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PREFIXSPAN ALGORITHM Mining Sequential Patterns Efficiently by Prefix- Projected Pattern Growth

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**Contet 1. Introduction 2. Problem statement 3. Existing Solutions**

4. Proposed Solution 5. Algorithm 6. Conclusion 7. References

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**Introduction Given a set of sequences, where**

each sequence consists of a list of elements and each element consists of set of items. <a(abc)(ac)d(cf)> - 5 elements, 9 items <a(abc)(ac)d(cf)> - 9-sequence <a(abc)(ac)d(cf)> ≠ <a(ac)(abc)d(cf)> id Sequence 10 <a(abc)(ac)d(cf)> 20 <(ad)c(bc)(ae)> 30 <(ef)(ab)(df)cb> 40 <eg(af)cbc>

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**Subsequence vs. super sequence**

Given two sequences α=<a1a2…an> and β=<b1b2…bm>. α is called a subsequence of β, denoted as α⊆ β, if there exist integers 1≤j1<j2<…<jn≤m such that a1⊆bj1, a2 ⊆bj2,…, an⊆bjn . β is a super sequence of α. Example: β =<a(abc)(ac)d(cf)> Correct : α1=<aa(ac)d(c)> α2=<(ac)(ac)d(cf)> α3=<ac> Not correct: α4=<df(cf)> α5=<(cf)d> α6=<(abc)dcf

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**Sequential Pattern Mining**

Find all the frequent subsequences, i.e. the subsequences whose occurrence frequency in the set of sequences is no less than min_support (user-specified). Solution – 53 frequent subsequences: <a><aa> <ab> <a(bc)> <a(bc)a> <aba> <abc> <(ab)> <(ab)c> <(ab)d> <(ab)f> <(ab)dc> <ac> <aca> <acb> <acc> <ad> <adc> <af> <b> <ba> <bc> <(bc)> <(bc)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> id Sequence 10 <a(abc)(ac)d(cf)> 20 <(ad)c(bc)(ae)> 30 <(ef)(ab)(df)cb> 40 <eg(af)cbc> min_support = 2

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Existing Solutions Apriori-like approaches (AprioriSome (1995.), AprioriAll (1995.), DynamicSome (1995.), GSP (1996.)): Potentially huge set of candidate sequences, Multiple scans of databases, Difficulties at mining long sequential patterns. FreeSpan (2000.) - pattern groth method (Frequent pattern-projected Sequential pattern mining) General idea is to use frequent items to recursively project sequence databases into a smaller projected databases , and grow subsequence fragments in each projected database. PrefixSpan (Prefix-projected Sequential pattern mining) Less projections and quickly shrinking sequence.

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Prefix Given two sequences α=<a1a2…an> and β=<b1b2…bm>, m≤n. Sequence β is called a prefix of α if and only if: bi= ai for i ≤ m-1; bm ⊆ am; Example : α =<a(abc)(ac)d(cf)> β =<a(abc)a>

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**Projection Given sequences α and β, such that β is a subsequence of α.**

A subsequence α’ of sequence α is called a projection of α w.r.t. β prefix if and only if: α’ has prefix β; There exist no proper super-sequence α’’ of α’ such that: α’’ is a subsequence of α and also has prefix β. Example: α =<a(abc)(ac)d(cf)> β =<(bc)a> α’ =<(bc)(ac)d(cf)>

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**Postfix Let α’ =<a1,a2…an> be the projection of α w.r.t. prefix**

β=<a1a2…am-1a’m> (m ≤n) . Sequence γ=<a’’mam+1…an> is called the postfix of α w.r.t. prefix β, denoted as γ= α/ β, where a’’m=(am - a’m). We also denote α =β ⋅ γ. Example: α’ =<a(abc)(ac)d(cf)>, β =<a(abc)a>, γ=<(_c)d(cf)>.

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**PrefixSpan – Algorithm**

Input of the algorithm : A sequence database S, and the minimum support threshold min_support. Output of the algorithm: The complete set of sequential patterns. Subroutine: PrefixSpan(α, L, S|α). Parameters: α: sequential pattern, L: the length of α; S|α: : the α-projected database, if α ≠<>; otherwise; the sequence database S. Call PrefixSpan(<>,0,S). id Sequence 10 <a(abc)(ac)d(cf)> 20 <(ad)c(bc)(ae)> 30 <(ef)(ab)(df)cb> 40 <eg(af)cbc>

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**PrefixSpan – Algorithm (2)**

Method: 1. Scan S|α once, find the set of frequent items b such that: b can be assembled to the last element of α to form a sequential pattern; or <b> can be appended to α to form a sequential pattern. 2. For each frequent item b: append it to α to form a sequential pattern α’ and output α’; output α’; 3. For each α’: construct α’-projected database S|α’ and call PrefixSpan(α’, L+1, S|α’).

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**PrefixSpan - Example 1. Find length1sequential patterns:**

2. Divide search space id Sequence 10 <a(abc)(ac)d(cf)> 20 <(ad)c(bc)(ae)> 30 <(ef)(ab)(df)cb> 40 <eg(af)cbc> <a> <b> <c> <d> <e> <f> <g> 4 3 1 <a><b><c><d><e><f> Prefix <a> <(abc)(ac)d(cf)> <(_d)c(bc)(ae)> <(_b)(df)cb> <(_f)cbc> <b> <(_c)(ac)d(cf)> <(_c)(ae)> <(df)cb> <c> <c> <(ac)d(cf)> <(bc)(ae)> <b> <bc> <d> <(cf)> <c(bc)(ae)> <(_f)cb> <e> <(_f)(ab)(df)cb> <(af)cbc> <f> <(ab)(df)cb> <cbc>

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**PrefixSpan – Example (2)**

Find subsets of sequential patterns: <d> <(cf)> <c(bc)(ae)> <(_f)cb> <a> <b> <c> <d> <e> <f> <_f> 1 2 3 <db> <dc> <db> <(_c)(ae)> <dc> <(bc)(ae)> <b> <b> <a> <e> <c> 2 1 <dcb> <dcb> <>

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**Conclusions PrefixSpan Efficient pattern growth method.**

Outperforms both GSP and FreeSpan. Explores prefix-projection in sequential pattern mining. Mines the complete set of patterns, but reduces the effort of candidate subsequence generation. Prefix-projection reduces the size of projected database and leads to efficient processing. Bi-level projection and pseudo-projection may improve mining efficiency.

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References Pei J., Han J., Mortazavi-Asl J., Pinto H., Chen Q., Dayal U., Hsu M., “PrefixSpan: Mining Sequential Patterns Efficiently by Prefix- Projected Pattern Growth”, 17th International Conference on Data Engineering (ICDE), April 2001. Agrawal R., Srikant R., “Mining sequential patterns”, Proceedings Int. Conf. Very Large Data Bases (VLDB’94), pp , 1999. Han J., Dong G., Mortazavi-Asl B., Chen Q., Dayal U., Hsu M.-C., ”Freespan: Frequent pattern-projected sequential pattern mining”, Proceedings 2000 Int. Conf. Knowledge Discovery and Data Mining (KDD’00), pp , 2000. Wojciech Stach, “http://webdocs.cs.ualberta.ca/~zaiane/courses/cmput /slides/PrefixSpan-Wojciech.pdf”.

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**Thank you for attention. Questions**

Thank you for attention. Questions? Lazar Arsić 2011/3301

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