Introduction to sequence pattern Prefixspan algorithm

Sequence

A sequence (sequence) is a set of well-ordered itemsets that are not necessarily contiguous, but still satisfy the order. The elements of a sequence pattern can also be an item set, such as a group page sequence. Sequential pattern mining can get deeper knowledge than associative mining.

sequence Pattern

Sequence patternmining, for frequent sequences, the typical application is still limited to discrete sequences, happens-after relationship and not just the consecutive Subsequences.

It can be used for purchase behavior prediction, fraud screening, fault prediction, Web user access prediction, human behavior law, etc.

The algorithm is a variety of class Apriori algorithm, there are Aprioriall, Apriorisome, GSP (generalized sequential Patterns), SPADE (sequential PAttern Discovery Using equivalence classes), Prefixspan.

the difference from the time series

Unlike time series mining, time series (or dynamic series) refers to the sequence of values of the same statistic indicator in the chronological order in which they occur. The main purpose of time series analysis is to predict the future based on the historical data. Common Ma, AR, ARMA, Garch models.

Example

<a (ABC) (AC) d (CF) >-9 items (items), 5 itemsets (itemsets), 1 sequence (sequence)

<a (ABC) (AC) d (cf) > = <a (CBA) (AC) d (CF) >

<a (ABC) (AC) d (CF) >≠<a (AC) (ABC) d (CF) >

Min Support (minimum supported) threshold-frequent subsequence frequency is not less than the minimum support level (Find all frequent subsequences,i.e. The subsequences whose occurrence Frequency in the set of sequences is noless than Min_support)

Supersequence: <a (ABC) (AC) d (CF) >

SUB-SEQUENCE:<AA (AC) d (c) >

sub-sequence:< (AC) (AC) d (CF) >

Sub-sequence:<ac>

<a (ABC) (AC) d (CF) >α1=<a> support (α1) = 4

< (AD) c (BC) (AE) >α2=<ac> support (α2) = 4

< (EF) (AB) (DF) cb>α3=< (AB) c> support (α3) = 2

<eg (AF) cbc>

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Prefixspan

prefix prefix

SEQ <a (ABC) a> is a prefix of Seq<a (ABC) (AC) d (cf);, but seq <a (ABC) c> are not. <a>, <aa>, <a (AB) >, <a (ABC) > are prefixes of sequence <a (ABC) (AC) d (CF) >, while <ab>, <a (BC) > are not.

suffix postfix

Seqβ<a (ABC) a> is a prefix and seqγ< (_c) d (CF) > is a postfix of Seqα<a (ABC) (AC) d (cf);. Denoteα=β⋅γ or γ=α/β

For sequence <a (ABC) (AC) d (CF);

< (ABC) (AC) d (CF) > is the suffix of the prefix <a>;

< (_BC) (AC) d (CF) > is the suffix of the prefix <aa>;

< (_c) (AC) d (CF) > is the suffix of the prefix <a (AB) >;

The "_" subscript represents the prefix.

Projection Projection

Projection is the projection database, which is a collection of all suffix sequences in the sequence database s that are relative to the alpha prefix.

algorithm

Sub-Program: Prefixspan (Α,l,) Parameters: Alpha refers to the sequence pattern of prefixes; L refers to the length of α; Refers to the projection database of Alpha. Algorithm: 1, scan, find frequent item set B: A) B can be the last set of items of α (e.g. AB + c=> ABC), OR: b) B can be appended to α to form a new sequence pattern (e.g. AB +_c + A (BC)); 2, for each frequent item B, append to α to form a new sequence pattern α ' (such as ABC or a (BC)); 3, for each α ', constructs α ' projection database, and calls Prefixspan (α ', l+1,) its process for depth-first search.

Advantages:

1) do not produce any of the Hou anthology, reduce space;

2) The projection database scale is decreasing (because the projection only occurs in the suffix section associated with the prefix);

3) The method of divide and conquer is used to improve the efficiency of the algorithm, and it is more stable in memory use than spade and GSP algorithm.

Disadvantages:

1) The main cost of the algorithm is the construction of the projection database, if the sequence of multiple and each sequence to establish a projection database, then the cost is relatively large (through a. Pseudo-projection technology (pseudo-projection) to reduce the number and size of projection database; b. Bi-level projection); 2) difficult to achieve.

Example

Part of the demonstration process

Final result

Code printing results (available for debug control)

Min_support:2 Input sequence:a (ABC) (AC) d (CF) Input Sequence: (AD) c (BC) (AE) Input Sequence: (EF) (AB) (DF) C B Input sequence:e g (AF) c b C frequence:a=4 b=4 c=4 d=3 e=3 f=3 g=1 support:a=4 b=4 c=4 d=3 e=3 f=3 Fullprefix~~~: A Lastprefix:a, PostFix: (ABC) (AC) d (CF) Lastprefix:a, PostFix: (_d) C (BC) (AE) Lastprefix:a, PostFix: (_b) (DF) C B Lastprefix:a, PostFix: (_f) c b C Input Sequence: (ABC) (AC) d (CF) Input Sequence: (_d) C (BC) (AE) Input Sequence: (_b) (DF) C B Input Sequence: (_f) c b C frequence:a=2 b=4 _b=2 c=4 _c=1 d=2 _d=1 e=1 f=2 _e=1 _f=1 support:a=2 b=4 _b=2 c=4 d=2 f=2 fullprefix~~~: AA Lastprefix:a, PostFix: (_BC) (AC) d (CF) Lastprefix:a, PostFix: (_e) Input Sequence: (_BC) (AC) d (CF) Input Sequence: (_e) Frequence:a=1 _b=1 c=1 _c=1 d=1 f=1 _e=1 Support fullprefix~~~: AB Lastprefix:b, PostFix: (_c) (AC) d (CF) Lastprefix:b, PostFix: (_c) (AE) Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: (_c) (AC) d (CF) Input Sequence: (_c) (AE) Input Sequence: Input Sequence:c frequence:a=2 c=2 _c=2 d=1 e=1 f=1 support:a=2 c=2 _c=2 fullprefix~~~: ABA Lastprefix:a, PostFix: (_c) d (CF) Lastprefix:a, PostFix: (_e) Input Sequence: (_c) d (CF) Input Sequence: (_e) Frequence:c=1 _c=1 d=1 f=1 _e=1 Support fullprefix~~~: ABC LASTPREFIX:C, postfix:d (CF) LASTPREFIX:C, PostFix: Input sequence:d (CF) Input Sequence: Frequence:c=1 d=1 f=1 _f=1 Support fullprefix~~~: A (BC) Lastprefix: _c, PostFix: (AC) d (CF) Lastprefix: _c, PostFix: (AE) Input Sequence: (AC) d (CF) Input Sequence: (AE) frequence:a=2 c=1 d=1 e=1 f=1 support:a=2 fullprefix~~~: A (BC) a Lastprefix:a, PostFix: (_c) d (CF) Lastprefix:a, PostFix: (_e) Input Sequence: (_c) d (CF) Input Sequence: (_e) Frequence:c=1 _c=1 d=1 f=1 _e=1 Support fullprefix~~~: (AB) Lastprefix: _b, PostFix: (_c) (AC) d (CF) Lastprefix: _b, PostFix: (DF) c B Input Sequence: (_c) (AC) d (CF) Input Sequence: (DF) c B Frequence:a=1 b=1 c=2 _c=1 d=2 f=2 support:c=2 d=2 f=2 fullprefix~~~: (AB) c LASTPREFIX:C, postfix:d (CF) LASTPREFIX:C, Postfix:b Input sequence:d (CF) Input sequence:b Frequence:b=1 c=1 d=1 f=1 _f=1 Support fullprefix~~~: (AB) d Lastprefix:d, PostFix: (CF) Lastprefix:d, PostFix: (_f) c b Input Sequence: (CF) Input Sequence: (_f) c b Frequence:b=1 c=2 f=1 _f=1 support:c=2 fullprefix~~~: (AB) DC Lastprefix:c, PostFix: (_f) LASTPREFIX:C, Postfix:b Input Sequence: (_f) Input sequence:b Frequence:b=1 _f=1 Support fullprefix~~~: (AB) F Lastprefix:f, PostFix: Lastprefix:f, Postfix:c b Input Sequence: Input Sequence:c b Frequence:b=1 c=1 Support fullprefix~~~: AC Lastprefix:c, PostFix: (AC) d (CF) Lastprefix:c, PostFix: (BC) (AE) LASTPREFIX:C, Postfix:b LASTPREFIX:C, Postfix:b C Input Sequence: (AC) d (CF) Input Sequence: (BC) (AE) Input sequence:b Input sequence:b C frequence:a=2 b=3 c=3 d=1 e=1 f=1 _f=1 support:a=2 b=3 c=3 fullprefix~~~: ACA Lastprefix:a, PostFix: (_c) d (CF) Lastprefix:a, PostFix: (_e) Input Sequence: (_c) d (CF) Input Sequence: (_e) Frequence:c=1 _c=1 d=1 f=1 _e=1 Support fullprefix~~~: ACB Lastprefix:b, PostFix: (_c) (AE) Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: (_c) (AE) Input Sequence: Input Sequence:c Frequence:a=1 c=1 _c=1 e=1 Support fullprefix~~~: ACC LASTPREFIX:C, postfix:d (CF) Lastprefix:c, PostFix: (AE) LASTPREFIX:C, PostFix: Input sequence:d (CF) Input Sequence: (AE) Input Sequence: Frequence:a=1 c=1 d=1 e=1 f=1 _f=1 Support fullprefix~~~: Ad Lastprefix:d, PostFix: (CF) Lastprefix:d, PostFix: (_f) c b Input Sequence: (CF) Input Sequence: (_f) c b Frequence:b=1 c=2 f=1 _f=1 support:c=2 Fullprefix~~~: ADC Lastprefix:c, PostFix: (_f) LASTPREFIX:C, Postfix:b Input Sequence: (_f) Input sequence:b Frequence:b=1 _f=1 Support fullprefix~~~: AF Lastprefix:f, PostFix: Lastprefix:f, Postfix:c b Input Sequence: Input Sequence:c b Frequence:b=1 c=1 Support Fullprefix~~~: b Lastprefix:b, PostFix: (_c) (AC) d (CF) Lastprefix:b, PostFix: (_c) (AE) Lastprefix:b, PostFix: (DF) c B Lastprefix:b, Postfix:c Input Sequence: (_c) (AC) d (CF) Input Sequence: (_c) (AE) Input Sequence: (DF) c B Input Sequence:c frequence:a=2 b=1 c=3 _c=2 d=2 e=1 f=2 support:a=2 c=3 _c=2 d=2 f=2 fullprefix~~~: BA Lastprefix:a, PostFix: (_c) d (CF) Lastprefix:a, PostFix: (_e) Input Sequence: (_c) d (CF) Input Sequence: (_e) Frequence:c=1 _c=1 d=1 f=1 _e=1 Support fullprefix~~~: BC LASTPREFIX:C, postfix:d (CF) LASTPREFIX:C, Postfix:b LASTPREFIX:C, PostFix: Input sequence:d (CF) Input sequence:b Input Sequence: Frequence:b=1 c=1 d=1 f=1 _f=1 Support fullprefix~~~: (BC) Lastprefix: _c, PostFix: (AC) d (CF) Lastprefix: _c, PostFix: (AE) Input Sequence: (AC) d (CF) Input Sequence: (AE) frequence:a=2 c=1 d=1 e=1 f=1 support:a=2 fullprefix~~~: (BC) A Lastprefix:a, PostFix: (_c) d (CF) Lastprefix:a, PostFix: (_e) Input Sequence: (_c) d (CF) Input Sequence: (_e) Frequence:c=1 _c=1 d=1 f=1 _e=1 Support fullprefix~~~: BD Lastprefix:d, PostFix: (CF) Lastprefix:d, PostFix: (_f) c b Input Sequence: (CF) Input Sequence: (_f) c b Frequence:b=1 c=2 f=1 _f=1 support:c=2 fullprefix~~~: BDC Lastprefix:c, PostFix: (_f) LASTPREFIX:C, Postfix:b Input Sequence: (_f) Input sequence:b Frequence:b=1 _f=1 Support fullprefix~~~: BF Lastprefix:f, PostFix: Lastprefix:f, Postfix:c b Input Sequence: Input Sequence:c b Frequence:b=1 c=1 Support Fullprefix~~~: C Lastprefix:c, PostFix: (AC) d (CF) Lastprefix:c, PostFix: (BC) (AE) LASTPREFIX:C, Postfix:b LASTPREFIX:C, Postfix:b C Input Sequence: (AC) d (CF) Input Sequence: (BC) (AE) Input sequence:b Input sequence:b C frequence:a=2 b=3 c=3 d=1 e=1 f=1 _f=1 support:a=2 b=3 c=3 Fullprefix~~~: CA Lastprefix:a, PostFix: (_c) d (CF) Lastprefix:a, PostFix: (_e) Input Sequence: (_c) d (CF) Input Sequence: (_e) Frequence:c=1 _c=1 d=1 f=1 _e=1 Support Fullprefix~~~: CB Lastprefix:b, PostFix: (_c) (AE) Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: (_c) (AE) Input Sequence: Input Sequence:c Frequence:a=1 c=1 _c=1 e=1 Support fullprefix~~~: CC LASTPREFIX:C, postfix:d (CF) Lastprefix:c, PostFix: (AE) LASTPREFIX:C, PostFix: Input sequence:d (CF) Input Sequence: (AE) Input Sequence: Frequence:a=1 c=1 d=1 e=1 f=1 _f=1 Support fullprefix~~~: D Lastprefix:d, PostFix: (CF) Lastprefix:d, Postfix:c (BC) (AE) Lastprefix:d, PostFix: (_f) c b Input Sequence: (CF) Input sequence:c (BC) (AE) Input Sequence: (_f) c b Frequence:a=1 b=2 c=3 e=1 f=1 _f=1 support:b=2 c=3 fullprefix~~~: DB Lastprefix:b, PostFix: (_c) (AE) Lastprefix:b, PostFix: Input Sequence: (_c) (AE) Input Sequence: Frequence:a=1 _c=1 e=1 Support Fullprefix~~~: DC Lastprefix:c, PostFix: (_f) Lastprefix:c, PostFix: (BC) (AE) LASTPREFIX:C, Postfix:b Input Sequence: (_f) Input Sequence: (BC) (AE) Input sequence:b Frequence:a=1 b=2 c=1 e=1 _f=1 support:b=2 fullprefix~~~: DCB Lastprefix:b, PostFix: (_c) (AE) Lastprefix:b, PostFix: Input Sequence: (_c) (AE) Input Sequence: Frequence:a=1 _c=1 e=1 Support Fullprefix~~~: E Lastprefix:e, PostFix: Lastprefix:e, PostFix: (_f) (AB) (DF) C B Lastprefix:e, Postfix:g (AF) c b C Input Sequence: Input Sequence: (_f) (AB) (DF) C B Input sequence:g (AF) c b C frequence:a=2 b=2 c=2 d=1 f=2 _f=1 g=1 support:a=2 b=2 c=2 f=2 fullprefix~~~: ea Lastprefix:a, PostFix: (_b) (DF) C B Lastprefix:a, PostFix: (_f) c b C Input Sequence: (_b) (DF) C B Input Sequence: (_f) c b C frequence:b=2 _b=1 c=2 d=1 f=1 _f=1 support:b=2 c=2 fullprefix~~~: EAB Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: Input Sequence:c Frequence:c=1 Support fullprefix~~~: EAC LASTPREFIX:C, Postfix:b LASTPREFIX:C, Postfix:b C Input sequence:b Input sequence:b C frequence:b=2 c=1 support:b=2 fullprefix~~~: EACB Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: Input Sequence:c Frequence:c=1 Support Fullprefix~~~: EB Lastprefix:b, PostFix: (DF) c B Lastprefix:b, Postfix:c Input Sequence: (DF) c B Input Sequence:c Frequence:b=1 c=2 d=1 f=1 support:c=2 fullprefix~~~: EBC LASTPREFIX:C, Postfix:b LASTPREFIX:C, PostFix: Input sequence:b Input Sequence: Frequence:b=1 Support fullprefix~~~: EC LASTPREFIX:C, Postfix:b LASTPREFIX:C, Postfix:b C Input sequence:b Input sequence:b C frequence:b=2 c=1 support:b=2 fullprefix~~~: ECB Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: Input Sequence:c Frequence:c=1 Support fullprefix~~~: EF Lastprefix:f, Postfix:c b Lastprefix:f, Postfix:c b C Input Sequence:c b Input Sequence:c b C frequence:b=2 c=2 support:b=2 c=2 fullprefix~~~: EFB Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: Input Sequence:c Frequence:c=1 Support fullprefix~~~: EFC LASTPREFIX:C, Postfix:b LASTPREFIX:C, Postfix:b C Input sequence:b Input sequence:b C frequence:b=2 c=1 support:b=2 fullprefix~~~: EFCB Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: Input Sequence:c Frequence:c=1 Support Fullprefix~~~: F Lastprefix:f, PostFix: Lastprefix:f, PostFix: (AB) (DF) C B Lastprefix:f, Postfix:c b C Input Sequence: Input Sequence: (AB) (DF) C B Input Sequence:c b C Frequence:a=1 b=2 c=2 d=1 f=1 support:b=2 c=2 fullprefix~~~: FB Lastprefix:b, PostFix: (DF) c B Lastprefix:b, Postfix:c Input Sequence: (DF) c B Input Sequence:c Frequence:b=1 c=2 d=1 f=1 support:c=2 fullprefix~~~: FBC LASTPREFIX:C, Postfix:b LASTPREFIX:C, PostFix: Input sequence:b Input Sequence: Frequence:b=1 Support Fullprefix~~~: FC LASTPREFIX:C, Postfix:b LASTPREFIX:C, Postfix:b C Input sequence:b Input sequence:b C frequence:b=2 c=1 support:b=2 fullprefix~~~: FCB Lastprefix:b, PostFix: Lastprefix:b, Postfix:c Input Sequence: Input Sequence:c Frequence:c=1 Support FULLPREFIXDB: A, AA, AB, ABA, ABC, A (BC), a (BC) A, (AB), (AB) c, (AB) d, (AB) DC, (AB) F, AC, ACA, ACB, ACC, AD, ADC, AF, B, BA, BC, (BC), (b c) A, BD, BDC, BF, C, CA, CB, CC, D, DB, DC, DCB, E, EA, EAB, EAC, EACB, EB, EBC, EC, ECB, EF, EFB, EFC, EFCB, F, FB, FBC, FC, FCB,