M2SP: Mining Sequential Patterns Among Several Dimensions

M2 SP: Mining Sequential Patterns among Several Dimensions M. Plantevit1 , Y.W. Choong2,3, A. Laurent1 , D. Laurent2 and M. Teisseire1 1LIRMM, Université Montpellier 2, CNRS, 161 rue Ada, 34392 Montpellier, France 2 LICP, Université de Cergy Pontoise, 2 av. Chauvin, 95302 Cergy-Pontoise, France 3 HELP University College, BZ-2 Pusat Bandar Damansara, 50490 Kuala Lumpur, Malaysia Abstract. Mining sequential patterns aims at discovering correlations between events through time, like A customer who bought a TV and a DVD player at the same time later bought a recorder. Even if many works have dealt with se- quential pattern mining, there is no work leading to rules like A customer who bought a surfboard together with a bag in NY later bought a wetsuit in SF. In this paper, we propose M 2 SP, a complete approach to mine such multidimensional sequential patterns. The main originality of our proposition is that we obtain not only intra-pattern sequences but also inter-pattern sequences. Moreover, we con- sider generalized multidimensional sequential patterns, called jokerized patterns, in which some of the dimension values may not be instanciated. Experiments on synthetic data are reported and show the scalability of our approach. Keywords: Data Mining, Sequential Patterns, Multidimensional Rules. 1 Introduction Mining sequential patterns aim at discovering correlations between events through time. For instance, rules that can be built are A customer who bought a TV and a DVD player at the same time later bought a recorder. Work dealing with this issue in the literature have proposed scalable methods and algorithms to mine such rules [9]. As for asso- ciation rules, the efficient discovery is based on the support which indicates to which extend data from the database contains the patterns. However, these methods only consider one dimension to appear in the patterns, which is usually called the product dimension. This dimension may also represent web pages for web usage mining, but there is normally a single dimension. Although some works from various studies claim to combine several dimensions, we argue here that they do not provide a complete framework for multidimensional sequential pattern mining [4,8,11]. The way we consider multidimensionality is indeed generalized in the sense that patterns must contain several dimensions combined over time. For instance we aim at building rules like A customer who bought a sur f board and a bag in NY later bought a wetsuit in SF. This rule not only combines two dimensions (City and Product) but it also combines them over time (NY appears before SF, surfboard appears before wetsuit). As far as we know, no method has been proposed to mine such rules.  In this paper, we present existing methods and their limits from several studies in recent years. We define then the basic concepts associated to our proposition M 2 SP and the algorithms to build such rules. Experiments are performed on synthetic data to as- sess our proposition. In our approach, sequential patterns are mined from a relational table, that can be seen as a fact table in a multidimensional database. This is why, contrary to the stan- dard terminology of the relational model, the attributes over which a relational table is defined are called dimensions. In order to mine such frequent sequences, we extend our approach so as to take into account partially instanciated tuples in sequences. More precisely, our algorithms are designed in order to mine frequent jokerized multidimensional sequences contain- ing as few ∗ as possible, i.e., replacing an occurrence of ∗ with any value from the corresponding domain cannot give a frequent sequence. This paper is organized as follows. Section 2 introduces a motivating example illus- trating the goal of our work. Section 3 presents existing works concerning sequential patterns, multidimensional approaches and multidimensional databases. Section 4 in- troduces our contribution, M2 SP, as an approach to discover multidimensional sequen- tial patterns. Section 5 introduces jokerized patterns. Section 6 presents the algorithms. Section 7 presents results of experiments performed on synthetic data. 2 Motivating Example In order to illustrate our approach, we consider the following example that will be used throughout the paper as a running example. Let us consider a relational table T in which transactions issued by customers are stored. More precisely, we assume that T is defined over six dimensions (or attributes) as shown in Fig. 1, and where: D is the date of transactions (considering three dates, denoted by 1, 2 and 3), CG is the category of customers (considering two categories, denoted by Educ and Ret, standing for educational and retired customers, respectively), A is the age of customers (considering three discretized values, denoted by Y (young), M (middle) and O (old)), C is the city where transactions have been issued (considering three cities, denoted by NY (New York), LA (Los Angeles) and SF (San Francisco)), P is the product of the transactions (considering four products, denoted by c, m, p and r), Q stands for the quantity of products in the transactions (considering nine such quantities). For instance, the first tuple of T (see Fig. 1) means that, at date 1, educational young customers bought 50 products c in New York. Let us now assume that we want to ex- tract all multidimensional sequences that deal with the age of customers, the products they bought and the corresponding quantities, and that are frequent with respect to the groups of customers and the cities where transactions have been issued. To this end, we consider three sets of dimensions as follows: (i) the dimension D, representing the date, (ii) the three dimensions A, P and Q that we call analysis dimensions (tuples over anal- ysis dimensions are those that appear in the items that constitute the sequential patterns to be mined), (iii) the two dimensions CG and C, that we call reference dimensions (the table is partitioned according to tuple values over reference dimensions and the support of a given multidimensional sequence is the ratio of the number of blocks supporting the sequence over the total number of blocks. Fig. 5 displays the corresponding blocks in our example). In this framework, h{(Y, c, 50), (M, p, 2)}, {M, r, 10)}i is a multidimensional sequence having support 13 , since the partition according to the reference dimensions contains 3 blocks, among which one supports the sequence. This is so because (Y, c, 50) and (M, p, 2) both appear at the same date (namely date 1), and (M, r, 10) appears later on (namely at date 2) in the first block shown in Figure 4. It is important to note that, in our approach, more general patterns, called joker- ized sequences, can be mined. The reason for this generalization is that considering partially instanciated tuples in sequences implies that more frequent sequences are mined. To see this, considering a support threshold of 32 , no sequence of the form h{(Y, c, µ)}, {(M, r, µ0 )}i is frequent. On the other hand, in the first two blocks of Fig. 5, Y associated with c and M associated with r appear one after the other, according to the date of transactions. Thus, we consider that the jokerized sequence, denoted by h{(Y, c, ∗)}, {(M, r, ∗)}i, is frequent since its support is equal to 32 . D CG C A P Q (Date) (Customer-Group) (City) (Age) (Product) (Quantity) 1 Educ NY Y c 50 1 Educ NY M p 2 1 Educ LA Y c 30 1 Ret. SF O c 20 1 Ret. SF O m 2 2 Educ NY M p 3 2 Educ NY M r 10 2 Educ LA Y c 20 3 Educ LA M r 15 Fig. 1. Table T 3 Related Work 3.1 Sequential Patterns An early example of research in the discovering of patterns from sequences of events can be found in [5]. In this work, the idea is the discovery of rules underlying the gen- eration of a given sequence in order to predict a plausible sequence continuation. This idea is then extended to the discovery of finding interesting patterns (or rules) embedded in a database of sequences of sets of events (items). A more formal approach in solving the problem of mining sequential patterns is the AprioriAll algorithm as presented in [6]. Given a database of sequences, where each sequence is a list of transactions ordered by transaction time, and each transaction is a set of items, the goal is to discover all se- quential patterns with a user-specified minimum support, where the support of a pattern is the number of data-sequences that contain the pattern. In [1], the authors introduce the problem of mining sequential patterns over large databases of customer transactions where each transaction consists of customer-id, transaction time, and the items bought in the transaction. Formally, given a set of sequences, where each sequence consists of a list of elements and each element consists of a set of items, and given a user-specified min support threshold, sequential pattern mining is to find all of the frequent subse- quences, i.e., the subsequences whose occurrence frequency in the set of sequences is no less than min support. Sequential pattern mining discovers frequent patterns ordered by time. An example of this type of pattern is A customer who bought a new television 3 months ago, is likely to buy a DVD player now. Subsequently, many studies have intro- duced various methods in mining sequential patterns (mainly in time-related data) but almost all proposed methods are Apriori-like, i.e., based on the Apriori property which states the fact that any super-pattern of a nonfrequent pattern cannot be frequent. An example using this approach is the GSP algorithm [9]. 3.2 Data to be Mined We assume that the reader is familiar with the relational model of databases [10], and we briefly recall the basic ingredients of this model used in this paper. Let U = {D 1 , . . . Dn } be a set of attributes, which we call dimensions in our approach. Each dimension D i is associated with a (possibly infinite) domain of values, denoted by dom(D i ). A relational table T over universe U is a finite set of tuples t = {d1 , . . . , dn } such that, for every i = 1, . . . , n, di ∈ dom(Di ). Moreover, given a table T over U, for every i = 1, . . . , n, we denote by DomT (Di ) (or simply Dom(Di ) if T is clear from the context) the active domain of Di in T , i.e., the set of all values of dom(Di ) that occur in T . Since we are interested in sequential patterns, we assume that U contains at least one dimension whose domain is totally ordered, corresponding to the time dimension. In our running example, we consider U = {D,CG, A, P, Q}, where D is the time dimen- sion. Moreover, considering the table T of Fig. 1, we have for instance Dom(D) = {1, 2, 3} and Dom(CG) = {Educ, Ret}. 3.3 Multidimensional Sequential Patterns As far as we know, three propositions have been studied in order to deal with several dimensions when building sequential patterns. Pinto et al. [8] is the first paper dealing with several dimensions in the framework of sequential patterns. For instance, purchases are not only described by considering the customer ID and the products, but also by considering the age, type of the customer (Cust-Grp) and the city where he lives, as shown in Fig. 1. Multidimensional sequen- tial patterns are defined over the schema A1 , ..., Am , S where the set of Ai stands for the dimensions describing the data and S stands for the sequence of items purchased by the customers ordered over time. A multidimensional sequential pattern is defined as (id1 ,(a1 , ..., am ),s) where ai ∈ Ai ∪ {∗}. id1 ,(a1 , ..., am ) is said to be a multidimensional pattern. For instance, the authors consider the sequence ((∗, NY, ∗),hb f i) meaning that customers from NY have all bought a product b and then a product f . Sequential pat- terns are mined from such multidimensional databases either (i) by mining all frequent sequential patterns over the product dimension and then regrouping them into multidi- mensional patterns, (ii) or by mining all frequent multidimensional patterns and then mining frequent product sequences over these patterns. Note that the sequences found by this approach do not contain several dimensions since the dimension time only con- cerns products. Dimension product is the only dimension that can be combined over time, meaning that it is not possible to have a rule indicating that when b is bought in Boston then c is bought in NY . Yu et Chen. [11] considers sequential pattern mining in the framework of Web Usage Mining. Even if they consider three dimensions (pages, sessions, days), these dimen- sions are very particular since they belong to a single hierarchized dimension. More- over, the sequences found describe correlations between objects over time by consider- ing only one dimension, which corresponds to the web pages. de Amo et al. [4] is based on first order temporal logic. This proposition is close to our approach, but there are some limits since (i) groups used to compute the support are predefined whereas we consider the fact that the user should be able to define them (see reference dimensions below), (ii) several attributes cannot appear in the sequences. The authors claim that they aim at considering several dimensions but they have only shown one dimension for the sake of simplicity. However, the paper does not provide any complete proposition to extend this point to real multidimensional patterns. 4 M2 SP: Mining Multidimensional Sequential Patterns As shown above, no work has been done in order to mine sequential patterns over several dimensions. The work described in [8] is said to be intra-pattern since sequences are mined within the framework of a single description (the so-called pattern). In this paper, we propose to generalize this work to inter-pattern multidimensional sequences. 4.1 Dimension Partition D CG C A P Q D CG C A P Q D CG C A P Q 1 Educ NY Y c 50 1 Educ LA Y c 30 1 Ret. SF O c 20 1 Educ NY M p 2 2 Educ LA Y c 20 1 Ret. SF O m 2 2 Educ NY M p 3 3 Educ LA M r 15 2 Educ NY M r 10 Fig. 2. block (Educ, NY ) Fig. 3. block (Educ, LA) Fig. 4. block (Ret., SF) Fig. 5. blocks defined on T over dimensions CG and C For each table defined on the set of dimensions D, we consider a partition of D into four sets: Dt for the temporal dimension, DA for the analysis dimensions, DR for the reference dimensions, DF for the ignored dimensions. Each tuple c = (d1 , . . . , dn ) can thus be denoted c = ( f , r, a,t) with f the restriction on DF of c, r the restriction on DR , a the restriction on DA , t the restriction on Dt .  Given a table T , the set of all tuples in T having the same value r on D R is said to be a block and we denote by BT,DR the set of blocks from table T . Fig. 5 shows blocks built from table T . Each block B in BT,DR is denoted by the tuple r that defines it. / DR = {CG,C}, DA = {A, P}, Dt = {D}. In our running example, we consider F = 0, When mining multidimensional sequential patterns, the set DR identifies the blocks of the database to be considered when computing the support. For this reason, this set is called reference. The support of a sequence is the proportion of blocks embedding it. Note that usual sequential patterns, and sequential patterns from [8] and [4], this set is reduced to one dimension (cid in [8] or IdG in [4]). The set DA describes the analy- sis dimensions, meaning that these dimensions will be found in the multidimensional sequential patterns. Note that usual sequential patterns only consider one analysis di- mension corresponding to the products purchased or the web pages visited. The set F describes the ignored dimensions, which are used neither to define the date, nor the blocks, and which are not present within the patterns mined. 4.2 Multidimensional Item, Itemset and Sequential Pattern Definition 1 (Multidimensional Item) A multidimensional item e defined on DA = {Di1 , . . . , Dim } is a tuple e = (di1 , . . . , dim ) such that ∀k ∈ [1, m], dik ∈ Dom(Dik ). Definition 2 (Multidimensional Itemset) A multidimensional itemset i defined on DA = {Di1 ,. . . ,Dim } is a non empty set of items i = {e1 , . . . , e p } where ∀ j ∈ [1, p], e j is a mul- tidimensional item defined on DA and ∀ j, k ∈ [1, p], e j 6= ek . Note that all items from an itemset are defined using the same dimensions (D A ). Also note that all pairs of multidimensional items from an itemset are different. Definition 3 (Multidimensional Sequence) A multidimensional sequence ς defined on DA = {Di1 ,. . . ,Dim } is an ordered non empty list of itemsets ς = hi1 , . . . , il i where ∀ j ∈ [1, l], i j is a multidimensional itemset defined on DA . In our running example, (Y, c, 50), (M, p, 2), (M, r, 10) are three multidimensional items on DA = {A, P, Q}. {(Y, c, 50), (M, p, 2)} and {(M, r, 10)} are multidimensional itemsets on DA . h{(Y, c, 50), (M, p, 2)}, {(M, r, 10)}i is a multidimensional sequence on DA . Definition 4 (Inclusion of sequence) A multidimensional sequence ς = ha1 , . . . , al i is said to be a subsequence of a sequence ς0 = hb1 , . . . , bl 0 i if there exist integers 1 ≤ j1 ≤ j2 ≤ . . . ≤ jl ≤ l 0 such that a1 ⊆ b j1 , a2 ⊆ b j2 , . . . , al ⊆ b jl . Let ς = h{(Y, c, 50)}, {(M, r, 10)}i and ς0 = h{(Y, c, 50), (M, p, 2)}, {(M, r, 10)}i be two multi- dimensional sequences. ς is a subsequence of ς0 . 4.3 Support Computing the support of a sequence amounts to counting the number of blocks that contain the sequence. A block supports a sequence if it is possible to find a set of tuples which satisfy it. For each itemset from the sequence, we must thus find a date from Dom(Dt ) such that all items from the itemset appear at this date. All itemsets must be retrieved at different dates from Dom(Dt ) such that the order of the itemsets from the sequence is satisfied. Definition 5 A table T supports a sequence hi1 , . . . , il i if ∀ j = 1 . . . l, ∃d j ∈ Dom(Dt ), for every item e in i j , ∃t = ( f , r, e, d j ) ∈ T with d1 < d2 < . . . < dl . The block (Educ, NY ) from Fig. 2 supports h{(Y, c, 50), (M, p, 2)}, {(M, r, 10)}i since {(Y, c, 50), (M, p, 2)} appears at date = 1 and {(M, r, 10)} appears at date = 2. The support of a sequence in a table T is the proportion of blocks of T that support it. Definition 6 (Sequence Support) Let DR be the reference dimensions and T be a ta- ble partitioned into the set of blocks BT,DR . The support of a sequence ς is: |{B∈BT,DR s.t. B supports ς}| support(ς) = |BT,DR | Definition 7 (Frequent Sequence) Let minsup ∈ [0, 1] be the minimum user-defined support value. A sequence ς is said to be frequent if support(ς) ≥ minsup. Note that an item e is said to be frequent if so is the sequence h{e}i. Let us consider DR = {CG,C}, DA = {A, P}, minsup = 15 , ς = h{(Y, c, 50), (M, p, 2)}, {(M, r, 10)}i. The three blocks of the partition of T from Fig. 5 must be scanned to compute support(ς). 1. block (Educ, NY) (Fig. 2). Two dates can be found in this block. At date 1, we have (Y, c, 50) and (M, p, 2) and at date 2, we have (M, r, 10). Thus this block supports ς. 2. block (Educ, LA) (Fig. 3). This block cannot be counted since it does not contain (M, p, 2). 3. block (Ret., SF) (Fig. 4). This block contains only one date. Therefore, it is not possible ς to be embedded in it. We have thus support(ς) = 13 ≥ minsup. 5 Jokerized Sequential Patterns Considering the definitions above, an item can only be retrieved if there exists a fre- quent tuple of values from domains of DA containing it. For instance, it can occur that neither (Y, r) nor (M, r) nor (O, r) is frequent whereas the value r is frequent. Thus, we introduce the joker value ∗. In this case, we consider (∗, r)which is said to be jokerized. Definition 8 (Jokerized Item) Let e = (d1 , . . . , dm ) a multidimensional item. We de- note by e[di /δ] the replacement in e of di by δ. e is said to be a jokerized multidimen- sional item if: (i) ∀i ∈ [1, m], di ∈ Dom(Di ) ∪ {∗}, and (ii) ∃i ∈ [1, m] such that di 6= ∗, and (iii) ∀di = ∗, @δ ∈ Dom(Di ) such that e[di /δ] is frequent. A jokerized item contains at least one specified analysis dimension. It contains a ∗ only if no specific value from the domain can be set. A jokerized sequence is a sequence containing at least one jokerized item. A block is said to support a sequence if a set of tuples containing the itemsets satisfying the temporal constraints can be found. Definition 9 (Support of a Jokerized Sequence) A table T supports a jokerized se- quence ς = his1 , . . . , isl i if ∀ j ∈ [1, l], ∃δ j ∈ Dom(Dt ), ∀e = (di1 , . . . , dim ) ∈ i j , ∃t = ( f , r, (xi1 , . . . , xim ), δ j ) ∈ T with di = xi or di = ∗ and δ1 < δ2 < . . . < δl . |{B∈BT,DR s.t. B supports ς}| The support of ς is the proportion of blocks containing ς. support(ς) = |BT,DR | 6 Algorithms 6.1 Mining 1-frequent items 1-frequent items are items built on the analysis dimensions. When considering no joker value, a single scan of the database results in the multidimensional items being frequent. When considering jokerized items, a levelwise algorithm is used in order to build the frequent multidimensional items having the smallest possible number of joker values. To this end, we consider a lattice which lower bound is the (∗, . . . , ∗) multidimensional item. This lattice is partially built from (∗, . . . , ∗) up to the frequent items containing as few ∗ as possible. At level i, i values are specified. Then items at level i are combined to build a set of candidates at level i + 1. Two frequent items are combined to build a candidate if they are on-compatible. Definition 10 (o n-compatibility) Let e1 = (d1 , . . . , dn ) and e2 = (d10 , . . . , dn0 ) be two dis- tinct multidimensional items where di and di0 ∈ dom(Di ) ∪ {∗}. e1 and e2 are said to be n-compatible if ∃∆ = {Di1 , . . . , Din−2 } ⊂ {D1 , . . . , Dn } such that for every j ∈ [1, n − 2], o di j = di0 j 6= ∗ with din−1 = ∗ and di0n−1 6= ∗ and din 6= ∗ and di0n = ∗. Definition 11 (Join) Let e1 = (d1 , . . . , dn ) and e2 = (d10 , . . . , dn0 ) be two on-compatible multidimensional items. We define e1 o n e2 = (v1 , . . . , vn ) where vi = di if di = di0 , vi = di if di0 = ∗ and vi = di0 if di = ∗. Let E and E 0 be two sets of multidimensional items of size n, we define Eon E 0 = {e o n e0 s.t. (e, e0 ) ∈ E × E 0 ∧ e and e’ are o n-compatible} Two multidimensional items defined on n dimensions are o n-compatible if they share n- 2 values. For instance, (NY,Y, ∗) and (∗,Y, r) are on-compatible. We have (NY,Y, ∗) o n (∗,Y, r) = (NY,Y, r). On the contrary, (NY, M, ∗) and (NY,Y, ∗) are not o n-compatible. Note that this method is close to the one used for iceberg cubes in [2,3]. F1i stands for the set of 1-frequent items having i dimensions which are specified (different from ∗). Frequent items of size 1 are obtained from the candidate items of size 1. More generally, the algorithm to build candidate items of size i are obtained by considering the frequent items of size i − 1: F11 = { f ∈ Cand11, support( f ) ≥ minsup}, Cand1i = F1i−1 on F1i−1 . 6.2 Mining Jokerized Multidimensional Sequences Once 1-frequent items are mined, the candidate sequences of size k (k ≥ 2) are generated and validated to keep the frequent items. This computation is based on usual algorithms such as PSP [7] by introducing the treatment of joker values. The support of a sequence ς in a table T according to the reference dimensions D R is given by Algorithm 1. The reference dimensions are used to compute the partition to be considered. This algorithm checks whether each block of the partition supports the sequence by calling the function supportTable (Algorithm 2). supportTable attempts to find a tuple from the block that matches the first item of the first itemset of the sequence in order to anchor the sequence. This operation is repeated recursively until all itemsets from the sequence are found (return true) or until there is no way to go on further (return false). Several possible anchors may be tested if the first ones do not fit.  Function supportcount Data : ς, T, DR , counting Result : support of ς Integer support ←− 0 ; BooleanseqSupported; BT,DR ←− {blocks o f T identi f ied over DR }; foreach B ∈ BT,DR do seqSupported ←− supportTable(ς, B, counting) ; if seqSupported then support ←− support + 1; support   return |B | T,DR Algorithm 1: Support of a sequence (supportcount) Function supportTable Data : ς, T, counting //counting indicates whether joker values are considered or not Result : Boolean ItemSetFound ←− f alse ; seq ←− ς ; itset ←− sequence. f irst() ; it ←− itset. f irst() if ς = 0/ then return (true) // End of Recursivity while t ←− T.next 6= 0/ do if supports(t, it, counting) then if (NextItem ←− itset.second()) = 0/ then ItemSetFound ←− true // Look for all the items from the itemset else // Anchoring on the item (date) T 0 ←− σdate=t.date (T ) while t 0 ←− T 0 .next() 6= 0/ ∧ ItemSetFound = f alse do if supports(t 0 , NextItem, counting) then NextItem ←− itset.next() if NextItem = 0/ then ItemSetFound ←− true if ItemSetFound = true then // Look now for the other itemsets return (supportTable(seq.tail(), σdate>t.date (T ), counting)) else itset ←− seq. f irst() C ←− σdate>t.date (T ) // Skip to next dates return(false) // Not found Algorithm 2: supportTable (Checks if a sequence ς is supported by a table T ) 7 Experiments In this section, we report experiments performed on synthetic data. These experiments aim at showing the interest and scalability of our approach, especially in the jokerized approach. As many databases from the real world include quantitative information, we have distinguished a quantitative dimension. In order to highlight the particular role of this quantitative dimension, we consider four ways of computing frequent sequential patterns: (i) no joker (M 2 SP), (ii) jokers on all dimensions but the quantitative one (M 2 SP-al pha), (iii) jokers only on the quantitative dimension (M 2 SP-mu), (iv) jokers on all dimensions (M 2 SP-al pha-mu). Note that case (iv) corresponds to the jokerized approach presented in Section 5. Our experiments can thus be seen as being conducted in the context of a fact table of a multidimensional database, where the quantitative dimension is the measure. In Figures 5-12, minsup is the minimum support taken into account, nb dim is the number of analysis dimensions being considered, DB size is the number of tuples, avg card is the average number of values in the domains of the analysis dimensions. Fig. 6 and 7 compare the behavior of the four approaches described above when the support changes. M 2 SP-al pha and M 2 SP-al pha-mu have a similar behavior, the difference being due to the verification of quantities in the case of M 2 SP-al pha. Note that these experiments are not led on the same minimum support values since no fre- quent items are found for M 2 SP and M 2 SP-mu if the support is too high. Fig. 8 shows the scalability of our approach since runtime grows almost linearly when the database size increases (from 1, 000 tuples up to 26, 000 tuples). Fig. 9 shows how runtime be- haves when the average cardinality of the domains analysis dimensions changes. When this average is very low, numerous frequent items are mined among few candidates. On the contrary, when this average is high, numerous candidates have to be considered from which few frequent items are mined. Between these two extrema, the runtime de- creases. Fig. 10 and 11 show the behavior of our approach when the number of analysis dimensions changes. The number of frequent items increases as the number of analy- sis dimensions grows, leading to an increase of the number of frequent sequences. Fig. 12 and 13 show the differential between the number of frequent sequences mined by our approach compared to the number of frequent sequences mined by the approach described in [8], highlighting the interest of our proposition. 8 Conclusion In this paper, we propose a novel definition for multidimensional sequential patterns. Contrary to the propositions from [4,8,11], several analysis dimensions can be found in the sequence. This allows the discovery of rules like A customer who bought a surf- board together with a bag in NY later bought a wetsuit in LA. We also define jokerized sequential patterns by introducing the joker value ∗ on analysis dimensions. Moreover, reference dimensions are defined in order to generalize the way clients are defined. The algorithms implementing our approach are given and evaluated with experiments performed on synthetic data. This work can be extended following several directions. For example, we can take into account approximate values on quantitative dimensions. In this case, we allow  50 10000 M2SP-mu M2SP-alpha-mu M2SP M2SP-alpha 45 40 8000 35 30 6000 Runtime (s) Runtime (s) 25 20 4000 15 10 2000 5 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Support Support Fig. 6. Runtime over Support Fig. 7. Runtime over Support (DB size=12000, nb dim=5, avg card=20) (DB size=12000, nb dim=5, avg card=20) 2000 10000 M2SP-alpha-mu M2SP-alpha-mu M2SP-alpha M2SP-alpha 1500 1000 Runtime (s) Runtime (s) 1000 100 500 10 0 1 0 5 10 15 20 25 5 10 15 20 25 30 35 40 45 Database Size (K-tuples) Average cardinality of analysis dimensions Fig. 8. Runtime over database size Fig. 9. Runtime over Average Cardinal- (minsup=0.5, nb dim=15, avg card = ity of Analysis Dimensions (minsup=0.8, 20) DB size=12000, nb dim=15) 1000 M2SP-alpha-mu 600 M2SP-alpha M2SP-alpha-mu M2SP-alpha 800 500 Number of Frequent Patterns 400 600 Runtime (s) 300 400 200 200 100 0 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Number of Analysis Dimensions Number of Analysis Dimensions Fig. 11. Number of Frequent patterns Fig. 10. Runtime over Number of Analysis over number of analysis dimensions Dimensions (minsup=0.5, DB size=12000, (minsup=0.5, DB size=12000, nb dim=15, nb dim=15, avg card=20) avg card=20)  1000 M2SP-alpha 2000 Han’s approach M2SP-alpha Han’s approach 800 Number of Frequent Patterns 1500 Number of frequent Patterns 600 1000 400 500 200 0 2 4 6 8 10 12 14 0 5 10 15 20 25 Number of Analysis Dimensions Database Size (K-tuples) Fig. 13. Number of Frequent Sequences Fig. 12. Number of Frequent Sequences over Number of Analysis Dimen- over Database Size (minsup=0.5, sions (minsup=0.5, DB size=12000, nb dim=15, avg card=20) avg card=20) the consideration of values that are not fully jokerized while remaining frequent. This proposition is important when considering data from the real world where the high number of quantitative values prevent each of them to be frequent. 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