FP-growth and PrefixSpan
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1 FP-growth and PrefixSpan
2 n Challenges of Frequent Pattern Mining n Improving Apriori n Fp-growth n Fp-tree n Mining frequent patterns with FP-tree n PrefixSpan
3 Challenges of Frequent Pattern Mining n Challenges n Multiple scans of transaction database n Huge number of candidates n Tedious workload of support counting for candidates n Improving Apriori: general ideas n Reduce passes of transaction database scans n Shrink number of candidates n Facilitate support counting of candidates
4 Transactional Database
5
6 Association Rule Mining n Find all frequent itemsets n Generate strong association rules from the frequent itemsets n Apriori algorithm is mining frequent itemsets for Boolean associations rules
7 Improving Apriori n Reduce passes of transaction database scans n Shrink number of candidates n Facilitate support counting of candidates n Use constraints
8 The Apriori Algorithm Example Database D TID Items Scan D itemset sup. {1} 2 {2} 3 {3} 3 {4} 1 {5} 3 C 1 L 1 itemset sup {1 2} 1 {1 3} 2 {1 5} 1 {2 3} 2 {2 5} 3 {3 5} 2 C 2 C 2 L 2 itemset sup Scan D {1 3} 2 {2 3} 2 {2 5} 3 {3 5} 2 C 3 itemset Scan D L 3 {2 3 5} itemset sup {2 3 5} 2 itemset sup. {1} 2 {2} 3 {3} 3 {5} 3 itemset {1 2} {1 3} {1 5} {2 3} {2 5} {3 5}
9 Apriori + Constraint Database D TID Items Scan D itemset sup. {1} 2 {2} 3 {3} 3 {4} 1 {5} 3 C 1 L 1 itemset sup {1 2} 1 {1 3} 2 {1 5} 1 {2 3} 2 {2 5} 3 {3 5} 2 C 2 C 2 L 2 itemset sup Scan D {1 3} 2 {2 3} 2 {2 5} 3 {3 5} 2 C 3 itemset Scan D L 3 {2 3 5} itemset sup {2 3 5} 2 itemset sup. {1} 2 {2} 3 {3} 3 {5} 3 itemset {1 2} {1 3} {1 5} {2 3} {2 5} {3 5} Constraint: Sum{S.price} < 5
10 Push an Anti-monotone Constraint Deep Database D TID Items Scan D itemset sup. {1} 2 {2} 3 {3} 3 {4} 1 {5} 3 C 1 L 1 itemset sup {1 2} 1 {1 3} 2 {1 5} 1 {2 3} 2 {2 5} 3 {3 5} 2 C 2 C 2 L 2 itemset sup Scan D {1 3} 2 {2 3} 2 {2 5} 3 {3 5} 2 C 3 itemset Scan D L 3 {2 3 5} itemset sup {2 3 5} 2 itemset sup. {1} 2 {2} 3 {3} 3 {5} 3 itemset {1 2} {1 3} {1 5} {2 3} {2 5} {3 5} Constraint: Sum{S.price} < 5
11 Transaction reduction n A transaction which does not contain frequent k-itemsets should be removed from the database for further scans
12 Partition: Scan Database Only Twice n Any itemset that is potentially frequent in DB must be frequent in at least one of the partitions of DB n Scan 1: partition database and find local frequent patterns n Scan 2: consolidate global frequent patterns n A. Savasere, E. Omiecinski and S. Navathe, VLDB 95 DB 1 + DB DB k = DB sup 1 (i) < σdb 1 sup 2 (i) < σdb 2 sup k (i) < σdb k sup(i) < σdb
13 Partitioning n First scan: Subdivide the transactions of database D into n non overlapping partitions If the minimum support in D is min_sup represented by frequency, then the minimum support for a partition is min_sup / number of transactions in D * number of transactions in that partition Local frequent items are determined A local frequent item my not by a frequent item in D n Second scan: Frequent items are determined from the local frequent items
14 Sampling n Pick a random sample S of D n Search for local frequent items in S n Use a lower support threshold n Determine frequent items from the local frequent items n Frequent items of D may be missed n For completeness a second scan is done
15 Is Apriori fast enough? n Basics of Apriori algorithm n Use frequent (k-1)-itemsets to generate k- itemsets candidates n Scan the databases to determine frequent k- itemsets
16 n It is costly to handle a huge number of candidate sets n If there are 10 4 frequent 1-itemsts, the Apriori algorithm will need to generate more than itemsets and test their frequencies
17 n To discover a 100-itemset n candidates have to be generated =1.27*10 30 (Do you know how big this number is?)... 7*10 27 number of atoms of a person 6*10 49 number of atoms of the earth number of the atom of the universe
18 Bottleneck of Apriori n Mining long patterns needs many passes of scanning and generates lots of candidates n Bottleneck: candidate-generation-and-test n Can we avoid candidate generation? n May some new data structure help?
19 Mining Frequent Patterns (FP) Without Candidate Generation n Grow long patterns from short ones using local frequent items n abc is a frequent pattern n Get all transactions having abc : DB abc n d is a local frequent item in DB abc à abcd is a frequent pattern
20 Construct FP-tree from a Transaction Database TID Items bought (ordered) frequent items 100 {f, a, c, d, g, i, m, p} {f, c, a, m, p} 200 {a, b, c, f, l, m, o} {f, c, a, b, m} 300 {b, f, h, j, o, w} {f, b} 400 {b, c, k, s, p} {c, b, p} 500 {a, f, c, e, l, p, m, n} {f, c, a, m, p} 1. Scan DB once, find frequent 1-itemset (single item pattern) 2. Sort frequent items in frequency descending order, f-list 3. Scan DB again, construct FP-tree Header Table Item frequency head f 4 c 4 a 3 b 3 m 3 p 3 F-list=f-c-a-b-m-p min_support = 3 {} f:4 c:1 c:3 a:3 b:1 b:1 p:1 m:2 p:2 b:1 m:1
21 Benefits of the FP-tree Structure n Completeness n Preserve complete information for frequent pattern mining n Never break a long pattern of any transaction n Compactness n Reduce irrelevant info infrequent items are gone n Items in frequency descending order: the more frequently occurring, the more likely to be shared n Never be larger than the original database (not count node-links and the count field) n There exists examples of databases, where compression ratio could be over 100
22 n The size of the FP-trees bounded by the overall occurrences of the frequent items in the database n The height of the tree is bound by the maximal number of frequent items in a transaction
23 Partition Patterns and Databases n Frequent patterns can be partitioned into subsets according to f-list f-list=f-c-a-b-m-p Patterns containing p Patterns having m but no p Patterns having c but no a nor b, m, p Pattern f n Completeness and non-redundency
24 Find Patterns Having p From p-conditional Database n Starting at the frequent item header table in the FP-tree n Traverse the FP-tree by following the link of each frequent item p n Accumulate all of transformed prefix paths of item p to form p s conditional pattern base {} Header Table Item frequency head f 4 c 4 a 3 b 3 m 3 p 3 m:2 c:3 a:3 f:4 c:1 b:1 p:2 m:1 b:1 b:1 p:1 Conditional pattern bases item cond. pattern base c f:3 a fc:3 b fca:1, f:1, c:1 m fca:2, fcab:1 p fcam:2, cb:1
25 From Conditional Pattern-bases to Conditional FP-trees n For each pattern-base Header Table Item frequency head f 4 c 4 a 3 b 3 m 3 p 3 n Accumulate the count for each item in the base n Construct the FP-tree for the frequent items of the pattern base m:2 c:3 a:3 {} f:4 c:1 b:1 p:2 m:1 b:1 b:1 p:1 m-conditional pattern base: fca:2, fcab:1 Ú {} f:3 c:3 Ú All frequent patterns relate to m m, fm, cm, am, fcm, fam, cam, fcam -> associations a:3 m-conditional FP-tree
26 Recursion: Mining Each Conditional FP-tree {} {} Cond. pattern base of am : (fc:3) f:3 f:3 c:3 a:3 m-conditional FP-tree Cond. pattern base of cm : (f:3) c:3 am-conditional FP-tree {} f:3 cm-conditional FP-tree {} Cond. pattern base of cam : (f:3) f:3 cam-conditional FP-tree
27 item conditional pattern base conditional FP-tree
28 Mining Frequent Patterns With FP-trees n Idea: Frequent pattern growth n Recursively grow frequent patterns by pattern and database partition n Method n For each frequent item, construct its conditional pattern-base, and then its conditional FP-tree n Repeat the process on each newly created conditional FP-tree n Until the resulting FP-tree is empty, or it contains only one path single path will generate all the combinations of its sub-paths, each of which is a frequent pattern
29
30 Experiments: FP-Growth vs. Apriori Data set T25I20D10K D1 FP-grow th runtime D1 Apriori runtime 70 Run time(sec.) Support threshold(%)
31 n Advantage when support decrease n No prove n advantage is shown by experiments with artificial data
32 Advantages of FP-Growth n Divide-and-conquer: n decompose both the mining task and DB according to the frequent patterns obtained so far n leads to focused search of smaller databases n Other factors n no candidate generation, no candidate test n compressed database: FP-tree structure n no repeated scan of entire database n basic ops counting local freq items and building sub FP-tree, no pattern search and matching
33 PrefixSpan Algorithm n Mining Sequential Patterns Efficiently by Prefix-Projected Pattern Growth n Concerned with finding statistically relevant patterns between data examples where the values are delivered in a sequence n Texts, Bioinformatics, DNA
34 Introduction n 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>
35 Subsequence, super sequence n Given two sequences α=<a 1 a 2 a n > and β=<b 1 b 2 b m >. n α is called a subsequence of β, denoted as α β, if there exist integers 1 j 1 <j 2 < <j n m such that a 1 b j1, a 2 b j2,, a n b jn. n β is a super sequence of α.
36 Example β =<a(abc)(ac)d(cf)> Correct (subsequences) α 1 =<aa(ac)d(c)> α 2 =<(ac)(ac)d(cf)> α 3 =<ac> Not correct (no subsequences) α 4 =<df(cf)> α 5 =<(cf)d> α 6 =<(abc)dcf
37 Sequential Pattern Mining n Find all the frequent subsequences, the subsequences whose occurrence frequency in the set of sequences is no less than min_support (user-specified). min_support = 2 id Sequence 10 <a(abc)(ac)d(cf)> 20 <(ad)c(bc)(ae)> 30 <(ef)(ab)(df)cb> 40 <eg(af)cbc>
38 Example 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 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>
39 Prefix Beginning of a sequence n Given two sequences α=<a 1 a 2 a n > and β=<b 1 b 2 b m >, m n. n Sequence β is called a prefix of α if and only if: b i = a i for i m-1; b m a m n Example : α =<a(abc)(ac)d(cf)> β =<a(abc)a>
40 Projection β subsequence and what follows after it n n 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 with reference to β prefix if and only if: n α has prefix β; n There exist no proper super-sequence α of α such that: α is a subsequence of α and also has prefix β. (β and what follows after it...) n Example: α =<a(abc)(ac)d(cf)> β =<(bc)a> α =<(bc)(ac)d(cf)>
41 Postfix What follows after a subsequence β n Let α =<a 1,a 2 a n > be the projection of α with prefix β=<a 1 a 2 a m-1 a m > (m n) n Sequence γ=<a m a m+1 a n > is called the postfix of α with prefix β, denoted as γ= α/ β, where a m =(a m - a m ). n We also denote α =β γ. Example (What follows after β...) : α =<a(abc)(ac)d(cf)>, β =<a(abc)a>, γ=<(_c)d(cf)>.
42 PrefixSpan Algorithm n Input of the algorithm : A sequence database S, the minimum support threshold min_support. n Output of the algorithm: The complete set of sequential patterns. id Sequence 10 <a(abc)(ac)d(cf)> 20 <(ad)c(bc)(ae)> 30 <(ef)(ab)(df)cb> 40 <eg(af)cbc>
43 PrefixSpan Algorithm n Subroutine: PrefixSpan(α, L, S α). n Parameters: n α: sequential pattern, n L: the length of α; n S α: n : if α <> the α-projected database; : if α =<> the sequence database S. α subsequence and what follows after it n Call: PrefixSpan(<>,0,S).
44 Method 1. Scan S α once, find the set of frequent items b such that: n n 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 ; 3. For each α : construct α -projected database S α and call PrefixSpan(α, L+1, S α ).
45 PrefixSpan - Example n 1. Find length1sequential patterns min_support = 2 <a><b><c><d><e><f>
46 n 2. Divide search space
47 n Find subsets of sequential patterns:
48 Method 1. Scan S α once, find the set of frequent items b such that: n n 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 ; 3. For each α : construct α -projected database S α and call PrefixSpan(α, L+1, S α ).
49 Prefixsan characteristics n No candidate sequence needs to be generated by PrefixSpan (see FP-growth) n Projected databases keep shrinking n The major cost of PrefixSpan is the construction of projected databases
50 n How to reduce this cost? n Different projection methods n Bi-level projection reduces the number and the size of projected databases n Pseudo-Projection reduces the cost of projection when projected database can be held in main memory
51 Bi-level Projection n Scan to get 1-length sequences Construct a triangular matrix instead of projecteddatabases for each length-1 patterns
52 n For each length-2 sequential pattern α, construct the α-projected database and find the frequent items n Construct corresponding S-matrix
53 Pseudo-Projection n n n Observation: postfixes of a sequence often appear repeatedly in recursive projected databases Method: instead of constructing physical projection by collecting all the postfixes, we can use pointers referring to the sequences in the database as a pseudo-projection Every projection consists of two pieces of information: pointer to the sequence in database and offset to the postfix in the sequence
54 n Efficient pattern growth method (FP-growth) n Outperforms both GSP and FreeSpan n Explores prefix-projection in sequential pattern mining n 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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