Progressive Partition Miner: An Efficient
Algorithm for Mining General Temporal
Association Rules
Chang-Hung Lee, Ming-Syan Chen, Senior Member, IEEE, and Cheng-Ru Lin
Abstract—In this paper, we explore a new problem of mining general temporal association rules in publication databases. In essence, a publication database is a set of transactions where each transaction T is a set of items of which each item contains an individual exhibition period. The current model of association rule mining is not able to handle the publication database due to the following fundamental problems, i.e., 1) lack of consideration of the exhibition period of each individual item and 2) lack of an equitable support counting basis for each item. To remedy this, we propose an innovative algorithm Progressive-Partition-Miner (abbreviated as PPM) to discover general temporal association rules in a publication database. The basic idea of PPM is to first partition the publication database in light of exhibition periods of items and then progressively accumulate the occurrence count of each candidate 2-itemset based on the intrinsic partitioning characteristics. Algorithm PPM is also designed to employ a filtering threshold in each partition to early prune out those cumulatively infrequent 2-itemsets. The feature that the number of candidate 2-itemsets generated by PPM is very close to the number of frequent 2-itemsets allows us to employ the scan reduction technique to effectively reduce the number of database scans. Explicitly, the execution time of PPM is, in orders of magnitude, smaller than those required by other competitive schemes that are directly extended from existing methods. The correctness of PPM is proven and some of its theoretical properties are derived. Sensitivity analysis of various parameters is conducted to provide many insights into Algorithm PPM.
Index Terms—Data mining, general temporal association rule, exhibition period, publication database.
æ
1
I
NTRODUCTIONT
HEdiscovery of association relationships among a hugedatabase has been known to be useful in selective marketing, decision analysis, and business management [7], [16]. A popular area of applications is the market basket analysis, which studies the buying behaviors of customers by searching for sets of items that are frequently purchased together (or in sequence). Let I ¼ fx1; x2; . . . ; xmg be a set of
items. A set X I with k ¼ jXj is called a k-itemset or simply an itemset. Let a database D be a set of transactions, where each transaction T is a set of items such that T I. A transaction T is said to support X if and only if X T . Conventionally, an association rule is an implication of the form X¼)Y , meaning that the presence of the set X implies the presence of another set Y , where X I, Y I , and XTY ¼ . The rule X¼)Y holds in the transaction set D with confidence c if c% of transactions in D that contain X also contain Y . The rule X¼)Y has supports in the transaction set D if s% of transactions in D contain XSY. For a given pair of confidence and support thresholds, the problem of mining association rules is to identify all association rules that have confidence and support greater than the corresponding minimum support threshold (de-noted as min supp) and minimum confidence threshold (denoted as min conf). Association rule mining algorithms
[2] work in two steps: 1) generate all frequent itemsets that satisfy min_supp and 2) generate all association rules that satisfy min_conf using the frequent itemsets. This problem can be reduced to the problem of finding all frequent itemsets for the same support threshold.
Since the early work in [2], several efficient algorithms to mine association rules have been developed in recent years. These studies cover a broad spectrum of topics including:
1. fast algorithms based on the level-wise Apriori
framework [4], [24], partitioning [20], [27], and sampling [31];
2. TreeProjection [1] and FP-growth algorithms [15]; 3. incremental updating [10], [18] and parallel
algo-rithms [3], [23];
4. mining of generalized and multilevel rules [14], [28]; 5. mining of quantitative rules [29];
6. mining of multidimensional rules [34];
7. constraint-based rule mining [32] and multiple
minimum supports issues [21];
8. associations among correlated or infrequent items [12]; and
9. temporal database discovery [5], [6], [9], [8], [22], [30]. While these are important results toward enabling the integration of association mining and fast searching algo-rithms, e.g., BFS and DFS which are classified in [16], we note that these mining methods cannot effectively be applied to the mining of a publication-like database which is of increasing popularity recently. In essence, a publica-tion database is a set of transacpublica-tions where each transacpublica-tion Tis a set of items of which each item contains an individual exhibition period. The current model of association rule . The authors are with the Department of Electrical Engineering, National
Taiwan University, Taipei, Taiwan, ROC.
E-mail: mschen@cc.ee.ntu.edu.tw, {chlee, owenlin}@arbor.ee.ntu.edu.tw. Manuscript received 25 Sept. 2001; revised 23 July 2002; accepted 5 Aug. 2002.
For information on obtaining reprints of this article, please send e-mail to: tkde@computer.org, and reference IEEECS Log Number 115072.
mining is not able to handle the publication database due to the following fundamental problems, i.e., 1) lack of consideration of the exhibition period of each individual item and 2) lack of an equitable support counting basis for each item. Note that the traditional mining process takes the same task-relevant tuples, i.e., the size of transaction set D, as a counting basis. Recall that the task of support specification is to specify the minimum transaction support for each itemset. However, since different items have different exhibition periods in a publication database, only consider-ing the occurrence count of each item might not lead to a fair measurement. This problem can be further explained by the two illustrative examples below.
Example 1.1. Consider an illustrative database of publica-tions in a bookstore, as shown in Fig. 1. Note that items A and B are exhibited from 1990 to 2001. However, item C is exhibited from 1992 to 2001 and item D is from 1994 to 2001. Even though each transaction item has a unique exhibition period, conventional mining algorithms, with-out further provision, tend to ignore such differences and determine frequent association rules with the same counting basis D.
Example 1.2.In a bookstore transaction database, as shown in Fig. 2, the minimum transaction support and confidence are assumed to be min supp ¼ 30% and min conf ¼ 75%, respectively. A set of time series database indicates the transaction records from January 2001 to March 2001. The publication date of each transaction item is also given. Based on the traditional mining techniques, the absolute support threshold is denoted as SA¼ d12 0:3e ¼ 4, where
12is the size of transaction set D. It can be seen that only B, C, D, E, and BC can be termed as frequent itemsets since the amounts of their occurrences in this transaction database are respectively larger than the absolute value of support threshold. Thus, only rule C¼)B is termed as a frequent association rule with support s ¼ 41:67% and confidence
c¼ 83:33%. However, some phenomena are observed
when we take the “item information” in Fig. 2 into consideration.
1. An early publication intrinsically possesses a
higher likelihood to be determined as a frequent itemset.For example, the sales volume of an early product, such as A, B, C, or D, is likely to be larger than that of a newly exhibited product, e.g., E or F, since an early product has a longer exhibition period. As a result, the association rules we usually get will be those with long-term products such as “milk and bread are frequently purchased
together,” which, while being correct by the definition, is of less interest to us in the association rule mining. In contrast, some more recent products, such as new books, which are really “frequent” and interesting in their exhibition periods, are less likely to be identified as frequent ones if a traditional mining process is employed. 2. Some discovered rules may be expired from users’
interest. Considering the generated rule C¼)B, both B and C were published from the very early dates of this mining transaction database. This information is very likely to have been explored in the previous mining database, such as the one from January 1996 to December 1997. Such mining results could be of less interest to our on-going mining works. For example, most researchers tend to pay more attention to the latest published papers. Note that one straightforward approach to addressing the above issues is to lower the value of the minimum support threshold required. However, this naive approach will cause another problem, i.e., those interesting rules with smaller supports may be overshadowed by lots of less important information with higher supports. As a consequence, we introduce the notion of exhibition period for each transaction item in this paper and develop an algorithm, Progressive Partition Miner (abbreviated as PPM), to address this problem. It is worth mentioning that the application domain of this study is not limited to the mining of a publication database. Other application domains include bookstore transaction databases, video and audio rental store records, stock market data, and transactions in electronic commerce, to name a few.
Explicitly, we explore in this paper the mining of general temporal association rules, i.e., ðX¼)Y Þt;n, where t is the latest-exhibition-start time of both itemsets X and Y and n denotes the end time of the publication database. In other words, ðt; nÞ is the maximal common exhibition period of itemsets X and Y . An association rule X¼)Y is termed to be a frequent general temporal association rule ðX¼)Y Þt;nif and only if its probability is larger than minimum support required, i.e., PðXt;n[ Yt;nÞ > min supp, and the conditional probability
PðYt;njXt;nÞ is larger than minimum confidence needed, i.e.,
PðYt;njXt;nÞ > min conf. Instead of using the absolute support
threshold SA¼ djDj min suppe as a minimum support
threshold for each item in Fig. 2, a relative minimum support, denoted by SR
X¼ djDXj min suppe, where jDXj indicates
the amount of partial transactions in the exhibition period of Fig. 1. An illustrative publication database where different items may
have different publication dates. Fig. 2. An illustrative transaction database and the corresponding item
itemset X, is given to deal with the mining of temporal association rules.
Example 1.3. Let us follow Example 1.2 and the given
minimum support and confidence thresholds. According to this newly identified support threshold SR
X, we have
temporal association rules as follows:
1. ðC¼)EÞ2;3with relative support 37.5 percent and confidence 75 percent,
2. ðE¼)CÞ2;3with relative support 37.5 percent and confidence 75 percent,
3. ðB¼)F Þ3;3 with relative support 75 percent and confidence 100 percent, and
4. ðF ¼)BÞ3;3 with relative support 75 percent and confidence 100 percent.
To deal with the mining of general temporal association rule ðX¼)Y Þt;n, an efficient algorithm, Progressive Partition Miner, is devised. The basic idea of PPM is to first partition the publication database in light of exhibition periods of items and then progressively accumulate the occurrence count of each candidate 2-itemset based on the intrinsic partitioning characteristics. Algorithm PPM is also designed to employ a filtering threshold in each partition to early prune out those cumulatively infrequent 2-itemsets. The feature that the number of candidate 2-itemsets generated by PPM is very close to the number of frequent 2-itemsets allows us to employ the scan reduction technique by generating Cks from C2
directly to effectively reduce the number of database scans. Experimental results show that PPM produces a significantly smaller amount of candidate 2-itemsets than Aprioriþ, i.e., an extended version of the Apriori algorithm. In fact, the number of the candidate itemsets Cks generated by PPM approaches
to its theoretical minimum, i.e., the number of frequent k-itemsets, as the value of the minimal support increases. Explicitly, the execution time of PPM is, in orders of magnitude, smaller than those required by Aprioriþ.
Sensi-tivity analysis on various parameters of the database is also conducted to provide many insights into Algorithm PPM.
The advantage of PPM over Aprioriþ becomes even more
prominent as the size of the database increases. This is indeed an important feature for PPM to be practically used for the mining of a time series database in the real world.
It is worth mentioning that the problem of mining general temporal association rules will be degenerated to the one of mining temporal association rules explored in prior works [5], [6], [8], [9], [30] if the exhibition period ðt; nÞ of association rule ðX¼)Y Þt;n is applied to a nonmaximal exhibition period of X¼)Y , such as ðj; nÞ, where j > t. Consider for example the database in Fig. 2, where ðC¼)BÞ1;3and ðC¼)EÞ2;3are two general temporal associa-tion rules in database D, while the temporal subset of ðC¼)BÞ1;3, e.g., ðC¼)BÞ2;3, can also be a temporal associa-tion rule as defined before [5], [6], [8], [9], [30], showing that the model we consider can be viewed as a general frame-work of prior studies. This is the very reason we use the term “general temporal association rule” in this paper.
We mention in passing that the works in [20], [27] are essentially based on a partition-based heuristic, i.e., if X is a frequent itemset in database D, which is divided into n partitions p1; p2; . . . ; pn, then X must be a frequent itemset in at least one of the
n partitions. However, these works were not applicable to handling the exhibition period of transaction items on mining
association rules. In addition, the Frequent Pattern growth (FP-growth), which constructs a highly compact data structure (an FP-tree) to compress the original transaction database, is a method of mining frequent itemsets without candidate generation [15]. However, in our opinion, FP-growth algorithms do not have obvious extensions to deal with this publication database problem, nor do those constraint-based rule mining methods that allow users to focus the search for rules by providing metarules [32]. Further, some methodologies were proposed to explore the problem of discovering temporal association relationships in the partial of database retrieved [5], [6], [8], [9], [11], [13], [17], [19], [26], [30], [33], i.e., to determine association rules from a given subset of database specified by time. These works, however, do not consider the individual exhibition period of each transaction item and are thus not applicable to solving the mining problems in a publication database. It is worth mentioning that, in this paper, we assume each item has the same cut-off date of the item exhibition period, i.e., the “n” of ðt; nÞ. This is different from the prior definition of “life span” in temporal association rule mining works [5] which may have different ending times of item exhibition periods. As will be seen later, the problem formulation with the same ending period enables us to derive very efficient and effective mining algorithms for temporal association rules.
On the other hand, some techniques were devised to use multiple minimum supports for frequent itemsets genera-tion [21]. However, it remains an open issue for how these techniques to be coupled with the corresponding minimum confidence thresholds when general temporal association rules we consider in this paper in a publication database are being generated. In this paper, we not only explore the new model of general temporal association rules in a publication database, but also propose an efficient Progressive Partition Miner methodology to perform the mining for this problem as well as conduct the corresponding performance studies. These features distinguish this paper from others.
The rest of this paper is organized as follows: Problem description is given in Section 2. Algorithm PPM is described in Section 3 with its correctness proven. Performance studies on various schemes are conducted in Section 4. This paper concludes with Section 5.
2
P
ROBLEMD
ESCRIPTIONTo facilitate our presentation, some definitions and symbols used are presented in Section 2.1. For further looking into the proposed problem of mining temporal association rules, the traversing of the search space is examined in Section 2.2. In addition, to assess the performance of PPM, we also present in Section 2.2 the concept of an extended version of the Apriori algorithm, called Aprioriþ, which will be employed in Section 4 later for performance comparison.
2.1 Preliminaries
Let n be the number of partitions with a time granularity, e.g., business-week, month, quarter, year, to name a few, in database D. In the model considered, dbt;n denotes the part of the
transaction database formed by a continuous region from partition Pt to partition Pn and jdbt;nj ¼Ph¼t;njPhj where
dbt;n D. An item xx:start;nis termed as a temporal item of x,
meaning that Px:startis the starting partition of x and n is the
Example 2.1.Consider the database in Fig. 2. Since database D records the transaction data from January 2001 to March 2001, database D is intrinsically segmented into three partitions P1, P2, and P3 in accordance with the
“month” granularity. As a consequence, a partial database db2;3 D consists of partitions P
2 and P3. A
temporal item E2;3denotes that the exhibition period of
E2;3is from the beginning time of partition P
2to the end
time of partition P3.
As such, we can define a maximal temporal itemset Xt;n
as follows:
Definition 1. An itemset Xt;n is called a maximal temporal
itemset in a partial database dbt;n if t is the latest starting
partition number of all items belonging to X in database D and nis the partition number of the last partition in dbt;nretrieved.
For example, as shown in Fig. 2, itemset DE2;3 is
deemed a maximal temporal itemset, whereas CD2;3is not.
In view of this, the exhibition period of an itemset is expressed in terms of Maximal Common exhibition Period (MCP) of the items that appear in the itemset. Let MCP ðxÞ denote the MCP value of item x. The MCP value of an itemset X is the shortest MCP among the items in itemset X. Consider three items C, E, and F in Fig. 2, for example. Their exhibition periods are as follows: MCP ðCÞ ¼ ð1; 3Þ,
MCPðEÞ ¼ ð2; 3Þ, and MCP ðF Þ ¼ ð3; 3Þ. Since itemset
CEF is termed to be CEF3;n¼ ðCEF Þ3;n
with considering the exhibition of CEF , we have MCP ðCEF Þ ¼ ð3; 3Þ.
In addition, jdbt;nj is the number of transactions in the
partial database dbt;n. The fraction of transaction T supporting
an itemset X with respect to partial database dbt;nis called the
support of Xt;n, i.e.,
supp X MCPðXÞ¼jfT 2 db
MCPðXÞjX T gj
jdbMCPðXÞj :
The support of a rule ðX¼)Y ÞMCPðXY Þis defined as suppðX¼)Y ÞMCPðXY Þ¼ supp ðX [YÞMCPðXY Þ: The confidence of this rule is defined as
confðX¼)Y ÞMCPðXY Þ¼
suppðXSYÞMCPðXY Þ supp Xð MCPðXY ÞÞ :
Consequently, a general temporal association rule
ðX¼)Y ÞMCPðXY Þ which holds in the transaction set D
can be defined as follows:
Definition 2. An association rule ðX¼)Y ÞMCPðXY Þis called a general temporal association rule in the transaction set D with
confðX¼)Y ÞMCPðXY Þ¼ c and
suppðX¼)Y ÞMCPðXY Þ¼ s
if c% of transactions in dbMCPðXY Þthat contain X also contain Y
and s% of transactions in dbMCPðXY Þ contain XSY, while
XTY ¼ .
For a given pair of min_conf and min_supp as the minimum thresholds required in the maximal common exhibition period of each association rule, the problem of mining general temporal association rules is to deter-mine all frequent general temporal association rules, e.g.,
ðX¼)Y ÞMCPðXY Þ2 dbMCPðXY Þ
which transaction itemsets X and Y have “relative” support and confidence greater than the corresponding thresholds. Thus, we have the following definition to identify the frequent general temporal association rules.
Definition 3.A general temporal association rule ðX¼)Y ÞMCPðXY Þ
is termed to be frequent if and only if
suppðX¼)Y ÞMCPðXY Þ> min supp and
confðX¼)Y ÞMCPðXY Þ> min conf:
Consequently, this rule mining of general temporal association can also be decomposed into to three steps:
1. Generate all frequent maximal temporal itemsets
(T Is) with their support values.
2. Generate the support values of all corresponding
temporal subitemsets (SIs) of frequent T Is.
3. Generate all temporal association rules that satisfy min_conf using the frequent T Is and/or SIs.
Example 2.2. Recall the illustrative general temporal
association rules, e.g., ðC¼)EÞ2;3 with relative support 37.5 percent and confidence 75 percent, in Example 1.3. In accordance with Definition 3, the implication ðC¼)EÞ2;3 is termed as a general temporal association rule if and only if suppððC¼)EÞ2;3Þ > min supp and confððC¼)EÞ2;3Þ > min conf. Consequently, we have to determine if suppðCE2;3Þ > min supp and suppðC2;3Þ >
min supp for discovering the newly identified associa-tion rule ðC¼)EÞ2;3. It is worth mentioning that though CE2;3 has to be a maximal temporal itemset, called T I, C2;3may not be a T I. We call C2;3is one of corresponding
temporal subitemsets, i.e., SI, of itemset CE2;3.
For better readability, a list of symbols used is given in Table 1. Then, the definition of a frequent maximal temporal itemset and the property of its corresponding subitemsets are given below.
Definition 4.A maximal temporal itemset XMCPðXÞ is termed
to be frequent when the occurrence frequency of XMCPðXÞ
is larger than the value of min supp required, i.e., suppðXMCPðXÞÞ > min supp, in transaction set dbMCPðXÞ.
Property 1. When a maximal temporal k-itemset XMCPðXkÞ
k
is frequent in data set dbMCPðXkÞ, each of its
correspond-ing subitemsets XMCPðXkÞ
i (1 i < k) is also frequent in
dbMCPðXkÞ.
Once F ¼ f XMCPðXÞ I j XMCPðXÞis frequentg, the set
of all frequent T Is and SIs together with their support values is known, deriving the desired association rules is
straightforward. For every XMCPðXÞ2 F , check the con-fidence of all rules ðX¼)Y ÞMCPðXY Þand drop those that do not satisfy sðXYMCPðXY ÞÞ=sðXMCPðXY ÞÞ min conf. This
problem can also be reduced to the problem of finding all frequent maximal temporal itemsets first and then generat-ing their correspondgenerat-ing frequent subitemsets for the same support threshold. Therefore, in the rest of this paper, we concentrate our discussion on the algorithms for mining frequent T Is and SIs. In fact, the process steps of generating frequent T Is and SIs can be further merged to one step in our proposed Algorithm PPM.
In addition, it is noted that users are likely to be interested in association rules whose exhibition periods are longer than a certain period. In view of this, we introduce a parameter of the minimum length of the exhibition period, denoted by min_leng, as a constraint in rule generation to reflect such a users’ requirement in the exhibition period. In other words, for each general temporal association rule ðX¼)Y ÞMCPðXY Þ produced, the value of MCP ðXY Þ should be larger than min_leng required, i.e., MCP ðXY Þ > min leng.
2.2 Traversing the Search Space
As explained, we have to find all maximal temporal itemsets that satisfy min supp first and then to calculate the occurrences of their corresponding subitemsets for producing all temporal association rules hidden in database D. However, if we use an existing algorithm to find all frequent T Is for this new problem, the downward closure property, which Apriori-based algorithms are based on, no longer holds. In addition, the candidate generation process is not intuitive at all. Note that, even though itemset Xt;nis
not a frequent itemset, it does not imply that Xtþ1;n, i.e., is
not a frequent itemset. In other words, even knowing Xt;nis
not frequent in dbt;n where MCP ðXÞ ¼ ðt; nÞ, we are not
able to assert whether XYtþ1;n is frequent or not when
MCPðY Þ ¼ ðt þ 1; nÞ. Specifically, to determine whether a general temporal association rule ðX¼)Y Þtþ1;n is frequent,
we have to find out the support values of Xtþ1;n and
XYtþ1;n where MCP ðXY Þ ¼ MCP ðY Þ ¼ ðt þ 1; nÞ.
It is worth mentioning that one may deal with the problem we consider with two naive procedures. The first one is to process the conventional mining algorithms in all kinds of combinatorial subdatabases, e.g., db1;3, db2;3, and
db3;3 in the foregoing example in Fig. 2, separately.
However, due to the huge search space involved, looking at all subsets of I , i.e., dbt;n for 1 t n, is too costly for
this approach to be practically used.
Further, since the downward level-wise property, which holds for Apriori-like algorithms, is not valid in this general temporal association rule mining problem, the second method is to expand each transaction item to be its combination with different exhibition periods. For instance, all temporal subitemsets of Xkt;n at level k with different exhibition periods, i.e., Xt;nk , Xtþ1;nk , Xktþ2;n; . . . ; Xn;nk , are taken as “temporal candidate k-itemsets” for producing any possible combination of general temporal association rules. Using this approach, the problem of mining temporal association rules can be implemented on an antimonotone Apriori-like heuristic. As in most previous works, the essential idea is to iteratively generate the set of candidate itemsets of length (k þ 1), i.e., Xkþ1r;n, from the set of frequent itemsets of length k, i.e., Xkr;n, (for k 1), and to check their corresponding occurrence frequencies in the database dbr;n.
This is the basic concept of an extended version of Apriori-based algorithm, called Apriori þ , whose performance will be comparatively evaluated with Algorithm PPM in our experimental studies later.
We next describe the search scenario of Apriori þ . For the special case I ¼ fA1;n, B1;n, X2;n, Y3;ng, we visualize the
search space that forms a lattice in Fig. 3. The frequent itemsets are located in the upper part of the figure whereas the infrequent ones are located in the lower part. Assume that the bold border separates the frequent itemsets from the infrequent ones. The basic principle of Apriori þ is to employ this border to efficiently prune the search space. As soon as the border line is found, we are able to restrict ourselves on determining the support values of the itemsets above the border and to ignore the itemsets below. However, it should be noted that a linearly growing number of temporal items still implies an exponential growing number of itemsets to be considered. In fact, as will be validated by experimental results later, the increase of candidates often causes a huge increase of execution time and a drastic performance degradation, meaning that without utilizing the partitioning and progressive support counting techniques we propose, a direct extension to priori work is not able to handle the general temporal association rule mining efficiently.
TABLE 1
3
M
ININGG
ENERALT
EMPORALA
SSOCIATIONR
ULESWe present an illustrative example for the operations of PPM in Section 3.1. A detailed description of Algorithm PPM is given in Section 3.2. The correctness of Algorithm PPM is proven in Section 3.3.
3.1 An Illustrative Example of Algorithm PPM
As explained above, a naive adoption of conventional methods to mine general temporal association rules will be prohibitively expensive. To remedy this, by partitioning a transaction database into several partitions, Algorithm PPM is devised to employ a filtering threshold in each partition to deal with the candidate itemset generation and process one partition at a time. For ease of exposition, the processing of a partition is termed a phase of processing. Explicitly, a progressive candidate set of itemsets is composed of the following two types of candidate itemsets, i.e., 1) the candidate itemsets that were carried over from the previous progressive candidate set in the previous phase and remain as candidate itemsets after the current partition is included into consideration (such candidate itemsets are called type candidate itemsets) and 2) the candidate itemsets that were not in the progressive candidate set in the previous phase, but are newly selected after only taking the current data partition into account (such candidate itemsets are called type candidate itemsets). Under PPM, the cumulative information in the prior phases is selectively carried over toward the generation of candidate itemsets in the subsequent phases. After the processing of a phase, Algorithm PPM outputs a progressive screen, denoted by P S, which consists of a
progressive candidate set of itemsets, their occurrence counts and the corresponding partial supports required.
The operation of algorithm PPM can be best understood by an illustrative example described below and its correspond-ing flowchart is depicted in Fig. 4. Recall the transaction database shown in Fig. 2, where the transaction database db1;3
is assumed to be segmented into three partitions P1, P2, and P3
which correspond to the three time granularities from January 2001 to March 2001. Suppose that min supp ¼ 30% and min conf¼ 75%. Each partition is scanned sequentially for the generation of candidate 2-itemsets in the first scan of the database db1;3. After scanning the first segment of four
transactions, i.e., partition P1, 2-itemsets fBD; BC; CD; ADg
are sequentially generated, as shown in Fig. 5. In addition, each potential candidate itemset c 2 C2 has two attributes:
1) c:start which contains the partition number of the corresponding starting partition when c was added to C2
and 2) c:count which contains the number of occurrences of c since c was added to C2. Since there are four transactions in P1,
the partial minimal support is d4 0:3e ¼ 2. Such a partial minimal support is called the filtering threshold in this paper. Itemsets whose occurrence counts are below the filtering threshold are removed. Then, as shown in Fig. 5, only fBD; BCg, marked by “ ,” remain as candidate itemsets (of type in this phase since they are newly generated) whose information is then carried over to the next phase P2 of
processing.
Similarly, after scanning partition P2, the occurrence
counts of potential candidate 2-itemsets are recorded (of type and type ). From Fig. 5, it is noted that since there are also four transactions in P2, the filtering threshold of
those itemsets carried out from the previous phase (that Fig. 3. Traversing the search space for existing algorithms, e.g., Apriori-like algorithms.
become type candidate itemsets in this phase) is dð4 þ 4Þ 0:3e ¼ 3 and that of newly identified candidate itemsets (i.e., type candidate itemsets) is d4 0:3e ¼ 2. It can be seen that we have three candidate itemsets in C2 after the
processing of partition P2, and one of them is of type and
two of them are of type .
Finally, partition P3is processed by Algorithm PPM. The
resulting candidate 2-itemsets are C2¼ fBC; CE; BF g as
shown in Fig. 5. Note that though appearing in the previous phase P2, itemset fDEg is removed from C2once P3is taken
into account since its occurrence count does not meet the filtering threshold then, i.e., 2 < 3. However, we do have one new itemset, i.e., BF , which joins the C2 as a type
candidate itemset. Consequently, we have three candidate 2-itemsetsgenerated by PPM and two of them are of type and one of them is of type . Note that only three candidate
2-itemsets are generated by PPM. The correctness of
Algorithm PPM will be formally proven later.
After generating C2from the first scan of database db1;3,
we employ the scan reduction technique [24] and use C2to
generate Ck(k ¼ 2; 3; . . . ; m), where Cmis the candidate
last-itemsets. Instead of generating C3 from L2? L2, a C2
generated by PPM can be used to generate the candidate
3-itemsetsand its sequential Ckÿ10 can be utilized to generate
Ck0. Clearly, a C30 generated from C2? C2, instead of from
L2? L2, will have a size greater than jC3j where C3 is
generated from L2? L2. However, since the jC2j generated by
PPM is very close to the theoretical minimum, i.e., jL2j, the
jC0
3j is not much larger than jC3j. Similarly, the jCk0j is close to
jCkj. Since C2¼ fBC; CE; BF g, no candidate k-itemset is
generated in this example where k 3. Thus, C0
k¼
fBC; CE; BF g and all C0
k can be stored in main memory.
Then, we can find Lks (k ¼ 1; 2; . . . ; m) together when the
second scan of the database db1;3 is performed. Note that
those generated itemsets C0
k¼ fBC; CE; BF g are termed to
be the candidate maximal temporal itemsets (T Is), i.e., BC1;3; CE2;3, and BF3;3, with a maximal exhibition period of
each candidate.
Before we process the second scan of the database db1;3to
generate Lks, all candidate SIs of candidate T Is can be
propagated based on Property 1, and then added into C0 k. For
instance, as shown in Fig. 5, both candidate 1-itemsets B1;3and
C1;3 are derived from BC1;3. Moreover, since BC1;3, for
example, is a candidate 2-itemset, its subsets, i.e., B1;3and
C1;3, should potentially be candidate itemsets. As a result,
nine candidate itemsets, i.e., fB1;3, B3;3, C1;3, C2;3, E2;3, F3;3,
BC1;3, BF3;3, CE2;3g, as shown in Fig. 5, are generated. Note
that since there is no candidate T I k-itemset (k 2) contain-ing A or D in this example, Ai;3and Di;3(1 i 3) are not
necessary to be taken as SI itemsets for generating general temporal association rules. In other words, we can skip them from the set of candidate itemsets C0
ks. Finally, all occurrence
counts of C0
ks can be calculated by the second database scan.
Note that itemsets BC1;3, BF3;3, and CE2;3 are termed as
frequent T Is, while B1;3, B3;3, C1;3, C2;3, E2;3, and F3;3 are
frequent SIs in this example.
As shown in Fig. 5, after all frequent T I and SI itemsets are identified, the corresponding general temporal association rules can be derived in a straightforward manner. Explicitly, the general temporal association rule of ðX ) Y Þi;nholds if confððX ) Y Þi;nÞ min conf.
3.2 Algorithm of PPM
Initially, a publication database D is partitioned into n partitions based on the exhibition periods of items and P S, i.e., progressive screen, is empty. Let C2 be the set of
progressive candidate 2-itemsets generated by database D. Three parameters, i.e., n, min supp, and min leng, are taken as the input values into Algorithm PPM. As mentioned above, the minimum support threshold required is denoted as min supp. In the process of general temporal association rule generation, we employ the parameter min leng to be a filtering threshold for frequent itemsets to satisfy the minimal length required for the exhibition period. The procedure of Algorithm PPM is outlined below, where Algorithm PPM is decomposed into five subprocedures for ease of description.
AlgorithmPPM (n, min_supp, min_leng):
Progressive Partition Miner
Initial Subprocedure: The database D is partitioned into npartitions and set P S ¼ ;
1. jdb1;nj ¼P
h¼1;njPhj; // db1;nis partitioned into
npartitions 2. P S ¼ ;;
Subprocedure I: Generate 2nd level candidate T Is with progressive screen
3. begin for h ¼ 1 to n // 1st scan of db1;n 4. begin for each 2-itemset Xt;n2 2 Phwhere
nÿ t > min leng
5. if (X22 P S)=
6. X2:count¼ NphðX2Þ;
7. X2:start¼ h;
8. if (X2:count min supp jPhj)
9. P S¼ P S [ X2;
10. if (X22 P S)
11. X2:count¼ X2:countþ NphðX2Þ;
12. if (X2:count <dmin supp Pm¼X2:start;hjPmje)
13. P S¼ P S ÿ X2;
14. end
15. end
16. select C2 from X2where X22 P S;
17. P S ¼ ;;
Subprocedure II: Generate candidate T Is and SIs with the scheme of database scan reduction
18. begin while (Ck6¼ ; & k 2)
19. Ckþ1¼ Ck? Ck; // where ? indicate the
operation of convolution 20. k¼ k þ 1; 21. end 22. Xt;nk ¼ fX t;n k XkjXk2 Ckg; // Candidate T Is generation 23. SIðXt;nk Þ ¼ fXt;nj subset of Xt;nk jj < kg;
// Candidate SIs of T Is generation 24. P S ¼ P S [ SIðXt;nk Þ;
25. select Xkt;ninto Ckwhere Xkt;n2 P S;
Subprocedure III: Generate all frequent T Is and SIs with the 2nd scan of database D
26. begin for h ¼ 1 to n
27. for each itemset Xkt;n2 Ck
28. Xt;nk :count¼ Xt;nk :countþ NphðX
t;n k Þ;
29. end
30. for each itemset Xkt;n2 Ck
31. if (Xkt;n:count dmin supp jdbt;nje)
32. Lk¼ Lk[ Xkt;n;
33. end
Subprocedure IV: Prune out the redundant frequent SIs from Lk
34. for each SI itemset Xkt;n2 Lk
35. if (=9 T I itemset Xjt;n Ljj j > k)
36. Lk¼ Lkÿ Xt;nk ;
37. end 38. return Lk;
Initially, the database db1;nis partitioned into n partitions
by executing the Subprocedure I (in Step 1), and P S, i.e., progressive screen, is empty (in Step 2). In essence, Subprocedure I first scans partition pi, for i ¼ 1 to n, to
find the set of all local frequent 2-itemsets in pi. Note that
P S is a superset of the set of all frequent 2-itemsets in D. Algorithm PPM constructs P S incrementally by adding candidate 2-itemset to P S and starts counting the number
of occurrences for each candidate 2-itemset X2 in P S
whenever X2is added to P S. If the cumulative occurrences
of a candidate 2-itemset X2 does not meet the partial
minimum support required, X2 is removed from the
progressive screen P S. From Step 3 to Step 15 of Subprocedure I, Algorithm PPM processes one partition at a time for all partitions. When processing partition Pi, each
potential candidate 2-itemset X2 is read and saved to P S,
where its exhibition period, i.e., n ÿ t, should be larger than the minimum constraint exhibition period min leng re-quired. The number of occurrences of an itemset X2and its
starting partition which keeps its first occurrence in P S are recorded in X2:countand X2:start, respectively. As such, in
the end of processing db1;h, only an itemset, whose
X2:count dmin supp Pm¼X2:start;hjPmje, will be kept in
P S. Note that a large amount of infrequent T I candidates will be further reduced with the early pruning technique by this progressive portioning processing. Next, in Step 16, we select C2from X22 P S and set P S ¼ ; in Step 17.
In Subprocedure II, with the scan reduction scheme [24], C2
produced by the first scan of database is employed to generate Cks ðk 3Þ in main memory from Step 18 to Step 21. Recall
that Xt;nk is a maximal temporal k-itemset in a partial database dbt;n. In Step 22, all candidate T Is, i.e., Xt;n
k s, are generated
from Xk2 Ckwith considering the maximal common
exhibi-tion period of itemset Xk, where MCP ðXkÞ ¼ ðt; nÞ. After
that, from Step 23 to Step 25, we generate all corresponding temporal subitemsets of Xt;nk , i.e., SIðXkt;nÞ, to join into P S.
Then, from Step 26 to Step 33 of Subprocedure III, we begin the second database scan to calculate the support of each itemset in P S and to find out which candidate itemsets are really frequent T Is and SIs in database D. As a result, those itemsets whose Xkt;n:count dmin supp jdbt;nje are
the frequent temporal itemsets Lks.
Finally, in Subprocedure IV, we have to prune out those redundant frequent SIs whose T I itemsets are not frequent in database D from the Lks. As will be proven in Section 3.3,
the output of Algorithm PPM consists of frequent itemsets Lks of database D. According to these output Lks in Step 38,
all kinds of general temporal association rules implied in database D can be generated in a straightforward method.
Note that PPM is able to filter out false candidate itemsets in Piwith a hash table. Same as in [24], using a hash table to
prune candidate 2-itemsets, i.e., C2, in each accumulative
ongoing partition set Piof transaction database, the CPU and
memory overhead of PPM can be further reduced. As will be validated by our experimental studies, PPM provides very efficient solutions for mining general temporal association rules. This feature is, as described earlier, very important for mining the publication-like databases whose data are being exhibited from different starting times.
In addition, the progressive screen produced in each processing phase constitutes the key component to realize the mining of general temporal association rules. Note that Algorithm PPM proposed has several important advan-tages, including 1) with judiciously employing progressive knowledge in the previous phases, PPM is able to reduce the amount of candidate itemsets efficiently, which in turn reduces the CPU and memory overhead, and 2) owing to the small number of candidate sets generated, the scan reduction technique can be applied efficiently. As a result, only two scan of the time series database is required.
3.3 Correctness of PPM
We now prove the correctness of Algorithm PPM. Let NphðXÞ be the number of transactions in partition Ph that
contain itemset X, and jPhj is the number of transactions in
partition Ph:Also, let dbi;jdenote the part of the transaction
database formed by a continuous region from partition Pito
partition Pj, and jdbi;jj ¼Ph¼i;jjPhj. We can then define the
region ratio of an itemset as follows:
Definition 5.A region ratio of an itemset X for the transaction database dbi;j, denoted by ri;jðXÞ; is
ri;jðXÞ ¼
P
h¼i;jNphðXÞ
jdbi;jj :
In essence, the region ratio of an itemset is the support of that itemset if only the part of transaction database dbi;jis considered. Lemma 1.A 2-itemset X2remains in the P S after the processing
of partition Pjif and only if there exists an i such that for any
integer t in the interval ½i; j, ri;tðX2Þ min supp, where
min suppis the minimal support required.
Proof of Lemma 1. We shall prove the “if” condition first. Consider the following two cases. First, suppose the 2-itemset X2is not in the progressive candidate set before
the processing of partition Pi. Since ri;iðX2Þ min supp,
itemset X2will be selected as a type candidate itemset by
PPM after the processing of partition Pi. On the other hand,
if the itemset X2is already in the progressive candidate set
before the processing of partition Pi, itemset X2 will
remain as a type candidate itemset by PPM. Clearly, for
the above two cases, itemset X2 will remain in P S
throughout the processing from Pi to Pj since for any
integer t in the interval ½i; j, ri;tðX2Þ min supp.
We now prove the “only if” condition, i.e., if X2
remains in P S after the processing of partition Pj, then
there exists an i such that for any t in the interval ½i; j, ri;tðX2Þ min supp. Note that itemset X2 can be either
type or type candidate itemset in the P S after the processing of partition Pj. Suppose X2 is a type
candidate itemset there, then this implication follows by setting j ¼ i since ri;iðX2Þ min supp. On the other hand,
suppose that X2 is a type candidate itemset after the
processing of Pj, which means itemset X2has become a
type candidate itemset in a previous phase. Then, we shall trace backward the type of itemset X2from partition
Pj(i.e., looking over Pj, Pjÿ1, Pjÿ2, etc.) until the partition
that records itemset X2 as a type candidate itemset is
first encountered. (It should be noted that there could be two discontinuous regions that record itemset X2 in the
P S, which means that an itemset may get on and off the progressive candidate set through the processing of partitions. This, in turn, means that an itemset may appear as a type candidate itemset more than once.) By referring the partition identified above as partition Pi, we
have, for any t in the interval ½i; j, ri;tðX2Þ min supp,
completing the proof of this lemma. tu
Lemma 1 leads to Lemma 2.
Lemma 2. An itemset X2remains in P S after the processing of
partition Pj if and only if there exists an i such that
ri;jðX2Þ min supp, where min supp is the minimal support
required.
Proof of Lemma 2.It can be seen that the proof of “only if” condition follows directly from Lemma 1. We now prove the “if” condition of this lemma. If there exists an i such that ri;jðX2Þ min supp, then we let u be the largest v
such that ri;vðX2Þ < min supp. If such a u does not exist,
it follows from Lemma 1 that itemset X2will remain in
P Safter the processing of partition Pj. If such a u exists,
we have ruþ1;jðX2Þ min supp since ri;uðX2Þ < min supp
and ri;jðX2Þ min supp. It again follows from Lemma 1
that itemset X2will remain in P S after the processing of
partition Pj. This lemma follows: tu
Lemma 2 leads to the following theorem that states the completeness of candidates 2-itemsets generated by the first scan of transaction database db1;nwith Algorithm PPM. Theorem 1. If there exits a frequent itemset Xt;n2 in the
transaction database dbt;n such that r
then X2 will be in the progressive candidate set of itemsets
produced by Algorithm PPM.
Proof of Theorem 1.Let n be the number of partitions of the transaction database. Since the itemset X2t;nis a frequent itemset, we have rt;nðX2Þ min supp, which is in essence
a special case of Lemma 2 for i ¼ t and j ¼ n, proving this
theorem. tu
Furthermore, we let Ci;j, i j, be the set of progressive
candidate itemsets generated by Algorithm PPM with respect to database dbi;jafter the processing of Pj. We then
have the following lemma.
Lemma 3.For i t j, then Ct;j Ci;j.
Proof of Lemma 3. Assume that there exists a 2-itemset
X22 Ct;j. From the“only if” implication of Lemma 2, it
follows that there exists an h such that rh;jðX2Þ s, where
t h j. Since i t j, we have i h j. Then, according to the “if” implication of Lemma 2, itemset X2 is also in Ci;j, i.e., X22 Ci;j. The fact that Ct;j Ci;j
follows. tu
Theorem 1 and Lemma 3 lead to the following theorem which states the correctness of Algorithm PPM.
Theorem 2. If there exists a frequent k-itemset Xt;nk in the transaction database dbt;n such that r
t;nðXkÞ s, then Xt;nk
will be produced by Algorithm PPM.
Proof of Theorem 2.Since itemset Xt;nk is frequent, we have
rt;nðXkÞ min supp. As mentioned above, all of its
sub-itemsets Xht;ns ðh < kÞ will be frequent with rt;nðXhÞ s.
Specifically, Xt;n2 s are in essence special cases of X t;n h s with
h¼ 2. Consequently, according to the implication of Theorem 1, X2s will be in the progressive candidate set of
itemsets, i.e., P S, produced by Algorithm PPM. As such, based on an antimonotone Apriori-like heuristic, i.e., if any length k itemset Xi;nk is not frequent in the database, its length
(k þ 1) superitemset Xi;nkþ1will never be frequent, the super-itemset Xkt;nof X2t;nwill be produced by Algorithm PPM,
proving this theorem. tu
Further, if there exists a frequent T I 3-itemset ABCt;n, for
example, in the transaction database dbt;n, meaning that
rt;nðABCÞ min supp, then we have rt;nðABÞ min supp,
rt;nðACÞ min supp, and rt;nðBCÞ min supp. According
to Theorem 1, we learn that all SIs of ABCt;n, i.e., ABt;n, ACt;n,
BCt;n, At;n, Bt;n, and Ct;n, will be in the progressive candidate
set of itemsets produced by Algorithm PPM. Consequently, Theorem 2 states the correctness of Algorithm PPM.
4
E
XPERIMENTALS
TUDIESTo assess the performance of Algorithm PPM, we performed several experiments on a computer with a CPU clock rate of 450 MHz and 512 MB of main memory. The transaction data resides in the NTFS file system and is stored on a 30GB IDE 3.5” drive with a measured sequential throughput of 10MB/second. The simulation program was coded in C++. The methods used to generate synthetic data are described in Section 4.1. The performance comparison of PPM and Aprioriþ is presented in Section 4.2. Section 4.3 shows the
I/O cost and CPU overhead for PPM and Aprioriþ. Results on scaleup experiments are presented in Section 4.4.
4.1 Generation of Synthetic Workload
For obtaining reliable experimental results, the method to generate synthetic transactions we employed in this study is similar to the ones used in prior works [4], [24]. Explicitly, we generated several different transaction databases from a set of potentially frequent itemsets to evaluate the perfor-mance of PPM. These transactions mimic the publication items in a publication database. Note that the efficiency of Algorithm PPM has been evaluated by some real databases, such as bookstore transaction databases and grocery sales data. However, we show the experimental results from synthetic transaction data so as to obtain results of different workload parameters. Each database consists of jDj transactions and, on the average, each transaction has jT j items. To simulate the characteristic of the exhibition period in each item, transaction items are uniformly distributed into database D with a random selection. In accordance with the exhibition periods of items, database D is divided into n partitions. Table 2 summarizes the meanings of various parameters used in the experiments. The mean of the correlation level is set to 0:25 for our experiments. Without loss of generality, we use the notation T x ÿ Iy ÿ
Dm to represent a database in which D ¼ m thousands,
jT j ¼ x, and jIj ¼ y. We compare relative performance of Aprioriþand PPM.
As mentioned before, the Aprioriþ algorithm is an
extended version of Apriori-like algorithms to deal with the mining problem in publication databases. As will be shown by our experimental results, with the progressive partition technique that carries cumulative information selectively, the execution time of PPM is, in orders of magnitude, smaller than that required by Aprioriþ.
4.2 Experiment One: Relative Performance
We first conducted several experiments to evaluate the
relative performance of Aprioriþ and PPM. As shown in
Fig. 6, the experimental results are consistent for various values of n, jLj and N on data set D100, e.g., T10-I4-D100(N20-L4-n20). For interest of space, we only report the results on jLj ¼ 2; 000 and N ¼ 10; 000 in the following experiments. In addition, the number of partitions in the database is selected as n ¼ 10. Fig. 6 shows the relative execution times for both two algorithms as the minimum support threshold is decreased from 1 percent support to 0.1 percent support. When the support threshold is high, there are only a limited number of frequent itemsets
TABLE 2
produced. However, as the support threshold decreases, the performance difference becomes prominent in that PPM significantly outperforms Aprioriþ. Explicitly, PPM is in
orders of magnitude faster than Aprioriþ, and the margin
grows as the minimum support threshold decreases. In fact, PPM outperforms Apriori in both CPU and I/O costs, which are evaluated next.
4.3 Experiment Two: Evaluation of I/O Cost and
CPU Overhead
To evaluate the corresponding of I/O cost, same as in [25], we assume that each sequential read of a byte of data consumes one unit of I/O cost and each random read of a byte of data consumes two units of I/O cost. Fig. 7a shows the number of
database scans and the I/O costs of Aprioriþand PPM over the data set T10-I4-D100. As shown in Fig. 7a, PPM outperforms Aprioriþ. Note that the large amount of database scans is the performance bottleneck when the database size does not fit into main memory. In view of that, PPM is advantageous since only two scan, of the publication database is required, which is independent of the variance in minimum supports.
As explained before, PPM substantially reduces the number of candidate itemsets generated. The effect is particularly important for the candidate 2-itemsets. The experimental results in Fig. 7b show the candidate itemsets generated by Aprioriþand PPM across the whole processing
on the data set T10-I4-D100 with minimum support threshold min supp¼ 0:2 percent. As shown in Fig. 7b, PPM leads to a Fig. 6. Relative performance studies.
96 percent candidate reduction rate in C2 when being
compared to Aprioriþ. This feature of PPM enables it to
efficiently reduce the CPU and memory overhead. Note that the number of candidate 2-itemsets produced by PPM approaches to its theoretical minimum, i.e., the number of large 2-itemsets. Recall that the C3 in Aprioriþ has to be
obtained by L2due to the large size of their C2. As shown in
Fig. 7b, the value of jCkj (k 3) is only slightly larger than that
of Aprioriþ, even though PPM only employs C
2to generate
Cks, thus fully exploiting the benefit of scan reduction.
4.4 Experiment Three: Scaleup Performance
In this experiment, we examine the scaleup performance of Algorithm PPM. The scale-up results for different selected data sets are obtained. Fig. 8 shows the scale-up perfor-mance of Algorithm PPM as the values of jDj increase. Three different minimum supports are considered. We obtained the results for the data set T10-I4-Dm when the number of customers increases from 100,000 to one million. The execution times are normalized with respect to the times for the 100,000 transactions data set in the Fig. 8a. Note that, as shown in Fig. 8a the execution time only slightly increases with the growth of the database size, showing good scalability of PPM.
To further understand the impact of jDj to the relative performance of algorithms PPM and Aprioriþ algorithms,
we conduct the scale-up experiments for both PPM and Aprioriþwith two minimum support thresholds 0.2 percent and 0.4 percent. The results are shown in Fig. 8b where the value in y-axis corresponds to the ratio of the execution time of PPM to that of Aprioriþ. Fig. 8b shows the referenced
ratio obtained from a publication-like database over data sets of T10-I4-Dm. The execution-time-ratio of PPM to Aprioriþdecreases when the amount of database jDj grows larger, meaning that the advantage of PPM over Aprioriþ increases as the database size increases.
5
C
ONCLUSIONIn this paper, we not only explored a new model of mining general temporal association rules, i.e., ðX ) Y ÞMCPðXY Þ, in a publication database, but also developed Algorithm PPM to generate the temporal association rules as well as conducted related performance studies. Under PPM, the cumulative information of mining previous partitions is selectively carried over toward the generation of candidate itemsets for the subsequent partitions. Algorithm PPM not only signifi-cantly reduced I/O and CPU cost by the concepts of Fig. 7. I/O cost and CPU overhead performance. (a) I/O cost performance over data set T10-I4-D100. (b) Number of candiate itemsets on T10-I4-D100.
Fig. 8. Scale-up performance of PPM and the execution time ratio between PPM and Aprioriþ. (a) Scale-up performance in various value ofjDj. (b) Execution time ratio in various value ofjDj.
progressive counting and scan reduction techniques, but also effectively controlled memory utilization by proper parti-tioning. Algorithm PPM is particularly powerful for efficient mining for a publication-like transaction database, such as bookstore transaction databases, video rental store records, library-book rental records, and transactions in electronic commerce. The correctness of PPM is proven and some of its theoretical properties are derived. Extensive simulations have been performed to evaluate performance of Algorithm PPM. Sensitivity analysis of various parameters was con-ducted to provide many insights into Algorithm PPM. It was noted that the improvement achieved by PPM increases as the size of the database increases.
A
CKNOWLEDGMENTSThe authors are supported in part by the Ministry of Education project no. 89-E-FA06-2-4-7 and the National Science Council, project nos. NSC 91-2213-E-002-034 and NSC 91-2213-E002-045, Taiwan, Republic of China.
R
EFERENCES[1] R. Agarwal, C. Aggarwal, and V.V.V. Prasad, “A Tree Projection Algorithm for Generation of Frequent Itemsets,” J. Parallel and Distributed Computing, 2000.
[2] R. Agrawal, T. Imielinski, and A. Swami, “Mining Association Rules between Sets of Items in Large Databases,” Proc. ACM SIGMOD, pp. 207-216, May 1993.
[3] R. Agrawal and J.C. Shafer, “Parallel Mining of Association Rules: Design, Implementation, and Experience,” IEEE Trans. Knowledge and Data Eng., pp. 487-499, Dec. 1996.
[4] R. Agrawal and R. Srikant, “Fast Algorithms for Mining
Association Rules in Large Databases,” Proc. 20th Int’l Conf. Very Large Data Bases, pp. 478-499, Sept. 1994.
[5] J.M. Ale and G. Rossi, “An Approach to Discovering Temporal Association Rules,” Proc. ACM Symp. Applied Computing, 2000. [6] C. Bettini, X.S. Wang, and S. Jajodia, “Mining Temporal
Relation-ships with Multiple Granularities in Time Sequences,” Bull. IEEE Computer Soc. Technical Committee on Data Eng., 1998.
[7] M.-S. Chen, J. Han, and P.S. Yu, “Data Mining: An Overview from Database Perspective,” IEEE Trans. Knowledge and Data Eng., vol. 8, no. 6, pp. 866-883, Dec. 1996.
[8] X. Chen and I. Petr, “Discovering Temporal Association Rules: Algorithms, Language, and System,” Proc. 2000 Int’l Conf. Data Eng., 2000.
[9] X. Chen, I. Petrounias, and H. Heathfield, “Discovery of
Association Rules in Temporal Databases,” Proc. Issues and Applications of Database Technology, 1998.
[10] D. Cheung, J. Han, V. Ng, and C.Y. Wong, “Maintenance of Discovered Association Rules in Large Databases: An Incremental Updating Technique,” Proc. 1996 Int’l Conf. Data Eng., pp. 106-114, Feb. 1996.
[11] G. Dong and J. Li, “Efficient Mining of Emerging Patterns: Discovering Trends and Differences,” Knowledge Discovery and Data Mining, pp. 43-52, 1999.
[12] E. Cohen et al., “Finding Interesting Associations without Support Pruning,” IEEE Trans. Knowledge and Data Eng., vol. 13, no. 1, pp. 64-78, Jan./Feb. 2001.
[13] J. Han, G. Dong, and Y. Yin, “Efficient Mining of Partial Periodic Patterns in Time Series Database,” Proc. 15th Int’l Conf. Data Eng., pp. 106-115, Mar. 1999.
[14] J. Han and Y. Fu, “Discovery of Multiple-Level Association Rules from Large Databases,” Proc. 21st Int’l Conf. Very Large Data Bases, pp. 420-431, Sept. 1995.
[15] J. Han and J. Pei, “Mining Frequent Patterns by Pattern-Growth: Methodology and Implications,” ACM SIGKDD Explorations, Dec. 2000.
[16] J. Hipp, U. Guntzer, and G. Nakhaeizadeh, “Algorithms for Association Rule Mining—A General Survey and Comparison,” ACM SIGKDD Explorations, vol. 2, no. 1, pp. 58-64, July 2000.
[17] E. Keogh, K. Chakrabarti, S. Mehrotra, and M. Pazzani, “Locally Adaptive Dimensionality Reduction for Indexing Large Time Series Databases,” Proc. ACM-SIGMOD Conf. Management of Data, 2001.
[18] C.-H. Lee, C.-R. Lin, M.-S. Chen, “Sliding-Window Filtering: An Efficient Algorithm for Incremental Mining,” Proc. ACM 10th Int’l Conf. Information and Knowledge Management, Nov. 2001.
[19] Y. Li, P. Ning, X.S. Wang, and S. Jajodia, “Discovering Calendar-Based Temporal Association Rules,” TIME, pp. 111-118, 2001. [20] J.-L. Lin and M.H. Dunham, “Mining Association Rules:
Anti-Skew Algorithms,” Proc. Int’l Conf. Data Eng., pp. 486-493, 1998. [21] B. Liu, W. Hsu, and Y. Ma, “Mining Association Rules with
Multiple Minimum Supports,” Proc. Int’l Conf. Knowledge Discovery and Data Mining, Aug. 1999.
[22] C.C. Liu, J.L. Hsu, and A.L.P. Chen, “Efficient Theme and Non-Trivial Repeating Pattern Discovering in Music Databases,” Proc. IEEE Int’l Conf. Data Eng., 1999.
[23] J.-S. Park, M.-S. Chen, and P.S. Yu, “Mining Association Rules with Adjustable Accuracy,” Proc. ACM Sixth Int’l Conf. Information and Knowledge Management, pp. 151-160, Nov. 1997.
[24] J.-S. Park, M.-S. Chen, and P.S. Yu, “Using a Hash-Based Method with Transaction Trimming for Mining Association Rules,” IEEE Trans. Knowledge and Data Eng., vol. 9, no. 5, pp. 813-825, Oct. 1997. [25] J. Pei, J. Han, and L.V.S. Lakshmanan, “Mining Frequent Itemsets
with Convertible Constraints,” Proc. Int’l Conf. Data Eng., 2001.
[26] J.F. Roddick and M. Spiliopoulou, “A Survey of Temporal
Knowledge Discovery Paradigms and Methods,” IEEE Trans. Knowledge and Data Eng., pending publication, 2000.
[27] A. Savasere, E. Omiecinski, and S. Navathe, “An Efficient Algorithm for Mining Association Rules in Large Databases,” Proc. 21st Int’l Conf. Very Large Data Bases, pp. 432-444, Sept. 1995. [28] R. Srikant and R. Agrawal, “Mining Generalized Association Rules,” Proc. the 21st Int’l Conf. Very Large Data Bases, pp. 407-419, Sept. 1995.
[29] R. Srikant and R. Agrawal, “Mining Quantitative Association Rules in Large Relational Tables,” Proc. ACM-SIGMOD Conf. Management of Data, 1996.
[30] A.U. Tansel and N.F. Ayan, “Discovery of Association Rules in Temporal Databases,” Proc. AAAI Knowledge Discovery in Data-bases, 1998.
[31] H. Toivonen, “Sampling Large Databases for Association Rules,” Proc. 22nd Very Large Data Base Conf., pp. 134-145, Sept. 1996. [32] A.K.H. Tung, J. Han, L.V.S. Lakshmanan, and R.T. Ng,
“Con-straint-Based Clustering in Large Databases,” Proc. 2001 Int’l Conf. Database Theory, Jan. 2001.
[33] R. Villafane, K.A. Hua, D. Tran, and B. Maulik, “Mining Interval Time Series,” Data Warehousing and Knowledge Discovery, pp. 318-330, 1999.
[34] C. Yang, U. Fayyad, and P. Bradley, “Efficient Discovery of Error-Tolerant Frequent Itemsets in High Dimensions,” Proc. Seventh ACM SIGKDD Int’l Conf. Knowledge Discovery and Data Mining, 2001. Chang-Hung Lee received the BS and PhD degrees in electrical engineering from the Na-tional Taiwan University, Taipei, in 1996 and 2002, respectively. Dr. Lee has published in ACM SIGKDD, IEEE ICDCS, IEEE ICDM, and SIAM SDM. He is currently a research staff member at BenQ Inc., Taipei, Taiwan, leading projects in wireless multimedia applications. His research interests include distributed database systems, data mining, mobile computing sys-tems, wireless multimedia, and flat panel display.
Ming-Syan Chen received the BS degree in electrical engineering from the National Taiwan University, Taipei, and the MS and PhD degrees in computer, information, and control engineering from The University of Michigan, Ann Arbor, in 1985 and 1988, respectively. Dr. Chen is currently a professor in the Electrical Engineering Department, National Taiwan University, Taipei. He was a research staff member at IBM Thomas J. Watson Research Center, Yorktown Heights, New York, from 1988 to 1996. His research interests include database systems, data mining, mobile computing systems, and multimedia networking, and he has published more than 160 papers in his research areas. In addition to serving as program committee members in many conferences, Dr. Chen served as an associate editor of IEEE Transac-tions on Knowledge and Data Engineering on data mining and parallel database areas from 1997 to 2001, an editor of Journal of Information Science and Engineering, a distinguished visitor of IEEE Computer Society for Asia-Pacific from 1998 to 2000, and program chair of PAKDD-02 (Pacific Area Knowledge Discovery and Data Mining), program vice-chair of VLDB-2002 (Very Large Data Bases) and ICPP 2003, general chair of Real-Time Multimedia System Workshop in 2001, program chair of IEEE ICDCS Workshop on Knowledge Discovery and Data Mining in the World Wide Web in 2000, and program cochairs of the International Conference on Mobile Data Management in 2003, International Computer Symposium (ICS) on Computer Networks, Internet and Multimedia in 1998 and 2000, and ICS on Databases and Software Engineering in 2002. He was a keynote speaker on Web data mining at the International Computer Congress in Hong Kong, 1999, a tutorial speaker on Web data mining in DASFAA-1999 and on parallel databases in the 11th IEEE International Conference on Data Engineering in 1995 and also a guest coeditor for IEEE Transactions on Knowledge and Data Engineering on a special issue for data mining in December 1996. He holds, or has applied for, 18 US patents and seven ROC patents in the areas of data mining, Web applications, interactive video playout, video server design, and concurrency and coherency control protocols. He received the Outstanding Innovation Award from IBM Corporate in 1994 for his contribution to parallel transaction design and implementation for a major database product, and has received numerous awards for his research, teaching, inventions, and patent applications. Dr. Chen is a senior member of the IEEE, the IEEE Computer Society, and a member of ACM.
Cheng-Ru Lin received the BS degree in electrical engineering from the National Taiwan University, Taipei, in 1999. He is currently a PhD candidate in Electrical Engineering Department, National Taiwan University, Taipei and is ex-pected to graduate in the summer of 2003. He has published in the IEEE Transactions on Knowledge and Data Engineering, IEEE Trans-actions on Parallel and Distributed Systems, ICDM, ACM SIGKDD, CIKM, and SIAM SDM, and also received the ACM SIGMOD Research Student Award. His research interests include distributed systems, databases, data cluster-ing, and data mining.
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