Boolean factors as a means of clustering of interestingness measures of association rules ?

Abstract

Measures of interestingness play a crucial role in association rule mining. An important methodological problem is to provide a reasonable classi cation of the measures. Several papers appeared on this topic. In this paper, we explore Boolean factor analysis, which uses formal concepts corresponding to classes of measures as factors, for the purpose of classi cation and compare the results to the previous approaches.

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An important problem in extracting association rules, well known since the early stage of association rule mining [ 32 ], is the possibly huge number of rules extracted from data. A general way of dealing with this problem is to de ne the concept of rule interestingness: only association rules that are considered interesting according to some measure are presented to the user. The most widely used measures of interestingness are based on the concept of support and condence. However, the suitability of these measures to extract interesting rules was challenged by several studies, see e.g. [ 34 ]. Consequently, several other interestingness measures of association rules were proposed, see e.g. [ 35 ], [ 23 ], [ 12 ], [ 38 ]. With the many existing measures of interestingness arises the problem of selecting an appropriate one.

To understand better the behavior of various measures, several studies of the properties of measures of interestingness appeared, see e.g. [ 12 ], [ 27 ], [ 23 ], [ 16 ]. Those studies explore various properties of the measures that are considered important. For example, Vaillant et al. [ 37 ] evaluated twenty interestingness measures according to eight properties. To facilitate the choice of the user-adapted interestingness measure, the authors applied the clustering methods on the decision matrix and obtained ve clusters. Tan et al. [ 35 ] studied twenty-one interestingness measures through eight properties and showed that no measure is adapted to all cases. To select the best interestingness measure, they use both a support-based pruning and standardization methods. By applying a new clustering approach, Huynh et al. [ 21 ] classifyed thirty-four interestingness measures with a correlation analysis. Geng and Hamilton [ 12 ] made a survey of thirtyeight interestingness measures for rules and summaries with eleven properties and gived strategies to select the appropriate measures. D. R. Feno [ 10 ] evaluated fteen interestingness measures with thirteen properties to describe their behaviour. Delgato et al. [ 9 ] provided a new study of the interestingness measures by means of the logical model. In addition, the authors proposed and justi ed the addition of two new principles to the three proposed by Piatetsky-Shapiro [ 32 ]. Finally, Heravi and Zaiane [ 22 ] studied fty-three objective measures for associative classi cation rules according to sixteen properties and explained that no single measure can be introduced as an obvious winner.

The assessment of measures according to their properties results in a measureproperty binary matrix. Two studies of this matrix were conducted. Namely, [ 17 ] describes how FCA can highlight interestingness measures with similar behavior in order to help the user during his choice. [ 16 ] and [ 14 ] attempted to nd natural clusters of measures using widely used clustering methods, the agglomerative hierarchical method (AHC) and the K-means method. A common feature of these methods is that they only produce disjoint clusters of measures. On the other hand, one could naturally expect overlapping clusters. The aim of this paper is to explore the possibility of obtaining overlapping clusters of measures using factor analysis of binary data and to compare the results with the results of other studies. In particular, we use the recently developed method from [ 3 ] and take the discovered factors for clusters. The method uses formal concepts as factors that makes it possible to interpret the factors easily. 2 2.1

Preliminaries

Binary (Boolean) data

Let X be a set of objects (such as a set of customers, a set of functions or the like) and Y be a set of attributes (such as a set of products that customers may buy, a set of properties of functions). The information about which objects have which attributes may formally be represented by a binary relation I between X and Y , i.e. I X Y , and may be visualized by a table (matrix) that contains 1s and 0s, according to whether the object corresponding to a row has the attribute corresponding to a column (for this we suppose some orders of objects and attributes are xed). We denote the entries of such matrix by Ixy. A data of this type is called binary data (or Boolean data). The triplet hX; Y; Ii is called a formal context in FCA but other terms are used in other areas.

Such type of data appears in two roles in our paper. First, association rules, whose interestingness measures we analyze, are certain dependencies over the binary data. Second, the information we have about the interestingness measures of association rules is in the form of binary data: the objects are interestingness measures and the attributes are their properties. 2.2

Association rules

An association rule [ 36 ] over a set Y of attributes is a formula

A ) B (1) where A and B are sets of attributes from Y , i.e. A; B Y . Let hX; Y; Ii be a formal context. A natural measure of interestingness of association rules is based on the notions of con dence and support. The con dence and support of an association rule A ) B in hX; Y; Ii is de ned by conf(A ) B) = jA# \ B#j jA#j and supp(A ) B) = jA# \ B#j ; jXj where C# for C Y is de ned by C# = fx 2 X j for each y 2 C : hx; yi 2 Ig. An association rule is considered interesting if its con dence and support exceed some user-speci ed thresholds. However, the support-con dence approach reveals some weaknesses. Often, this approach as well as algorithms based on it lead to the extraction of an exponential number of rules. Therefore, it is impossible to validate it by an expert. In addition, the disadvantage of the support is that sometimes many rules that are potentially interesting, have a lower support value and therefore can be eliminated by the pruning threshold minsupp. To address this problem, many other measures of interestingness have been proposed in the literature [ 13 ], mainly because they are e ective for mining potentially interesting rules and capture some aspects of user interest. The most important of those measures are subject to our analysis and are surveyed in Section 3.1. Note that association rules are attributed to [ 1 ]. However, the concept of association rule itself as well as various measures of interestingness are particular cases of what is investigated in depth in [ 18 ], a book that develops logico-statistical foundations of the GUHA method [ 19 ]. 2.3

Factor analysis of binary (Boolean) data

Let I be an n a decomposition m binary matrix. The aim in Boolean factor analysis is to nd

I = A The inner dimension, k, in the decomposition may be interpreted as the number of factors that may be used to describe the original data. Namely, Ail = 1 if and only if the lth factor applies to the ith object and Blj = 1 if and only if the jth attribute is one of the manifestations of the lth factor. The factor model behind (2) has therefore the following meaning: The object i has the attribute j if and only if there exists a factor l that applies to i and for which j is one of its particular manifestations. We refer to [ 3 ] for further information and references to papers that deal with the problem of factor analysis and decompositions of binary matrices.

In [ 3 ], the following method for nding decompositions (2) with the number k of factors as small as possible has been presented. The method utilizes formal concepts of the formal context hX; Y; Ii as factors, where X = f1; : : : ; ng, Y = f1; : : : ; mg (objects and attributes correspond to the rows and columns of I). Let

F = fhC1; D1i; : : : ; hCk; Dkig be a set of formal concepts of hX; Y; Ii, i.e. hCl; Dli are elements of the concept lattice B(X; Y; I) [ 11 ]. Consider the n k binary matrix AF and a k m binary matrix BF de ned by (AF )il = 1 iff i 2 Cl and (BF )lj = 1 iff j 2 Dl: (3) Denote by (I) the smallest number k, so-called Schein rank of I, such that a decomposition of I exists with k factors. The following theorem shows that using formal concepts as factors as in (3) enables us to reach the Schein rank, i.e. is optimal [ 3 ]:

Theorem 1. For every binary matrix I, there exists F

I = AF BF and jF j = (I).

B(X; Y; I) such that

As has been demonstrated in [ 3 ], a useful feature of using formal concepts as factors is the fact that formal concepts may easily be interpreted. Namely, every factor, i.e. a formal concept hCl; Dli, consists of a set Cl of objects (objects are measures of interestingness in our case) and a set Dl of attributes (properties of measures in our case). Cl contains just the objects to which all the attributes from Dl apply and Dl contains all attributes shared by all objects from Cl. From a clustering point of view, the factors hCl; Dli may thus be seen as clusters Cl with their descriptions by attributes from Dl. The factors thus have a natural, easy to understand meaning. Since the problem of computing the smallest set of factors is NP-hard, a greedy approximation algorithm was proposed in [3, Algorithm 2]. This algorithm is utilized below in our paper.

Clustering interestingness measures using Boolean factors 3.1

Measures of interestingness

In the following, we present the interestingness measures reported in the literature and recall nineteen of their most important properties that were proposed in the literature.

To identify interesting association rules and to enable the user to focus on what is interesting for him, about sixty interestingness measures [ 20 ], [ 35 ], [ 10 ] were proposed in the literature. All of them are de ned using the following parameters: p(XY ), p(XY ), p(XY ) and p(XY ), where p(XY ) = nXY represents n the number of objects satisfying XY (the intersection of X and Y ), and X is the negation of X. The following are important examples of interestingness measures: Lift [ 6 ]: Given a rule X ! Y , lift is the ratio of the probability that X and Y occur together to the multiple of the two individual probabilities for X and Y , i.e.,

Lift (X ! Y ) = p( Xp()XYp()Y ) : If this value is 1, then X and Y are independent. The higher this value, the more likely that the existence of X and Y together in a transaction is not just a random occurrence, but because of some relationship between them. Correlation coe cient [ 31 ]: Correlation is a symmetric measure evaluating the strength of the itemsets' connection. It is de ned by

p(XY ) p(X)p(Y ) :

Correlation = pp(X)p(Y )p(X)p(Y ) A correlation around 0 indicates that X and Y are not correlated. The lower is its value, the more negatively correlated X and Y are. The higher is its value, the more positively correlated they are.

Conviction [ 6 ]: Conviction is one of the measures that favor counter-examples. It is de ned by

Conviction = p(X)p(Y ) p(XY ) Conviction which is not a symmetric measure, is used to quatify the deviation from independence. If its value is 1, then X and Y are independent. MGK [ 15 ]: MGK is an interesting measure, which allows the extraction of negative rules.

MGK = p(Y =X) p(Y ) ; if X favorise Y

1 p(Y )

MGK = p(Y =pX(Y) )p(Y ) ; if X defavorise Y It takes into account several situations of references: in the case where the rule is situated in the attractive zone (i.e. p(Y =X) > p(Y )), this measure evaluates the distance between independence and logical implication. Thus, the higher the value of MGK is close to 1, the more the rule is close to the logical implication and the higher the value of MGK is close to 0, the more the rule is close to the independence. In the case where the rule is located in the repulsive zone (i.e. p(Y =X) < p(Y )), MGK evaluates this time a distance between the independence and the incompatibility. Thus, the closer the value of MGK is to 1, the more similar to incompatibility the rule is; and the closer the value of MGK is to 0, the closer to the independence the rule is.

As was mentioned above, several studies [ 35 ], [ 23 ], [ 25 ], [ 13 ] were reported in the literature on the various properties of interestingess measures to be able to characterize and evaluate the interestingness measures. The main goal of researchers in the domain is then to provide a user assistance in choosing the best interestingness measure meeting his needs. For that, formal properties have been developed [ 32 ], [ 24 ], [ 35 ], [ 12 ], [ 4 ] in order to evaluate the interestingness measures and to help users understanding their behavior. In the following, we present nineteen properties reported in the literature. 3.2

Properties of the measures

The measure-property matrix describing interestingness measures by their properties is depicted in Figure 2. It consists of 62 measures (61 measures from [ 14 ] plus one more that has been studied recently) described by 21 properties because the three-valued property P14 is represented by three yes-no properties No. Property Ref. P1 Intelligibility or comprehensibility of measure [ 25 ] P2 Easiness to x a threshold to the rule [ 23 ] P3 Asymmetric measure. [ 35 ], [ 23 ] P4 Asymmetric measure in the sense of the conclusion negation. [ 23 ], [ 35 ] P5 Measure assessing in the same way X ! Y and Y ! X in the logical [ 23 ] implication case.

P6 Measure increasing function the number of examples or decreasing func- [ 32 ], [ 23 ] tion the number of counter-examples.

P7 Measure increasing function the data size. [ 12 ], [ 35 ] P8 Measure decreasing function the consequent/antecedent size. [ 23 ], [ 32 ] P9 Fixed value a in the independence case. [ 23 ], [ 32 ] P10 Fixed value b in the logical implication case. [ 23 ] P11 Fixed value c in the equilibrium case. [ 5 ] P12 Identi ed values in the attraction case between X and Y . [ 32 ] P13 Identi ed values in the repulsion case between X and Y . [ 32 ] P14 Tolerance to the rst counter-example. [ 23 ], [ 38 ] P15 Invariance in case of expansion of certain quantities. [ 35 ] PP1167 DDeessiirreedd rreellaattiioonnsshhiipp bbeettwweeeenn XX !! YY aanndd XX !! YY raunlteisn.omic rules. [[3355]] P18 Desired relationship between X ! Y and X ! Y rules. [ 35 ] P19 Antecedent size is xed or random. [ 23 ] P20 Descriptive or statistical measure. [ 23 ] P21 Discriminant measure. [ 23 ]

P14:1, P14:2, and P14:3. We computed the decomposition of the matrix using Algorithm 2 from [ 3 ] and obtained 28 factors (as in the case below, several of them may be disregarded as not very important; we leave the details for a full version of this paper). In addition, we extended the original 62 21 binary matrix by adding for every property its negation, and obtained a 62 42 binary matrix. The reason for adding negated properties is due to our goal to compare the results with the two clustering methods mentioned above and the particular role of the properties and their negations in these clustering methods. From the 62 42 matrix, we obtained 38 factors, denoted F1; : : : ; F38. The factors are presented in Figures 3 and 4. Figure 3 depicts the object-factor matrix describing the interestingness measures by factors, Figure 4 depicts the factor-property matrix explaining factors by properties of measures. Factors are sorted from the most important to the least important, where the importance is determined by the number of 1s in the input measure-property matrix covered by the factor [ 3 ]. The rst factors cover a large part of the matrix, while the last ones cover only a small part and may thus be omitted [ 3 ], see the graph of cumulative cover of the matrix by the factors in Figure 5. 4

Interpretation and comparison to other approaches The aim of this section is to provide an interpretation of the results described in the previous section and compare them to the results already reported in the literature, focusing mainly on [ 14 ]. As was described in the previous section, 38 factors were obtained. The rst 21 of them cover 94 % of the input measureproperty matrix (1s in the matrix), the rst nine cover 72 %, and the rst ve Fig.2. Input binary matrix describing interestingness measures by their properties. Fig. 3. Interestingness measures described by factors obtained by decomposition of the input matrix from Figure 2 extended by negated properties. Fig. 4. Factors obtained by decomposition of the input matrix from Figure 2 extended by negated properties. The factors are described in terms of the original and negated properties. 0 cover 52.4 %. Another remark is that the rst ten factors cover the whole set of measures.

Note rst that the Boolean factors represent overlapping clusters, contrary to the clustering using the agglomerative hierarchical method and the K-means method performed in [ 14 ]. Namely, the clusterings are depicted in Figure 6 describing the Venn diagram of the rst ve Boolean factors (plus the eighth and part of the sixth and tenth to cover the whole set of measures) and Figure 7, which is borrowed from [ 14 ], describing the consensus on the classi cation obtained by the hierarchical and K-means clusterings. This consensus refunds the classes C1 to C7 of the extracted measures, which are common to both techniques.

Due to lack of space, we focus on the rst four factors since they cover nearly half of the matrix (45.1 %), and also because most of the measures appear at least once in the four factors.

Factor 1. The rst factor F1 applies to 20 measures, see Figure 3, namely: correlation, Cohen, Pavillon, conviction, Bayes factor, Loevinger, collective strength, information gain, Goodman, interest, Klosgen, Mgk, YuleQ, relative risk, one way support, two way support, YuleY, Zhang, novelty, and odds ratio. These measures share the following 9 properties: P4, P7, P9, not P11, P12, P13, not P19, not P20, P21, see Figure 4.

Interpretation. The factor applies to measures whose evolutionary curve increases w.r.t the number of examples and have a xed point in the case of independence (this allows to identify the attractive and repulsive area of a rule). The factor also applies only to descriptive and discriminant measures that are not based on a probabilistic model.

Comparison. When looking at the classi cation results reported in [ 14 ], F1 covers two classes from [ 14 ]: C6 and C7, which together contain 15 measures. Those classes are closely related within the dendrogram obtained with the agglomerative hierarchical clustering method used in [ 14 ]. The 5 missing measures form a class obtained with K-means method in [ 14 ] with Euclidian distance.

Factor 2. F2 applies to 18 measures, namely: con dence, causal con dence, Ganascia, causal con rmation, descriptive con rmation, cosine, causal dependency, Laplace, least contradiction, precision, recall, support, causal con rmed con dence, Czekanowski, negative reliability, Leverage, speci city, and causal support. These measures share the following 11 properties: P4, P6, not P9, not P12, not P13, P14.2, not P15, not P16, not P19, not P20, P21.

Interpretation. The factor applies to measures whose evolutionary curve increases w.r.t. the number of examples and has a variable point in the case of independence, which implies that the attractive and repulsive areas of a rule are not identi able. The factor also applies only to measures that are not discriminant, are indi erent to the rst counter-examples, and are not based on a probabilistic model.

Comparison. F2 corresponds to two classes, C4 and C5 reported in [ 14 ]. C4 [ C5 contains 22 measures. The missing measures are: Jaccard, Kulczynski, examples and counter-examples rate and Sebag. Those measures are not covered by F2 since they are not indi erent to the rst counter-examples.

Factor 3. F3 applies to 10 measures, namely: coverage, dependency, weighted dependency, implication index, Jmeasure, Pearl, prevalence, Gini, variation support, and mutual information. These measures share the following 10 properties: not P6, not P8, not P10, not P11, not P13, not P14.1, not P15, not P16, not P17, not P19.

Interpretation. The factor applies to measures whose evolutionary curve does not increase w.r.t. the number of examples.

Comparison. F3 corresponds to class C3 reported in [ 14 ], which contains 8 measures. The two missing measures, variation support and Pearl, belong to the same classes obtained by both K-means and the hierarchical method. Moreover, these two missing measures are similar to those from C3 obtained by the hierarchical method since they merge with the measures in C3 at the next level of the generated dendrogram. Here, there is a strong correspondence between results obtained using Boolean factors and the ones reported in [ 14 ].

Factor 4. F4 applies to 9 measures, namely: con dence, Ganascia, descriptive con rmation, IPEE, IP3E, Laplace, least contradiction, Sebag, and examples and counter-examples rate. These measures share the following 12 properties: P3, P4, P6, P11, not P7, not P8, not P9, not P12, not P13, not P15, not P16, not P18.

Interpretation. The factor applies to measures whose evolutionary curve increases w.r.t. the number of examples and has a xed value in the equilibrium case. As there is no xed value in the independence case, we can not get an identi able area in the case of attraction or repulsion.

Comparison. F4 mainly applies to measures of class C5 obtained in [ 14 ]. The two missing measures, IPEE et IP3E, belong to a di erent class. 5

Conclusions and further issues

We demonstrated that Boolean factors provide us with clearly interpretable meaningful clusters of measures among which the rst ones are highly similar to other clusters of measures reported in the literature. Contrary to other clustering methods, Boolean factors represent overlapping clusters. We consider this an advantage because overlapping clusters are a natural phenomenon in human classi cation. We presented preliminary results on clustering the measures using Boolean factors. Due to limited scope, we presented only parts of the results obtained and leave other results for a full version of this paper.

An interesting feature of the presented method, to be explored in the future, is that the method need not start from scratch. Rather, one or more clusters, that are considered important classes of measures, may be supplied at the start and the method may be asked to complete the clustering. Another issue left for future research is the bene t of the clustering of measures for a user who is interested in selecting a type of measure, rather than a particular measure of interestingness of association rules. In the intended scenario, a user may use various interestingness measures that belong to di erent classes of measures.

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