5878

0.6298

0.4104

Average

0.7959

0.8358

0.6548

0.6131

0.5280

0.3348

106

N. Azam, J.T. Yao / Knowledge-Based Systems 72 (2014) 96–107

Fig. 2. Summary of results for collaborative based recommendations.

Fig. 3. Summary of results for demographic based recommendations.

about the remaining cases which are being classified into the boundary region. Fig. 3 summarizes the results of the demographic based approach. The Pawlak model provides better accuracy and weighted accuracy results for Task 1. However, the difference between the two weighted accuracies on testing data is not very significant. One may accept this performance decrease for GTRS keeping in view the improvements it provides in the generality aspect. On Task 2 the GTRS not only matches the testing accuracy of the Pawlak model but also provides comparatively better weighted accuracy results. The generality of GTRS is superior in all the cases. The results presented in this section suggest and advocate for the use of GTRS in recommender systems to obtain recommendations. 7. Conclusion Recommender systems are becoming a popular tool in E-business. We examine the role of Game-theoretic rough sets or GTRS

in the capacity of an intelligent component for making recommendations in recommender systems. Two properties of recommendations, i.e., accuracy and generality, are being focused. In an ideal situation, the recommender systems are expected to provide highly accurate recommendations for majority of the users. This however is not always possible and one has to rely on some mechanism for determining a suitable tradeoff between the two properties. The role and use of GTRS is highlighted for obtaining and finding an effective and balanced solution by considering and implementing a game between the properties of accuracy and generality. Experimental results on movielen dataset suggest that the GTRS not only improves the generality of recommendations but also provides more accurate recommendations in some cases. Furthermore, it outperforms the standard Pawlak model in majority of the cases. These results encourage and advocate for the use of GTRS as an alternative way for obtaining recommendation in recommender systems. The ultimate success of any recommender system is in the appreciation of its users and the ability of the system to serve their

N. Azam, J.T. Yao / Knowledge-Based Systems 72 (2014) 96–107

distinct needs. While we try to optimize the parameters in the global perspective, there is a good reason to tune the parameters for each individual users. This consideration will open the door to extensive future theoretical and empirical research, bringing personalization to the GTRS based recommendations. Acknowledgements This work was partially supported by a Discovery Grant from NSERC Canada and the University of Regina Verna Martin Memorial Scholarship Program.

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