Agrawal et al, Mining sequential patterns, Data Eng., 1995

Mining Sequential PatternsRakesh Agrawal, Ramakrishana Srikant (1995)

IBM Almaden Research Center 650 Harry Road, San Jose, CA 95120

데이터 연구실 이세린지도 교수 박종수

2014. 3. 28

Contents

• Abstract1. Introduction2. Finding Sequential Patterns3. The Sequence Phase4. Performance5. Conclusions and Future Work

2

Abstract

• Introduces the problem of mining sequential pat-terns over a large database.

• Presents 3 algorithms to solve this problem.

• Shows their results of performance and scale-up experiments.

3

1. Introduction

• 1.1 Problem Statement

• The problem of mining sequential patterns is to find the maximal sequences among all sequences that have a certain user-specified minimum sup-port.

• Each such maximal sequence represents a se-quential pattern.

4

1. Introduction

• 1.2 Related Work– R. Agrawal, T. Imielinski, and A. Swami, “Mining association rules between

sets of items in large databases” (1993)– T. G. Dietterich and R. S. Michalski, “Discovering patterns in sequences of

events, Artificial Intelligence” (1985)– A. Califano and I. Rigoutsos, “Flash: A fast look-up algorithm for string homol-

ogy” (1993)– S. Wu and U. Manber, “Fast text searching allowing errors” (1992)– M. Waterman, “Mathematical Methods for DNA Sequence Analysis” (1989)– S. Altschul, W. Gish, W. Miller, E. Myers, and D. Lipman, “A basic local align-

ment search tool” (1990)– M. Roytberg, “Computer Applications in the Biosciences: A search for common

patterns in many sequences” (1992)– M. Vingron and P. Argos, “Computer Applications in the Biosciences: A fast

and sensitive multiple sequence alignment algorithm” (1992)– J. T.-L. Wang, G.-W. Chrin, T. G. Marr, B. Shapiro, D. Shasha, and K. Zhang.

“Combinatorial pattern discovery for scientific data: Some preliminary re-sults” (1994)

5

1. Introduction

• 1.2 Related Work– Finding of items bought together in a transac-

tion. (Intra-transaction patterns)– AI prediction of the sequential pattern.– Finding matches for pattern in text subse-

quences.– Discovering similarities in a database of ge-

netic sequences.

6

Compari-son

1. Introduction

• 1.2 Related Work

7

• Element is a character• Not guaranteed to be complete.• Algorithm was main-memory based and was tested

against a database of 150 sequences

“Combinatorial pattern discovery for scien-tific data: Some preliminary results” (1994)

• Element is a set of character• Guarantee that we have discovered all sequential patterns

of interest that are present in a specified minimum number of sequences.

• Our Solution is targeted at millions of customer sequences.

“Mining sequential pattern” (1995)

1. Introduction

• 1.3 Organization of the Paper

• Section 2. Gives this problem decomposition.• Section 3. Examines the sequence phase in detail and presents al-

gorithms for this phase. • Section 4. Empirically evaluate the performance of these algo-

rithms and study their scale-up properties.• Section 5. conclusion - summary and directions for future work.

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Terminology

9

Length of a sequence

the number of itemsets in the sequence.

K-sequence a sequence of length k.

x.y. sequence formed by the concatenation of two se-quences x and y.

Support for an item

fraction of customers who bought the items in I in a single transaction. (Thus, the itemset I and the 1-sequence <i> have the same support.)

litemset the itemset with minimum support.

Large se-quence

must be a list of litemsets.

Maximal se-quence

Final sequence pattern that satisfy the minimum sup-port.

2. Finding Sequential Patterns

• 2.1 The Algorithm• 1. Sort Phase• Converts the original transaction database into a

database of customer sequences.

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• 2.1 The Algorithm• 2. Litemset Phase• Find the set of all litemsets L including the set of

all 1-sequences.

• The set of litemsets is mapped to a set of con-tiguous integers.

11


• 2.1 The Algorithm• 3. Transformation Phase• To process repetitive determination in the follow-

ing step faster,• Each transaction is replaced by the set of all

litemsets contained in that transaction.

12


• 2.1 The Algorithm• 4. Sequence Phase• Use the set of litemsets to find the desired se-

quences.

• 5. Maximal Phase• Find the maximal sequences among the set of

large sequences.

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3. The Sequence Phase

• Make multiple passes over the data to generate candidate sequences from seed set of large se-quences.

14

Candi-date Seq.

<1>

<2>

<3>

<4>

<5>

Cus-tomer Id

Transformed Cus-tomer Seq. (After

Mapping)

1 <{1} {5}>

2 <{1} {2, 3, 4}>

3 <{1, 3}>

4 <{1} {2, 3, 4}, {5}>

5 <{5}>

25% (Support > 1.25)

Sup-port

4

2

3

2

3


• 2 Families of algorithms

Count-all AprioriAll

Count-some

15

AprioriSome

DynamicSome

AprioriSome DynamicSome

Both have forward phase and Backward phase .

using only the large se-quences and then makes a pass over the data to find

their support.

on-the-fly using the large sequences and the cus-tomer sequences read

from the database.


• 3.1 Algorithm AprioriAll

16

Large 1-sequence

<1>

<2>

<3>

<4>

<5>

litemsetphase

<1>

<2>

<3>

<4>

<5>

①

②Candi-

date Seq.Sup-port

<1 2> 2

<1 3> 2

<1 4> 2

<1 5> 1

︙ ︙<5 4> 0

Large 2-se-

quence

Sup-port

<1 2> 2

<1 3> 3

<1 4> 2

<1 5> 2

<2 3> 2

<2 4> 2

<3 4> 2

③Maximal Se-

quences

<1 2 3 4>

<1 3 5>

<4 5>


• 3.1 Algorithm AprioriAll• 3.1.1 Apriori Candidate Generation

17

p q3-seq

<1 2 3>

<1 2 4>

<1 3 4>

<1 3 5>

<2 3 4>

Candidate Seq.

<1 2 3 4>

<1 2 4 3>

<1 3 4 5>

<1 3 5 4>

3-seq

<1 2 3>

<1 2 4>

<1 3 4>

<1 3 5>

<2 3 4>

Candidate Seq.

Support

<1 2 3 4>

2

<1 2 4 3>

0

<1 3 4 5>

1

<1 3 5 4>

0

𝐶𝑘𝐿𝑘−1 𝐿𝑘−1

Join


• 3.2 Algorithm AprioriSome

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• 3.2 Algorithm AprioriSome• In the forward pass, we only count sequences of

certain lengths.

length1 length2 length3 length4 length5 length6

Forward phase

Backward phase

• Forward phase procedure

𝐶1 𝐿1 𝐶2 𝐿2 𝐶3 𝐶4 𝐿4 …

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pruning pruning


• 3.2 Algorithm AprioriSome• Next() takes as parameter the length of se-

quences counted in the last pass and returns the length of sequences to be counted in the next pass.

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𝒉𝒊𝒕𝒌=|𝐿𝑘|/¿𝐶𝑘∨¿


• 3.3 Algorithm DynamicSome

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𝐶1 𝐶𝑠𝑡𝑒𝑝∧𝐿1 𝐿𝑠𝑡𝑒𝑝

Has to be initialized

If step = 3,After initialization of 1, 2, 3, Generate 6, 9, 12 …

* Backward phase is same as AprioriSome.



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𝐶1𝐿1 𝐶2 𝐿2 𝐶3 𝐿3 𝐶4 𝐿4 …

𝐶1𝐿1 𝐶2 𝐿2 𝐶3 𝐶4 𝐿4 …

𝐶1𝐿1 𝐶2 𝐿2 𝐶4 𝐿4 …

Apriori-All

AproriSome

DynamicSome



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otf-generates (On-the-fly) gen-erates more candidates than apriori-generate.

Avoid overlap-ping

𝐿𝑘⋈𝐿 𝑗=𝐶𝑘+ 𝑗

3 Algorithm Example

• AprioriAll / AprioriSome / DynamicSome ①

24

Large 1-sequence

<1>

<2>

<3>

<4>

<5>

Candidate2-se-

quence

Support

<1 2> 2

<1 3> 4

<1 4> 3

<1 5> 3

︙ ︙

<5 4> 0

Large 2-se-

quence

Sup-port

<1 2> 2

<1 3> 4

<1 4> 3

<1 5> 3

<2 3> 2

<2 4> 2

<3 4> 3

<3 5> 2

<4 5> 2

Customer Sequences

<(1 5) (2) (3) (4)>

<(1) (3) (4) (3 5)>

<(1) (2) (3) (4)>

<(1) (3) (5)>

<(4) (5)>

3 Algorithm Example

• AprioriAll ②

25

Large 2-se-

quence

Sup-port

<1 2> 2

<1 3> 4

<1 4> 3

<1 5> 3

<2 3> 2

<2 4> 2

<3 4> 3

<3 5> 2

<4 5> 2 Maximal Se-quences

<1 2 3 4>

<1 3 5>

<4 5>

Large3-se-

quence

Support

<1 2 3> 2

<1 2 4> 2

<1 3 4> 3

<1 3 5> 2

<2 3 4> 2

Candidate 3-se-

quence

Support

<1 2 3> 2

<1 2 4> 2

<1 2 5> 0

︙ ︙

<3 4 5> 1

Candidate 4-sequence

Sup-port

<1 2 3 4> 2

<1 2 4 3> 0

<1 3 4 5> 1

<1 3 5 4> 0

Large4-sequence

Sup-port

<1 2 3 4> 2

3 Algorithm Example

• AprioriSome ②

26

Large 2-se-

quence

Sup-port

<1 2> 2

<1 3> 4

<1 4> 3

<1 5> 3

<2 3> 2

<2 4> 2

<3 4> 3

<3 5> 2

<4 5> 2

Candidate 3-se-

quence

<1 2 3>

<1 2 4>

<1 2 5>

︙

<3 4 5>


Sup-port

<1 2 3 4> 2

<1 2 3 5> 0

︙ ︙

<2 3 4 5> 0

Large4-sequence

Sup-port

<1 2 3 4> 2

Large3-se-

quence

Support

<1 2 3> 2

<1 2 4> 2

<1 3 4> 3

<1 3 5> 2

<2 3 4> 2

Candidate 3-se-

quence

Support

<1 2 3> 2

<1 2 4> 2

<1 2 5> 0

︙ ︙

<3 4 5> 1

3 Algorithm Example

• AprioriSome ③

27

Maximal Se-quences

<1 2 3 4>

<1 3 5>

<4 5>

Large4-sequence

Sup-port

<1 2 3 4> 2

3 Algorithm Example

• DynamicSome ② (step = 2)

28

Large 2-se-

quence

Sup-port

<1 2> 2

<1 3> 4

<1 4> 3

<1 5> 3

<2 3> 2

<2 4> 2

<3 4> 3

<3 5> 2

<4 5> 2

Large4-sequence

Sup-port

<1 2 3 4> 2


Sup-port

<1 2 3 4> 2

<1 2 3 5> 0

︙ ︙

<2 3 4 5> 0

3 Algorithm Example

• DynamicSome ③

29

Maximal Se-quences

<1 2 3 4>

<1 3 5>

<4 5>

Large3-se-

quence

Support

<1 2 3> 2

<1 2 4> 2

<1 3 4> 3

<1 3 5> 2

<2 3 4> 2

Candidate 3-se-

quence

Support

<1 2 3> 2

<1 2 4> 2

<1 2 5> 0

︙ ︙

<3 4 5> 1

Large4-sequence

Sup-port

<1 2 3 4> 2

4. Performance

• 4.1 Generation of Synthetic Data

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• Customer-sequence sizes are typically clustered around a mean and a few customers may have many transactions.

• Transaction sizes are usually clustered around a mean and a few transactions have many items.

• Setting: = 5,000 = 25,000N = 10,000

4. Performance

• 4.2 Relative Performance• Decreased support by 1% to 0.2%.

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Advantages Disadvantages

Avoids counting many non-maximal sequences

Faster for lower supports.

1.Number of candidates generated using AprioriSome can be larger. ( ⊇ )

2.Have to generate skipped candidates of some lengths and stay memory res-ident.

4. Performance

• 4.2 Relative Performance• Observation:- Execution time support- DynamicSome performs worse.- AprioriSome shows:

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4. Performance

• 4.3 Scale-up• Scale-up experiments for the AprioriSome algo-

rithm.(AprioriSome and AprioriAll results to be very similar.)

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4. Performance

• 4.3 Scale-up

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5. Conclusions and Future Work

• Introduced a new problem of mining sequential patterns from a database of customer sales transactions.

• Presented 3 algorithms for solving this problem.• AprioriSome and AprioriAll have comparable perfor-

mance. AprioriSome performs a little better for the lower values of

the minimum number of customers that must support a se-quential pattern.

Both scale linearly with the number of customer transactions. Both have excellent scale-up properties with respect to the

number of transactions in a customer sequence and the number of items in a transaction.

• AprioriAll is preferred in some cases that need detail counts of the number of people.

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5. Conclusions and Future Work

In the future,• Extension of the algorithms to discover

sequential patterns across item cate-gories.

• Transposition of constraints into the dis-covery algorithms. There could be item constraints or time constraints.

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Thank You

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Agrawal et al, Mining sequential patterns, Data Eng., 1995

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