Mining Fuzzy Sequential Patterns with Fuzzy Time-Intervals in Quantitative Sequence Databases

Transcription

1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 18, No 2 Soia 2018 Print ISSN: ; Online ISSN: DOI: /cait Mining Fuzzy Sequential Patterns with Fuzzy Time-Intervals in Quantitative Sequence Dataases Truong Duc Phuong 1, Do Van Thanh 2, Nguyen Duc Dung 3 1 Hanoi Metropolitan University, Vietnam 2 Faculty o Inormation Technology, Nguyen Tat Thanh University, Vietnam 3 Institute o Inormation Technology, Vietnamese Academy o Science and Technology, Vietnam s: tdphuong@daihocthudo.edu.vn dvthanh@ntt.edu.vn nddung@ioit.ac.vn Astract: The main ojective o this paper is to introduce uzzy sequential patterns with uzzy time-intervals in quantitative sequence dataases. In the uzzy sequential pattern with uzzy time-intervals, oth quantitative attriutes and time distances are represented y linguistic terms. A new algorithm ased on the Apriori algorithm is proposed to ind the patterns. The mined patterns can e applied to market asket analysis, stock market analysis, and so on. Keywords: Data mining, uzzy sequential pattern, uzzy time-interval, sequence dataase. 1. Introduction Mining sequential patterns are one o the most important domains in data mining. Mining sequential patterns rom transaction dataases (events present or not) is the irst introduced in 1995 [1] and there are many related works [8-10, 17, 22, 23]. The sequential pattern presented a relationship etween events in a chronological sequence with the same oject. For instance, I a customer uys a Laptop and later a Modem, then he will uy a Printer. Mining uzzy sequential patterns in sequence dataases, in which values o attriutes are numeric or categorical, is also introduced in [2, 5, 12, 14, 15, 18]. In these works, the values are transormed into uzzy sets. A relationship etween events in the uzzy sequential patterns is like I a customer uys a Large numer o Laptops and later an Average numer o Modems, then he will uy a Small numer o Printers. In many cases, the values o time distances among the events in a sequence are preerred. The time-interval sequential patterns in transaction dataases were investigated y authors such as C h e n and H u a n g [6], C h e n, C h i a n g and K o [7], C h a n g, C h u e h and Lin [3], C h a n g, C h u e h and Luo [4]. In [7], timeinterval sequential patterns were presented such as Laptop, I 1, Modem, I 2, Printer, which meant I a customer uys a Laptop and later a Modem an interval o I 1, then 3

2 he will uy a Printer an interval o I 2, where I 1 and I 2 were predetermined time ranges. For instance, I 1 was ranged rom 3 to 5 days, I 2 rom 10 to 12 days. To solve sharp oundary prolems when a time interval is near the oundary o two time ranges in [7], the uzzy theory was applied to time intervals [6]. The uzzy timeinterval sequential patterns in [6] showed the relationship among events such as Laptop, Short, Modem, Long, Printer, which meant I a customer uys a Laptop and later a Modem an interval o Short, then he will uy a Printer an interval o Long, where Short and Long were linguistic terms or time intervals. In [6], time intervals among events were transormed into uzzy sets and the two algorithms were proposed. They were the FTI-Apriori algorithm ased on the idea o an Apriori algorithm and the FTI-PreixSpan algorithm ased on the PreixSpan algorithm. Papers [3, 4] also revealed uzzy time-interval sequential patterns in sequence dataases such as Laptop, μ Laptop_Modem, Modem, μ Modem_Printer, Printer that meant I a customer uys a Laptop and later a Modem an interval o μ Laptop_Modem, then he will uy a Printer an interval o μ Modem_Printer, where μ Laptop_Modem and μ Modem_Printer were trapezoidal uzzy numers which present time intervals etween events o pair (Laptop, Modem) and (Modem, Printer). The uzzy numers were computed y requency o time intervals o events in a sequence. For example, μ Laptop_Modem = (6, 6, 12, 12) and μ Modem_Printer = (2, 2, 7, 15). The main idea o the SPFTI algorithm [3] and the ISPFTI algorithm [4] were taking the advantage o the idea o Apriori algorithm, ut the trapezoidal uzzy numers were received rom dataases unlike predetermined uzzy sets in [6]. In [21], we considered to mine uzzy association rules with uzzy time-intervals rom quantitative dataases. The results showed that the patterns such as Laptop_ Large, Short, Modem_ Average, Long, Printer_ Small that meant I a Large numer o Laptops are sold and later an Average numer o Modems an interval o Short, then a Small numer o Printers will e sold an interval o Long. In the pattern, Laptop_ Large, Modem_ Average, and Printer_ Small were the uzzy sets o attriutes; Short and Long were the pre-deined uzzy sets or time intervals. Works [3, 4, 6, 7] have indicated only the uzzy time-interval sequential patterns in transaction sequence dataases, not yet in quantitative sequence dataases. While paper [21] has een applied to only quantitative dataases, not yet quantitative sequence dataases. So, the main ojective o the paper is to introduce an algorithm or mining uzzy sequential patterns with uzzy time-intervals in quantitative sequence dataases. They are like Laptop_ Large, Short, Modem_ Average, Long, Printer_ Small that means I a customer uys a Large numer o Laptops and later an Average numer o Modems an interval o Short, then he will uy a Small numer o Printers an interval o Long, where Short and Long are predetermined linguistic terms or time intervals. These patterns are dierent rom the ones in [3, 4, 6, 7] y quantitative attriutes and rom the ones in [21] y interesting in the ojects. The main idea o the algorithm is to convert quantitative attriutes and time intervals to linguistic terms in the same way as in [21] and then improving the Apriori algorithm [1] to ind uzzy sequential patterns with uzzy time-intervals. The comparison o datasets and patterns o the selected algorithms is descried in Tale 1. 4

3 Tale 1. Comparison o datasets and patterns o the selected algorithms Algorithm Dataset Pattern Description AprioriAll [1] FGBSPMA [5] I-Apriori algorithm, I-PreixSpan [7] FTI-Apriori, FTI-PreixSpan [6] SPFTI [3], ISPFTI [4] FTQ [21] Proposed Transactional sequence dataases Quantitative sequence dataases Transactional sequence dataases Transactional sequence dataases Transactional sequence dataases Quantitative (not sequence) dataases Quantitative sequence dataases Laptop, Modem, Printer Laptop_ Large,Modem_ Average, Printer_ Small Laptop, I 1, Modem, I 2, Printer Laptop, Short, Modem, Long, Printer Laptop,μ Laptop_Modem,Modem, μ Modem_Printer, Printer Laptop_ Large,Short,Modem_ Average, Long, Printer_ Small Laptop_ Large, Short, Modem_ Average, Long, Printer_ Small I a customer uys a Laptop and later a Modem, then he will uy a Printer I a customer uys a Large numer o Laptops and later an Average numer o Modems, then he will uy a Small numer o Printers I a customer uys a Laptop and later a Modem an interval o I 1, then he will uy a Printer an interval o I 2 I a customer uys a Laptop and later a Modem an interval o Short, then he will uy a Printer an interval o Long I a customer uys a Laptop and later a Modem an interval o μ Laptop_Modem then he will uy a Printer an interval o μ Modem_Printer I a Large numer o Laptops are sold and later an Average numer o Modems an interval o Short, then a Small numer o Printers will e sold an interval o Long I a customer uys a Large numer o Laptops and later an Average numer o Modems an interval o Short, then he will uy a Small numer o Printers an interval o Long The rest o the paper is organized as ollows: Section 2 deines prolems. Section 3 develops the FSPFTIM algorithm to ind out the uzzy sequential patterns with uzzy time-intervals and gives an example. Section 4 presents experimental results. Conclusions are drawn in Section Prolem deinition This section presents some deinitions related to the proposed prolem. Deinition 1. Let E={e 1, e 2,, e u} e a set o attriutes, s = (a 1, q 1, t 1), (a 2, q 2, t 2), (a 3, q 3, t 3),, (a n, q n, t n) e a quantitative sequence, where a k E is an attriute (1 k n), t k is time o a k, t k 1 t k with 2 k n and a k(t k) = q k, q k is numeric or categorical. A transaction sequence is Sid, s where s is a quantitative sequence and Sid is the identiier o the sequence. A quantitative sequence dataase is a set o all transaction sequences. Example 1. A quantitative sequence dataase is shown in Tale 2. In Tale 2, E = {a,, c, d, e,, g, h, i} is set o attriutes, time is started at 0. The irst transaction sequence is 1, (a, 2, 1), (, 2, 4), (e, 5, 29). It means that this sequence includes Sid with value 1 and three transactions: The a with value 2 5

4 (quantitative o a is 2) at time 1, the with value 2 at time 4 and the e with value 5 at time 29. Tale 2. An example o a quantitative sequence dataase Sid Quantitative sequence 1 (a, 2, 1), (, 2, 4), (e, 5, 29) 2 (d, 2, 1), (a, 5, 2), (d, 4, 24) 3 (, 1, 1), (d, 2, 11), (e, 5, 28) 4 (, 6, 1), (, 6, 5), (c, 1, 19), (c, 2, 25) 5 (a, 1, 4), (, 1, 5), (d, 2, 10), (e, 5, 28) 6 (a, 2, 0), (, 1, 5), (e, 1, 30) 7 (i, 5, 2), (a, 3, 17), (h, 2, 17) 8 (c, 6, 3), (i, 5,10), (, 3, 18) 9 (h, 3, 4), (a, 1, 10), (, 6, 21) 10 (a, 2, 0), (g, 5, 0), (, 2, 3), (e, 1, 30) e1 e2 eu Deinition 2. Let FE = F, F,..., F e a set o uzzy sets o attriutes o e E,,1,,2,...,, k e k e k e F h h h k h e a set o uzzy sets o e k attriute (k=1, 2,..., u), k k k k e where k e h, j is the j-th uzzy set (1 j h k), h k is the numer o uzzy sets o e k; h k, k is called a uzzy attriute. Each uzzy set has its memership unction : X[0, 1]. Sequence s=(a 1, q 1, t 1), (a 2, q 2, t 2),, (a n, q n, t n) is called a uzzy one, where a if a i (1 i n) is a uzzy set, q i is the value o the memership unction ai o a i at q i (q i = ai (q i)), a i is called a uzzy attriute. Sid, s is called a uzzy transaction sequence. A uzzy sequence dataase is a set o all uzzy transaction sequences. Example 2. Given uzzy sets are in [13], xm A K, i m is i m-th linguistic value x (1 i m K) o the x m attriute (x me), K is the numer o partitions o x m, m is the memership unction o x A, that is determined as ollows: m (1) K im xm K K, i ( v) max1 v a m im K /,0 where, K a mi (ma mi)( i 1) / ( K 1), im k = (ma mi)/(k 1), where mi is the minimum value o x m attriute domain, and ma is the maximum value. With K=3 or all attriutes in the dataase o the Example 1, we get memership unctions as in the Fig. 1 and the uzzy transaction sequence dataase D is descried in Tale 3. m K,i m k j 6

5 1 x, i K m Fig. 1. The memership unctions o xm with K=3 Tale 3. The uzzy transaction sequence dataase D Sid Fuzzy sequence 1 a a e (, 0.5,1), (, 0.5,1), (, 0.6, 4), (, 0.4, 4), (,1, 29) ,1 3,2 3,1 3,2 3,3 (,1,1), (,1, 2), (,1, 24) d a d 3,1 3,3 3,1 (,1,1), (,1,11), (,1, 28) d e 3,1 3,1 3,3 (,1,1), (,1, 5), (,1,19), (, 0.6, 25), (, 0.4, 25) c c c 3,3 3,3 3,1 3,1 3,2 (,1, 4), (,1, 5), (,1,10), (,1, 28) a d e 3,1 3,1 3,1 3,3 (, 0.5, 0), (, 0.5, 0), (,1, 5), (,1, 30) a a e 3,1 3,2 3,1 3,1 (,1, 2), (,1,17), (,1,17) i a h 3,1 3,2 3,1 (,1, 3), (,1,10), (,1,18) c i 3,3 3,1 3,1 (,1, 4), (,1,10), (,1, 21) h a 3,3 3,1 3,3 (, 0.5, 0), (, 0.5, 0), (,1, 0), (, 0.6, 3), (, 0.4, 3), (,1, 30) a a g e 3,1 3,2 3,1 3,1 3,2 3,1 x Each uzzy tuple Ki,, q, t in the D aove means that the uzzy attriute unction o m is the i m-th uzzy set o x attriute, q[0, 1] is the value o the memership a, at q and t is the time o the event. For example, with ( 3,2, 0.5,1), x i K m the value o memership unction o the uzzy set 3 at a=2 is 0.5, the transaction time is 1. Deinition 3. Let LT={lt j j=1, 2,..., p} e uzzy sets o time-intervals, lt : X [0,1] e the memership unction o the uzzy set lt j [13]. Then, j 0 mi α = 1, lt 1, 2, lt 2,, r 1, lt r 1, r is called a uzzy sequence with uzzy timeintervals i i, 1 i r is a uzzy set and lt ilt, 1 i r 1. A uzzy sequence with uzzy time-intervals whose length is r reerred to r-uzzy sequential pattern with uzzy time-intervals. Example 3. Let LT={Short, Medium, Long} e uzzy sets o time intervals. Its memership unctions as in the Fig. 2 [6] and the uzzy sequence dataase D in a Tale 3, then = 3,1, Short,, e 3,1 Medium, 3,1 is a uzzy sequence with uzzy time-intervals whose length is 3 or 3-uzzy sequence with uzzy time-intervals. a, 2 ma 7

6 1 Short Medium Long 8 Fig. 2. The memership unctions o the uzzy sets in LT Deinition 4. Given a uzzy sequence B = ( 1, q 1, t 1), ( 2, q 2, t 2),, ( r, q r, t r) and a uzzy sequence with uzzy time-intervals α = 1, lt 1, 2, lt 2,, r 1, lt r 1, r, we deine: The support o B or α, denoted y ( ), is q1 i r 1, (2) ( ) r B qi min lt (t 1 t ) i j j j r i jr A uzzy sequence B elongs to a uzzy transaction sequence S =Sid, s where s = (a 1, q 1, t 1), (a 2, q 2, t 2),.., (a n, q n, t n) i there exists a integer r such that k = a ik q k = q ik t k = t ik, 1kr, and i 1< i 2 < < i r. The support o the uzzy transaction sequence S or α, denoted y Supp S(α), is the maximum o the supports o B which elong to S, (3) Supp S(α) = max BS B. The support o a uzzy sequence with uzzy time-intervals α, denoted Supp(α), is the average o supports o all uzzy transaction sequences in D or α. NS Supp S i ( ) i1 (4) Supp(α)=, NS where S i is the i-th uzzy transaction sequence in D, NS is the numer o uzzy transaction sequences in D. A uzzy sequential pattern with uzzy time-intervals is the uzzy sequence with uzzy time-intervals whose support is not less than a user-deined threshold. Example 4. Given the uzzy transaction sequence dataase D, the memership unctions o uzzy sets o LT mentioned in the Example 3, we compute the support o two uzzy sequences with uzzy time-intervals, 3 3, Short,, 2 0 3, Medium, c, 1 3. B a, 1 3, Short,, 1 3 and a Case 1. = 3, 1, Short, 3, 1. Supp S1 () = Short(4 1)= =0.308, ecause the irst a uzzy transaction sequence has only the uzzy sequence ( 3, 1, 0.5, 1), ( 3, 1, 0.667, 4) itting. Similarly, the supports o the 5th uzzy transaction sequence and the 6th uzzy transaction sequence are Supp S5 () = 1 1 Short(5 4) = 1 1 1=1 and Supp S6 () = Short(5 0)= =0.385, respectively. Not all the rest uzzy 8

7 transaction sequences support or. So that, the support o the uzzy sequence with uzzy time-intervals is computed as ollows: Supp()=( )/10 = c Case 2. β = 3, 3, Short, 3, 2, Medium, 3, 1. In D, there is only the 4th uzzy transaction sequence, S 4, containing uzzy sequences itting β, c c S 4 = 4, ( 3,3, 1, 1), ( 3,2, 0.333, 5), ( 3,3, 0.667, 5), ( 3,1, 1, 19), ( 3,1, 0.6, 25), c ( 3,2, 0.4, 25) has two uzzy sequences, itting β that are B 1, B 2: c B 1= ( 3,3, 1, 1), ( 3,2, 0.333, 5), ( 3,3, 1, 19) has the support or β, B ( ) = min{ Short(4), Medium(14)} = min{0.846, 0.923}= 1 = =0.285, c B 2 = ( 3,3, 1, 1), ( 3,2, 0.333, 5), ( 3,1, 0.6, 25) has the support or β, B ( ) = min{ Short(4), Medium(20)} = 0.2 min{0.846, 0.615}= 2 = = Hence, the support o S 4 or β is Supp S4 () = max{ B ( ), B ( ) }= Consequently, Supp(β)=( )/10 = I the user-deined threshold is o value 0.1 then the uzzy sequence with uzzy a time-intervals 3, 1, Short, 3, 1 is a uzzy sequential pattern with uzzy timeintervals ecause its support is greater than min_sup, whereas the uzzy c sequence with uzzy time-intervals 3, 3, Short, 3, 2, Medium, 3, 1, having the support o < 0.1, is not a sequential pattern. 3. Algorithm o Mining Fuzzy Sequential Patterns with Fuzzy Time- Intervals FSPFTIM Algorithm 3.1. The prolem Input: D is a quantitative sequence dataase, min_sup is a user-deined threshold, FE is a set o uzzy sets with its memership unctions o attriutes in D, LT is a set o uzzy sets with its memership unctions o time intervals. Output: k-uzzy sequential patterns with uzzy time intervals, k > Proposed approach First, all the quantitative attriutes in D are partitioned into uzzy sets, then the transaction sequences in D are transormed into uzzy transaction sequences ased on these uzzy sets and their corresponding memership unctions. And we get a uzzy sequence dataase, called D. Next, the FSPFTIM Algorithm is applied to ind out uzzy sequential patterns with uzzy time-intervals. This algorithm is improved rom the Apriori algorithm

8 The algorithm is an iterative process consisting o several steps. At the k-th step, the algorithm generates a set denoted C k o candidate k-uzzy sequences with uzzy time intervals and then calculates the support o these candidate sequences. The sequences having the support greater than min_sup will e added into L k as a set o k-uzzy sequential patterns with uzzy time-intervals. This process will e ended when it is unale to generate any new set o uzzy sequential patterns with uzzy timeintervals. More speciically, the process o generating the sets C k is as ollows: For k=1. All uzzy attriutes o D are added into the set C 1 o candidate 1-uzzy sequences with uzzy time-intervals. By calculating the support o sequences in C 1, the set L 1 o 1-uzzy sequential patterns with uzzy time-intervals is created. For k=2. The set C 2 o 2-uzzy sequences with uzzy time-intervals are generated y joining two sequences in L 1 together and with LT in the orm o L 1LTL 1. For example, i L 1={ a, }, LT={lt 1, lt 2, lt 3}, then there are nine candidate 2-uzzy sequences with uzzy time-intervals such as a, lt 1, a, a, lt 2, a, a, lt 3, a, a, lt 1,, a, lt 2,, a, lt 3,,, lt 1,,, lt 2,,, lt 3,. For k>2. Let 1, lt 1, 2, lt 2,, lt k 2, k 1 and 2, lt 2, 3, lt 3,, lt k 1, k e 2 (k 1)-uzzy sequential patterns with uzzy time-intervals in L k 1, then α = 1, lt 1, 2, lt 2, 3, lt 3,, k 1, lt k 1, k is a candidate k-uzzy sequence with uzzy time-intervals [6]. In the same way, all candidate k-uzzy sequences with uzzy time-intervals are generated and set C k is created. The support o each candidate k-uzzy sequence with uzzy time-intervals in C k is calculated y the ormula (4) and the set L k is created. The result o the algorithm is a set o all k-uzzy sequential patterns with uzzy time-intervals L k or k> FSPFTIM Algorithm Pseudo code o the algorithm is shown in Fig. 3. In the algorithm, the operator * joins the values into a uzzy sequence with uzzy time-intervals. For example, e 1*Short*e 2 is a presentation o the uzzy sequence with uzzy time-intervals e 1, Short, e 2 where Short LT and e 1, e 2 are uzzy sets o quantitative attriutes. At line 2, each uzzy tuple, considered as a 1-uzzy sequence with uzzy timeintervals, is added into C 1. I the support o a sequence is greater than or equal to min_sup then the sequence will e added into L 1 at line 3. The set o candidates 2-uzzy sequences with uzzy time-intervals, C 2=L 1LTL 1, is generated y the lines The support o candidate 2-uzzy sequences with uzzy time-intervals in C 2 are calculated (lines 13-15) and 2-uzzy sequences with uzzy time-intervals having the support greater than or equal to min_sup, are added into L 2 (line 16). In the lines 17-24, candidate k-uzzy sequences with uzzy time-intervals are added into C k (with k >2) and y calculating their support, k-uzzy patterns with uzzy time-intervals are included in L k. The loop or generating candidate sequences is ended when L k is empty. The result o the algorithm is the set o all k-uzzy sequential patterns with uzzy time-intervals (line 24) with k>1. 10

9 Procedure FSPMFTI (D, min_sup) 1. Creating a uzzy sequence dataase D rom the sequence dataase D 2. C1{e e is an attriute o D } 3. L1{C1 Supp() min_sup} 4. C2; 5. or each e1l1 6. or each e2l1 7. or each ltdlt 8. e1*ltd*e2; 9. add to C2; 10. end or 11. end or 12. end or 13. or each C2 14. Computing the support o (Supp()); 15. end or 16. L2{C2 Supp() min_sup} 17. or (k>2;lk 1 ;k++) 18. Ckuzzy_apriori_gen(Lk 1); 19. or each cck 20. Computing the Supp() 21. end or 22. Lk{Ck Supp() min_sup} 23. end or 24. return Lk //k>1 Procedure uzzy_apriori_gen(lk 1) //generating the candidates o Ck 25. Ck; 26. or each alk or each Lk ; 29. or (i=2; i<=k 2; i++) 30. i (ai i 1 or alti lti 1) 31. reak; 32. end i 33. c*ai*alti; 34. end or 35. i (i=k 1 and ak 1=k 2) 36. a1*alt1 * c*ak 1*ltk 2*k 1; 37. add to Ck; 38. end i 39. end or 40. end or 41. return Ck Fig. 3. The FSPFTIM Algorithm The unction generating candidate k-uzzy sequences with uzzy time-intervals in C k rom L k-1 is presented in the lines The check o the conditions or comining two (k 1)-uzzy sequential patterns with uzzy time-intervals in L k 1 to creates a candidate k-uzzy sequence with uzzy time-intervals in lines A candidate k-uzzy sequence with uzzy time-intervals α is created and added into C k in lines At line 41, the set o candidate k-uzzy sequences with uzzy timeintervals, C k, is created. 11

10 The complexity o the FSPFTIM Algorithm The parameters used to evaluate the computational complexity o the proposed algorithm are as ollows: N is the numer o transaction sequences in D; M is the total numer o attriutes in D; l is the average length o transaction sequences in D; h is the numer average o o uzzy sets associated with each attriute in D; LT is the numer o uzzy sets o time-intervals LT; min_supp is a user-deined threshold. Based on the similar way as in [20], the computational complexity o the FSPFTIM algorithm is O(N. +M.N. 2 + L 1 2. ( 2 ). N. LT ) + k=3 L k 1 2. ( k ). N LT ) Details are given in the Appendix An example o executing the FSPFTIM Algorithm Input: min_sup = 0.11; Quantitative sequence dataase D in Tale 2; Fuzzy sets o attriutes and their memership unctions in the Example 2; Set o uzzy sets o time-intervals LT={Short, Medium, Long} and their memership unctions in the Example 3. Output: k-uzzy sequential patterns with uzzy time-intervals, k >1. Execution: D is created as the Tale 3; a a a c c c 3,1, 3,2, 3,3, 3,1, 3,2, 3,3, 3,1, 3,2, 3,3, C 1= d e e g h h i ; 3,1, 3,1, 3,3, 3,1, 3,3, 3,1, 3,1, 3,3, 3,1 a a d e e i,,,,,, ; = L 1= 3,1 3,2 3,1 3,3 3,1 3,1 3,3, 3, 1 C 2 (included 147 elements) = a a a a a a 3,1, Short, 3,1, 3,1, Medium, 3,1, 3,1, Long, 3,1, a a a a a a 3,1, Short, 3,2, 3,1, Medium, 3,2, 3,1, Long, 3,2, ;..., i i i i i i 3,1, Short, 3,1, 3,1, Medium, 3,1, 3,1, Long, 3,1 12

11 L 2= C 3= a a e e 3,1, Short, 3,1, 3,1, Long, 3,3, 3,1, Long, 3,3,. d e d e 3,1, Short, 3,1, 3,1, Long, 3,1, 3,1, Medium, 3,3 a e a d 3,1, Short, 3,1,Long, 3,1, 3,1, Short, 3,1, Short, 3,1, ; a e a d e 3,1, Short, 3,1,Long, 3,3, 3,1, Short, 3,1, Medium, 3,3 L 3= a d e 3,1 3,1 3,3 C 4=; L 4=. Output= 4. Experimental results, Short,, Medium, ; a a e 3,1, Short, 3,1, 3,1, Long, 3,3, e d 3,1, Long, 3,3, 3,1, Short, 3,1,. e d e 3,1, Long, 3,1, 3,1, Medium, 3,3, a d e 3,1, Short, 3,1, Medium, 3,3 The algorithm is implemented in the C# programming language and run on Chip Intel Core i5 2.5 GHz, RAM 4 GB, Windows 7 OS Datasets Two datasets used or experiment o the proposed algorithm are S100I1000T3D341K and Online Retail_France. These datasets are shown in Tale 4. The dataset S100I1000T3D341K is generated according to the theory o R. Agrawal and R. Srikant [19]. Input parameters or generating data include the numer o transactions D, the average o transactions T, the numer o attriutes I, and the numer o transaction sequences S. The length o the transactions is generated y the Poisson distriution in which the parameter is the average o transactions (T). The quantitative values and the time are random integers rom 1 up to The Online Retail_France dataset is gathered rom Online Retail [24] with the Country= France rom 01/12/2010 to 09/12/2011. The dataset includes inormation as ollows: Customer ID; Invoice Date; Stock Code; Quantity. 13

12 Tale 4. Datasets 14 Dataset Numer o attriutes (I) Numer o transactions (D) Numer o transaction sequences (S) Average o transactions (T) Average o transaction sequences S100I1000T3D341K Online Retail_France The test dataset is transormed into a quantitative sequence dataase itted or the input o the algorithm as ollows: CusId: Customer ID; Time: is an integer numer starting at 1, which is the numer o days o Invoice Date dier rom the irst day (01/12/2010); StockCode: Descriptions o stock codes ought; Quantity: Quantitative value o each StockCode. Each StockCode is partitioned into uzzy sets and their memership unctions are x deined according to the Formulae (1) in Example 2 with K=3 and A m, 3,1 x m _ Small x m, x A3,2 x m _ Average A m. 3,3 x m _ Large The ma, mi are the maximum, minimum o the quantitative values o x m StockCode. Quantitative values o the StockCode are used to calculate uzzy values. Fuzzy sets o the time-intervals and their memership unctions are deined as in ormula (1) in the Example 2, too. The ma, mi are the irst time and the last time in t t t order and A3,1 Short, A3,2 Medium, A3,3 Long where t is time-interval. The time distances are used to calculate uzzy time-intervals Results Relationships etween the numer o uzzy sequential patterns with uzzy timeintervals and min_sup, and etween the execution time and min_sup according to the dierent numers o partitions o quantitative attriutes. In this case, the numer o uzzy sets o time intervals is ixed. Namely, the time intervals are partitioned into 3 uzzy sets (K t=3). The relationships etween the numer o uzzy sequential patterns with uzzy time-intervals and min_sup, and etween the execution time and min_sup according to dierent numers o partitions o quantitative attriutes or the S100I1000T3D341K dataset are shown in the Figs 4a and, respectively, and or the Online Retail_France datasets are also shown in the Figs 5a and. From these Figs, we can see that when the numer o partitions o time intervals is ixed, the numer o uzzy sequential patterns with uzzy time-intervals as well as the execution time will e changed in inverse direction with changes o min_sup. This is true or oth the S100I1000T3D341K and Online Retail_France datasets, and does not depend on the numer o partitions o quantitative attriutes (K). Also, as K increases, the numer o uzzy sequential patterns with uzzy time-intervals decreases, ut with min_sup greater than a certain threshold, when K increases then the execution time increases. In addition, the dierence etween the numers o

13 sequence patterns, and etween the times o execution (depending on the numer o partitions o the quantitative attriutes) will e narrowed very rapidly according to the increase o min_sup. Fig. 4. Relationship etween the numer o uzzy sequential patterns with uzzy time-intervals and min_sup (a), Relationship etween the execution time and min_sup () according to the dierent numers o partitions o quantitative attriutes (K) or the S100I1000T3D341K dataset Fig. 5. Relationship etween the numer o uzzy sequential patterns with uzzy time-intervals and min_sup (a), Relationship etween the execution time and min_sup () according to dierent numers o partitions o quantitative attriutes (K) or the Online Retail_France dataset Relationships etween the numer o uzzy sequential patterns with uzzy timeintervals and min_sup, and etween the execution time and min_sup according to the dierent numers o partitions o time-intervals In this case, the numer o partitions o quantitative attriutes is 3 (K=3). Figs 6a and 7a, show the relationship etween the numer o uzzy sequential patterns with uzzy time-intervals and min_sup according to the dierent numers o partitions o the time-intervals (K t) or the S100I1000T3D341K and Online Retail_France datasets, respectively. Similarly, Figs. 6 and 7 show the relationship etween the time o execution and min_sup or the two test datasets mentioned aove. 15

14 The relationship etween the numer o uzzy sequential patterns with uzzy time-intervals and min_sup, and etween the time o execution and min_sup according to changes o the numer o partitions o time-intervals in the context the numer o partitions o the quantitative attriutes is ixed they are very similar to the relationship mentioned in Section For example, Figs 6 and 7 also show that when the numer o partitions o quantitative attriutes is ixed, i the numer o partitions o time-intervals increases then the execution time also increases. Fig. 6. Relationship etween the numer o uzzy sequential patterns with uzzy time-intervals and min_sup (a), etween the execution time and min_sup () according to the dierent numers o partitions o time-intervals (Kt) or the S100I1000T3D341K dataset Fig. 7. Relationship etween the numer o uzzy sequential patterns with uzzy time-intervals and min_sup (a), etween the execution time and min_sup () according to the dierent numers o partitions o time-intervals (Kt) or the Online Retail_France dataset 4.3. Meaning o uzzy sequential patterns with uzzy time-intervals Tale 5 shows all uzzy sequence patterns with uzzy time-intervals that were ound out using the FSPFTIM algorithm in the Online Retail_France dataset when the quantitative attriutes are partitioned into three linguistic terms {Small, Average, Large}, the time interval into three linguistic terms {Short, Medium, Long}, the user deined threshold min_sup is Some o these patterns can e explained as ollows: 16

15 POSTAGE _Small, Short, POSTAGE _Small means I a customer uys a Small numer o POSTAGE products, then he will uy a Small numer o these products an interval o Short ; POSTAGE _Small, Medium, RABBIT-NIGHT-LIGHT_ Small means I a customer uys a Small numer o POSTAGE products, then he will uy a Small numer o RABBIT-NIGHT-LIGHT an interval o Medium ; POSTAGE _Small, Medium, POSTAGE _Small, Short, POSTAGE _Small means I a customer uys a Small numer o POSTAGE products and later a Small numer o POSTAGE products an interval o Medium, then he will uy a Small numer o POSTAGE products an interval o Short. Tale 5.The uzzy sequential patterns with uzzy time-intervals in Online Retail_France dataset Fuzzy sequential patterns with uzzy time-intervals Support POSTAGE_Small, Short, POSTAGE_Small POSTAGE_Small, Medium, POSTAGE_Small POSTAGE_Small, Short, RABBIT-NIGHT-LIGHT_Small POSTAGE_Small, Medium, RABBIT-NIGHT-LIGHT_Small POSTAGE_Average, Short, POSTAGE_Small POSTAGE_Small, Short, POSTAGE_Small, Short, POSTAGE_Small POSTAGE_Small, Medium, POSTAGE_Small, Short, POSTAGE_Small Conclusions and uture work The paper proposes the FSPFTIM algorithm to mine uzzy sequential patterns with uzzy time-intervals in quantitative sequence dataases. The proposed algorithm is presented in oth idea and pseudo code. An example o application o the algorithm is also illustrated in this paper. The experimental results show the inluence o some input parameters to the numer o uzzy sequential patterns with uzzy time-intervals and the execution time. In this paper, the FSPFTIM algorithm is developed ased on the Apriori algorithm with the approach o search in readth irst. In the uture, we will continue to work on developing more eicient algorithms to mine uzzy sequence patterns with uzzy time intervals. These algorithms will e developed ased on the approach o search in depth irst, such as ased on the projected dataase organization [10] or the FPTree algorithm [11]. R e e r e n c e s 1. A g r a w a l, R., R. S r i k a n t. Mining Sequential Patterns. In: Proc. o 11th International Conerence on Data Engineering, Cao, H., et al. A Fuzzy Sequential Pattern Mining Algorithm Based on Independent Pruning Strategy or Parameters Optimization o Ball Mill Pulverizing System. Inormation Technology and Control, Vol. 43, 2014, No 3, pp C h a n g, C.-I., H.-E. C h u e h, N. P. Lin. Sequential Patterns Mining with Fuzzy Time-Intervals. In: 6th International Conerence on FSKD 09, Vol. 3, C h a n g, C.-I., H.-E. C h u e h, d Y.-C. Luo. An Integrated Sequential Patterns Mining with Fuzzy Time-Intervals In: International Conerence on Systems and Inormatics (ICSAI 12),

16 5. C h e n, R.-S. et al. Discovery o Fuzzy Sequential Patterns or Fuzzy Partitions in Quantitative Attriutes. In: ACS/IEEE International Conerence on Computer Systems and Applications, 2001, pp C h e n, Y.-L., T. C.-K. H u a n g. Discovering Fuzzy Time-Interval Sequential Patterns in Sequence Dataases. IEEE Transactions on Systems, Man, and Cyernetics, Part B (Cyernetics), Vol. 35, 2005, No 5, pp C h e n, Y.-L., M.-C. C h i a n g, M.-T. K o. Discovering Time-Interval Sequential Patterns in Sequence Dataases. Expert Systems with Applications, Vol. 25, 2003, No 3, pp F o w k e s, J., C. S u t t o n. A Susequence Interleaving Model or Sequential Pattern Mining. arxiv preprint arxiv: , G a r o a l a k i s, M. N., R. R a s t o g i, K. S h i m. SPIRIT: Sequential Pattern Mining with Regular Expression Constraints. VLDB, Vol. 99, Han, J., et al. Preixspan: Mining Sequential Patterns Eiciently y Preix-Projected Pattern Growth. In: Proc. o 17th International Conerence on Data Engineering, Han, J., J. Pei, Y. Yin. Mining Frequent Patterns without Candidate Generation. In: ACM Sigmod Record, Vol. 29, H o n g, T.-P., C.-S. Kuo, S.-C. Chi. Mining Fuzzy Sequential Patterns rom Quantitative Data. In: Proc. o Conerence IEEE SMC 99, Vol. 3, H u, Y.-C., et al. A Fuzzy Data Mining Algorithm or Finding Sequential Patterns. International Journal o Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 11, 2003, No 2, pp H u, Y.-C., G.-H. T z e n g, C.-M. C h e n. Deriving Two-Stage Learning Sequences rom Knowledge in Fuzzy Sequential Pattern Mining. Inormation Sciences, Vol. 159, 2004, No 1, pp H u a n g, T., et al. Extracting Various Types o Inormative We Content via Fuzzy Sequential Pattern Mining. Asia-Paciic We (APWe) and We-Age Inormation Management (WAIM) Joint Conerence on We and Big Data. Springer, Cham, 2017, pp H u a n g, T. C.-K. Discovery o Fuzzy Quantitative Sequential Patterns with Multiple Minimum Supports and Adjustale Memership Functions. Inormation Sciences, Vol. 222, 2013, pp K i e u, T., et al. Mining Top-K Co-Occurrence Items With Sequential Pattern. Expert Systems with Applications, Vol. 85, 2017, pp Kuo, R. J., C. M. C h a o, C. Y. Liu. Integration o k-means Algorithm and AprioriSome Algorithm or Fuzzy Sequential Pattern Mining. Applied Sot Computing, Vol. 9, 2009, No 1, pp A g r a w a l, R., R. S r i k a n t. Fast Algorithms or Mining Association Rules. In: J. Bocca, M. Jarke, C. Zaniolo, Eds. Proc. o 20th International Conerence on Very Large Data Bases (VLDB 94). Santiago, Morgan Kaumann, 1994, pp Tan, P.-N., M. S t e i n a c h, V. K u m a r. Association Analysis: Basic Concepts and Algorithms. Introduction to Data Mining, 2005, pp P h u o n g, T. D., D. V. T h a n h, N. D. D u n g. An Eective Algorithm or Association Rules Mining rom Temporal Quantitative Dataases. Indian Journal o Science and Technology, Vol. 9, 2016, No W r i g h t, A. P., et al. The Use o Sequential Pattern Mining to Predict Next Prescried Medications. Journal o Biomedical Inormatics, Vol. 53, 2015, pp Y u, C.-C., Y.-L. C h e n. Mining Sequential Patterns rom Multidimensional Sequence Data. IEEE Transactions on Knowledge and Data Engineering, Vol. 17, 2005, No 1, pp UCI-Machine Learning Repository. June

17 Appendix The computational complexity o the FSPFTIM algorithm is estimated as ollows: Creating the uzzy sequence dataase D : Each uzzy transaction sequence in D has uzzy attriutes, so the computational complexity o converting the quantitative sequence dataase D to the uzzy sequence dataase D' is O(N.). Generating C 1 and calculating the support o sequences in C 1: C 1 includes the M.h candidate 1-uzzy sequences with uzzy time-intervals (in this case, they are the uzzy attriutes in D ). To calculate the support or each candidate sequence in C 1, it is necessary to look at all uzzy transaction sequences in D as well as all uzzy attriutes in these transaction sequences. So, the computational complexity is O(M.h). O(N.) = O(M.N. 2 ). Generating C 2 and calculating the support o sequences in C 2 (or creating L 2 or short): C 2 the set o candidate 2-uzzy sequences with uzzy time-intervals is created y joining two sequences in L 1 together and with LT, namely, C 2=L 1LTL 1. The numer o candidate sequences in C 2 can e L 1 2. LT. To calculate the support o each candidate sequence in C 2, we have to look at all uzzy transaction sequences in D as well as all 2-uzzy sequences in these transaction sequences, i.e., the computational complexity o calculating the support o each candidate sequence in C 2 is O(N.( 2 )). Hence the computational complexity o creating L2 is O( L1 2. LT ). O(N.( 2 )) = O L1 2. ( 2 ). N. LT ). Generating C k and calculating the support o sequences in C k with k>2 (or creating L k or short): C k is generated y merging two (k 1)-uzzy sequential patterns with uzzy time-intervals in L k 1, so the numer o sequences in C k can e L k 1 2. LT. Check o merging conditions requires a maximum o 2k 5 comparisons, including k 2 comparisons or uzzy attriutes and k 3 comparisons or uzzy time-intervals. Hence, the cost o generating C k is O((2k 5). L k 1 2. LT ). And similar to the aove, the computational complexity o each sequence in C k is O(N.( k )). Thereore the computational complexity o creating L k (k >2) is O((2k 5). L k 1 2. LT ). O(N.( k )) = O((2k 5). Lk 1 2.( k ). N LT ). Finally, the computational complexity o the proposed algorithm is O(N. +M.N. 2 + L 1 2. ( 2 ). N. LT ) + k=3 (2k 5). L k 1 2. ( k ). N LT ) = O(N. +M.N. 2 + L 1 2. ( 2 ). N. LT ) + k=3 L k 1 2. ( k ). N LT ) Note: L k (k 1) depends so much on min_sup. Received ; Second Version ; Accepted