Mining Quantitative Association Rules on Overlapped Intervals

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1 Mining Quantitative Association Rules on Overlapped Intervals Qiang Tong 1,3, Baoping Yan 2, and Yuanchun Zhou 1,3 1 Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China {tongqiang, yczhou}@sdb.cnic.cn 2 Computer Network Information Center, Chinese Academy of Sciences, Beijing, China ybp@mimi.cnc.ac.cn 3 Graduate School of the Chinese Academy of Sciences, Beijing, China Abstract. Mining association rules is an important problem in data mining. Algorithms for mining boolean data have been well studied and documented, but they cannot deal with quantitative and categorical data directly. For quantitative attributes, the general idea is partitioning the domain of a quantitative attribute into intervals, and applying boolean algorithms to the intervals. But, there is a conflict between the minimum support problem and the minimum confidence problem, while existing partitioning methods cannot avoid the conflict. Moreover, we expect the intervals to be meaningful. Clustering in data mining is a discovery process which groups a set of data such that the intracluster similarity is maximized and the intercluster similarity is minimized. The discovered clusters are used to explain the characteristics of the data distribution. The present paper will propose a novel method to find quantitative association rules by clustering the transactions of a database into clusters and projecting the clusters into the domains of the quantitative attributes to form meaningful intervals which may be overlapped. Experimental results show that our approach can efficiently find quantitative association rules, and can find important association rules which may be missed by the previous algorithms. 1 Introduction Mining association rules is a key data mining problem and has been widely studied [1]. Finding association rules in binary data has been well investigated and documented [2, 3, 4]. Finding association rules in numeric or categorical data is not as easy as in binary data. However, many real world databases contain quantitative attributes and current solutions to this case are so far inadequate. An association rule is a rule of the form X Y, where X and Y are sets of items. It states that when X occurs in a database so does Y with a certain probability. X is called the antecedent of the rule and Y the consequent. There X. Li, S. Wang, and Z.Y. Dong (Eds.): ADMA 2005, LNAI 3584, pp , c Springer-Verlag Berlin Heidelberg 2005

2 44 Q. Tong, B. Yan, and Y. Zhou are two important parameters associated with an association rule: support and confidence. Support describes the importance of the rule, while confidence determines the occurrence probability of the rule. The most difficult part of an association rule mining algorithm is to find the frequent itemsets. The process is affected by the support parameter designated by the user. A well known application of association rules is in market basket data analysis, which was introduced by Agrawal in 1993 [2]. In the problem of market basket data analysis, the data are boolean, which have values of 1 or 0. The classical association rule mining algorithms are designed for boolean data. However, quantitative and categorical attributes widely exist in current databases. In [5], Srikant and Agrawal proposed an algorithm dealing with quantitative attributes by dividing quantitative attributes into equi-depth intervals and then combining adjacent partitions when necessary. In other words, for a depth d, the first d values of the attribute are placed in one interval, the next d in a second interval, and so on. There are two problems with the current methods of partitioning intervals: MinSup and MinConf [5]. If a quantitative attribute is divided into too many intervals, the support for a single interval can be low. When the support of an interval is below the minimum support, some rules involving the attribute may not be found. This is the minimum support problem. Some rules may have minimum confidence only when a small interval is in the antecedent, and the information loss increases as the interval size becomes larger. This is the minimum confidence problem. The critical part of mining quantitative association rules is to divide the domains of the quantitative attributes into intervals. There are several classical dividing methods. The equi-width method divides the domain of a quantitative attribute into n intervals, and each interval has the same length. In the equidepth method, there are equal number of items contained in each interval. The equi-width method and the equi-depth method are so straightforward that the partitions of quantitative attributes may not be meaningful, and cannot deal with the minimum confidence problem. In [5], Srikant and Agrawal introduced a measure of partial completeness which quantified the information lost due to partitioning, and developed an algorithm to partition quantitative attributes. In [6], Miller and Yang pointed out the pitfalls of the equi-depth method, and presented several guiding principles for quantitative attribute partitioning. In selecting intervals or groups of data to consider, they wanted to have a measure of interval quality to reflect the distance among data points. They took the distance among data into account, since they believed that putting closer data together was more meaningful. To achieve this goal, they presented a more general form of an association rule, and used clustering to find subsets that made sense by containing a set of attributes that were close enough. They proposed an algorithm which used birch [7] to find clusters in the quantitative attributes and used the discovered clusters to form items, then fed the items into the classical boolean algorithm apriori [3]. In their algorithm, clustering was used to determine sets of dense values in a single attribute or over a set of attributes that were to be treated as a whole.

3 Mining Quantitative Association Rules on Overlapped Intervals 45 Although Miller and Yang took the distance among data into account and used a clustering method to make the intervals of quantitative attributes more meaningful, they did not take the relations among other attributes into account by clustering a quantitative attribute or a set of quantitative attributes alone. We believe that their technique still falls shot of a desirable goal. Based on the above analysis, we find that the partitioning method can be further improved. On the one hand, since clustering an attribute or a set of attributes alone is not good enough, we believe that the relations among attributes should be taken into account. We tend to cluster all attributes together, and project the clusters into the domains of the quantitative attributes. On the other hand, the projection of the clusters on a specific attribute can be overlapped. We think this is reasonable. Moreover, this is a good resolution to the conflict between the minimum support problem and the minimum confidence problem. A small interval may result in the minimum support problem, while a large interval may lead to the minimum confidence problem. When several overlapped intervals coexist in the domain of a quantitative attribute, and different intervals are used for different rules, the conflict between the two problems which confuses the quantitative attributes partitioning does not exist. In this paper, we propose an approach which first applys a clustering algorithm to all attributes, and projects the discovered clusters into the domains of all attributes to form intervals (the intervals may be overlapped), then uses a boolean association rule mining algorithm to find association rules. The rest of the paper is organized as follows. In Section 2, we introduce some definitions of the quantitative association rule mining problem and review the background in brief. In Section 3, we present our approach and our algorithm. In Section 4, we give the experimental results and our analysis. Finally, in Section 5, we give the conclusions and the future work. 2 Problem Description Now, we give a formal statement of the problem of mining quantitative association rules and introduce some definitions. Let I = {i 1, i 2,..., i n } be a set of attributes, and R be the set of real numbers, I R = X R R, that is I R = {(x, l, u) x I, l R, u R, l x u}. A triple (x, l, u) I R denotes either a quantitative attribute x with a value interval [l, u], or a categorical attribute with a value l (l = u). Let D be a set of transactions, where each transaction T is a set of attribute values. X I R, if (x, l, u) X, (x, v) T, l v u, we say transaction T supports X. A quantitative association rule is an implication of the form X Y, where X I R, Y I R, and attribute(x) attribute(y ) =. If s percent of transactions in D support X Y, and c percent of transactions which support X also support Y, we say that the association rule has support s and confidence c respectively. The problem of mining quantitative association rules is the process of finding association rules which meet the minimum support and the minimum confidence at a given transaction database which contains quantitative and/or categorical attributes.

4 46 Q. Tong, B. Yan, and Y. Zhou Clustering can be considered the most important unsupervised learning technique, which deals with finding a structure in a collection of unlabeled data. A cluster is therefore a collection of objects which are similar to each other and are dissimilar to the objects belonging to other clusters [8]. In this paper, a cluster is a set of transactions. An important component of a clustering algorithm is the distance measure among data points. In [6], Miller and Yang defined two thresholds on the cluster size and the diameter. First, the diameter of a cluster should be less than a specific value to ensure that the cluster is sufficiently dense. Second, the number of transactions contained in a cluster should be greater than the minimum support to ensure that the cluster is frequent. Since our clustering approach is different, our definition of the diameter of a cluster is also different. Definition 1. We adopt Euclidean distance as the distance metric, and the distance between two transactions is defined as d(i i, I j ) = n (I ik I jk ) 2 (1) k=1 Definition 2. A cluster C = {I 1, I 2,..., I m } is a set of transactions, and the gravity center of C is defined as C g = 1 m m I i (2) Definition 3. The diameter of a cluster C = {I 1, I 2,..., I m } is defined as D g (C) = 1 m i=1 m (I i C g ) T (I i C g ) (3) i=1 Definition 4. The number of transactions contained in a cluster C is denoted as C, d 0 and s 0 are thresholds for association rule mining, C and D g (C) should satisfy the following formula C s 0, D g (C) d 0 (4) The first criterion ensures that the cluster contains enough number of transactions to be frequent. The second criterion ensures that the cluster is dense enough. To ensure that clusters are isolated from each other, we will rely on a clustering algorithm to discover clusters which are as isolated as possible. 3 The Proposed Approach In this section, we describe our approach of mining quantitative association rules. We divide the problem of mining quantitative association rules into several steps:

5 Mining Quantitative Association Rules on Overlapped Intervals Map the attributes of the given database to I R = I R R. For ordered categorical attributes, map the values of the attribute to a set of consecutive integers, such that the order of the attributes is preserved. For unordered categorical attributes, we define the distance between two different attributes as a constant value. For boolean attributes, map the values of the attributes to 0 and 1. For quantitative attributes, we keep the original values or transform the values to a standard form, such as Z-Score. We adopt various mapping methods to fit the clustering algorithm. For different data sets, we may use different mapping methods. 2. Apply a clustering algorithm to the new database produced by the first step. In the clustering algorithm, by dealing with the transactions as n- dimension vectors, we take all attributes into account. In this paper, we adopt a common clustering algorithm k-means to identify transaction groups that are compact (the distance among transactions within a cluster is small) and isolated (relatively separable from other groups). By clustering all attributes together, the relations among all attributes are considered, and the clusters may be more meaningful. Besides, we also use Definition 4 as the principle for evaluating the quality of the discovered clusters. 3. Project the clusters into the domains of the quantitative attributes. The projections of the clusters will form overlapped intervals. We make an interval x [l, u] a new boolean attribute. The two-dimension example of the projection is shown in Figure Mine association rules by using a classical boolean algorithm. Since the quantitative attributes have been booleanized, we can use a boolean algorithm (such as apriori) to find frequent itemsets, and then use the frequent itemsets to generate association rules. I2 u1 u3 C1 C3 l1 u2 l3 C2 l2 l1 l2 u1 u2 l3 u3 I1 Fig. 1. Projecting the clusters into the domains of quantitative attributes to form intervals, which may be overlapped

6 48 Q. Tong, B. Yan, and Y. Zhou 4 Experimental Results Our experimental environment is an IBM Netfinity 5600 server with dual PIII 866 CPUs and 512M memory, which runs Linux operating system. The experiment has been done over a real data set of bodyfat [10]. The attributes in the bodyfat dataset are: density, age, weight, height, neck, chest, abdomen, hip, thigh, knee, ankle, biceps, forearm and wrist. All of the attributes are quantitative attributes. There are 252 records of various people in the dataset. Our purpose is to find association rules over all attributes. For our algorithm, the parameters needed from the user are the minimum support, the minimum confidence, and the number of clusters. In our experiment, we use minimum support of 10%, minimum confidence of 60%, and the clusters of six. We first use a common clustering algorithm (k-means) to find clusters, then project the clusters into the domains of the quantitative attributes, and finally use a boolean association rule mining algorithm (apriori) to find association rules. Some of the rules which we have found are listed in Figure 2. From the above rules, we can see that the intervals are overlapped, which cannot be discovered by the previous partitioning methods. The equi-width method cannot divide some quantitative attributes properly (such as density), because the attributes range only in a very small domain, while the equi-depth method may put far apart transactions into the same interval. As shown in Figure 1, our partitioning method projects the clusters into the domains of the quantitative attributes, and forms overlapped intervals. Our method considers both the distance among transactions and the relations among attributes. For previous methods, if an interval is small, it may not meet the minimum support; if an interval is large, it may not meet the minimum confidence. In our method, since the intervals can be overlapped, we can avoid the conflict between the minimum support problem and the minimum confidence problem. Moreover, since our intervals tend to be less than those of the previous methods, the boolean association rule mining algorithm works more efficiently. ID Rules 1 Age[40, 74]&Weight[178, 216] Abdomen[88.7, 113.1] 2 Age[34, 42]&Weight[195.75, ] Chest[99.6, 115.6] 3 Weight[219, ] Hip[105.5, 147.7]&Chest[108.3, 136.2] 4 Weight[154, 191]&Height[65.5, 77.5] Density[1.025, 1.09] 5 Abdomen[88.6, 111.2]&Hip[101.8, 115.5] Weight[196,224] 6 Biceps[24.8, 38.5] & Forearm[22, 34.9] Wrist[15.8, 18.5] 7 Thigh[54.7, 69]&Knee[34.2, 42.2] Ankle[21.4, 33.9] 8 Weight[118.5, ]&Height[64, 73.5] Density[1.047, 1.11] Fig. 2. Some of the rules discovered by our algorithm with the parameters (minimum support = 10%, minimum confidence = 60%, and the number of clusters k = 6)

7 Mining Quantitative Association Rules on Overlapped Intervals 49 5 Conclusions and Future Work In this paper, we have proposed a novel approach to efficiently find quantitative association rules. The critical part of quantitative association rule mining is to partition the domains of quantitative attributes into intervals. The previous algorithms dealt with this problem by dividing the domains of quantitative attributes into equi-depth or equi-width intervals, or using a clustering algorithm on a single attribute (or a set of attributes) alone. They cannot avoid the conflict between the minimum support problem and the minimum confidence problem, and risk missing some important rules. In our approach, we treat a transaction as an n-dimension vector, and apply a common clustering algorithm to the vectors, then project the clusters into the domains of the quantitative attributes to form overlapped intervals. We finally use a classical boolean algorithm to find association rules. Our approach takes the relations and the distances among attributes into account, and can resolve the conflict between the minimum support problem and the minimum confidence problem by allowing intervals to be overlapped. Experimental results show that our approach can efficiently find quantitative association rules, and can find important association rules which may be missed by the previous algorithms. Since our approach adopts a common clustering algorithm and a classical boolean association rule mining algorithm rather than integrates the two algorithms together, we believe that our approach can be further improved by integrating the clustering algorithm and the association rule mining algorithm tightly in our future work. Acknowledgement This work is partially supported by the National Hi-Tech Development 863 Program of China under grant No. 2002AA104240, and the Informatization Project under the Tenth Five-Year Plan of the Chinese Academy of Sciences under grant No. INF105-SDB. We thank Mr. Longshe Huo and Miss Hong Pan for their helpful suggestions. References 1. Han, J., Kamber, M.: Data Mining Concepts and Techniques. China Machine Press and Morgan Kaufmann Publishers (2001) 2. Agrawal, R., Imielinski, T. and Swami, A.: Mining Association Rules between Sets of Items in Large Databases. In Proc. of the 1993 ACM SIGMOD International Conf. on Management of Data, Washington, D.C., May (1993) Agrawal, R. and Srikant, R.: Fast Algorithms for Mining Association Rules in Large Databases. In Proc. of 20th International Conf. on Very Large Data Bases, Santiago, Chile, September (1994) Han, J., Pei, J., Yin, Y. and Mao, R.: Mining Frequent Patterns without Candidate Generation: A Frequent-Pattern Tree Approach. Data Mining and Knowledge Discovery (2004) 8, 53 87

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