Data Mining Concepts & Techniques
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1 Data Mining Concepts & Techniques Lecture No. 05 Sequential Pattern Mining Naeem Ahmed Department of Software Engineering Mehran Univeristy of Engineering and Technology Jamshoro
2 Outline Data Mining: Sequential Pattern Mining Basic concepts BIDE+ algorithm Acknowledgements: Introduction to Data Mining Tan, Steinbach, Kumar
3 Sequence Data Certain order in which data (objects) are related Sequence Database: Timeline Object Timestamp Events A 10 2, 3, 5 A 20 6, 1 A 23 1 B 11 4, 5, 6 B 17 2 B 21 7, 8, 1, 2 B 28 1, 6 C 14 1, 8, 7 Object A: Object B: Object C: 1 7 8
4 Example: Sequence Data Sequence Database Sequence Element (Transaction) Event (Item) Customer Purchase history of a given customer A set of items bought by a customer at time t Books, diary products, CDs, etc Web Data Browsing activity of a particular Web visitor A collection of files viewed by a Web visitor after a single mouse click Home page, index page, contact info, etc Event data History of events generated by a given sensor Events triggered by a sensor at time t Types of alarms generated by sensors Genome sequences DNA sequence of a particular species An element of the DNA sequence Bases A,T,G,C Element (Transaction) Sequence E1 E2 E1 E3 E2 E2 E3 E4 Event (Item)
5 Sequence Formal Definition: A sequence is an ordered list of elements (transactions) s = < e 1 e 2 e 3 > Each element contains a collection of events (items) e i = {i 1, i 2,, i k } Each element is attributed to a specific time or location Length of a sequence, s, is given by the number of elements of the sequence A k-sequence is a sequence that contains k events (items)
6 Web sequence: Example: Sequence < {Homepage} {Electronics} {Digital Cameras} {Canon Digital Camera} {Shopping Cart} {Order Confirmation} {Return to Shopping} > Sequence of initiating events causing the nuclear accident at 3-mile Island: ( < {clogged resin} {outlet valve closure} {loss of feedwater} {condenser polisher outlet valve shut} {booster pumps trip} {main waterpump trips} {main turbine trips} {reactor pressure increases}> Sequence of books checked out at a library: <{Fellowship of the Ring} {The Two Towers} {Return of the King}>
7 Subsequence Formal Definition: A sequence <a 1 a 2 a n > is contained in another sequence <b 1 b 2 b m > (m n) if there exist integers i 1 < i 2 < < i n such that a 1 b i1, a 2 b i1,, a n b in Data sequence Subsequence Contain? < {2,4} {3,5,6} {8} > < {2} {3,5} > Yes < {1,2} {3,4} > < {1} {2} > No < {2,4} {2,4} {2,5} > < {2} {4} > Yes The support of a subsequence w is defined as the fraction of data sequences that contain w A sequential pattern is a frequent subsequence (i.e., a subsequence whose support is minsup)
8 Sequential Pattern Mining Pattern: an arrangement or sequence regularly found in comparable objects or events Given: a database of sequences a user-specified minimum support threshold, minsup Task: Find all subsequences with support minsup
9 Sequential Pattern Mining: Challenge Given a sequence: <{a b} {c d e} {f} {g h i}> Examples of subsequences: <{a} {c d} {f} {g} >, < {c d e} >, < {b} {g} >, etc. How many k-subsequences can be extracted from a given n-sequence? Answer n k : 9 = 4 = 126
10 Sequential Pattern Mining: Challenge A huge number of possible sequential patterns are hidden in databases A mining algorithm should find the complete set of patterns, when possible, satisfying the minimum support (frequency) threshold be highly efficient, scalable, involving only a small number of database scans be able to incorporate various kinds of user-specific constraints
11 Sequential Pattern Mining: Example Object Timestamp Events A 1 1,2,4 A 2 2,3 A 3 5 B 1 1,2 B 2 2,3,4 C 1 1, 2 C 2 2,3,4 C 3 2,4,5 D 1 2 D 2 3, 4 D 3 4, 5 E 1 1, 3 E 2 2, 4, 5 Minsup = 50% Examples of Frequent Subsequences: < {1,2} > s=60% < {2,3} > s=60% < {2,4}> s=80% < {3} {5}> s=80% < {1} {2} > s=80% < {2} {2} > s=60% < {1} {2,3} > s=60% < {2} {2,3} > s=60% < {1,2} {2,3} > s=60%
12 Extracting Sequential Patterns Given n events: i 1, i 2, i 3,, i n Candidate 1-subsequences: <{i 1 }>, <{i 2 }>, <{i 3 }>,, <{i n }> Candidate 2-subsequences: <{i 1, i 2 }>, <{i 1, i 3 }>,, <{i 1 } {i 1 }>, <{i 1 } {i 2 }>,, <{i n-1 } {i n }> Candidate 3-subsequences: <{i 1, i 2, i 3 }>, <{i 1, i 2, i 4 }>,, <{i 1, i 2 } {i 1 }>, <{i 1, i 2 } {i 2 }>,, <{i 1 } {i 1, i 2 }>, <{i 1 } {i 1, i 3 }>,, <{i 1 } {i 1 } {i 1 }>, <{i 1 } {i 1 } {i 2 }>,
13 Sequential Pattern Mining: Algorithms Concept introduction and an initial Apriori-like algorithm Agrawal & Srikant. Mining sequential patterns, ICDE 95 Apriori-based method: GSP (Generalized Sequential Patterns: Srikant & EDBT 96) Pattern-growth methods: FreeSpan & PrefixSpan (Han et al.@kdd 00; Pei, et al.@icde 01) Vertical format-based mining: SPADE (Zaki@Machine Leanining 00) Constraint-based sequential pattern mining (SPIRIT: Garofalakis, Rastogi, Shim@VLDB 99; Pei, Han, CIKM 02) Mining closed sequential patterns: CloSpan (Yan, Han & 03)
14 Generalized Sequential Pattern (GSP) Step 1: Make the first pass over the sequence database D to yield all the 1-element frequent sequences Step 2: Repeat until no new frequent sequences are found Candidate Generation: Merge pairs of frequent subsequences found in the (k-1)th pass to generate candidate sequences that contain k items Candidate Pruning: Prune candidate k-sequences that contain infrequent (k-1)-subsequences Support Counting: Make a new pass over the sequence database D to find the support for these candidate sequences Candidate Elimination: Eliminate candidate k-sequences whose actual support is less than minsup
15 Candidate Generation Base case (k=2): Merging two frequent 1-sequences <{i 1 }> and <{i 2 }> will produce two candidate 2-sequences: <{i 1 } {i 2 }> and <{i 1 i 2 }> General case (k>2): A frequent (k-1)-sequence w 1 is merged with another frequent (k-1)-sequence w 2 to produce a candidate k-sequence if the subsequence obtained by removing the first event in w 1 is the same as the subsequence obtained by removing the last event in w 2 The resulting candidate after merging is given by the sequence w 1 extended with the last event of w 2. If the last two events in w 2 belong to the same element, then the last event in w 2 becomes part of the last element in w 1 Otherwise, the last event in w 2 becomes a separate element appended to the end of w 1
16 Candidate Generation Examples Merging the sequences w 1 =<{1} {2 3} {4}> and w 2 =<{2 3} {4 5}> will produce the candidate sequence < {1} {2 3} {4 5}> because the last two events in w 2 (4 and 5) belong to the same element Merging the sequences w 1 =<{1} {2 3} {4}> and w 2 =<{2 3} {4} {5}> will produce the candidate sequence < {1} {2 3} {4} {5}> because the last two events in w 2 (4 and 5) do not belong to the same element We do not have to merge the sequences w 1 =<{1} {2 6} {4}> and w 2 =<{1} {2} {4 5}> to produce the candidate < {1} {2 6} {4 5}> because if the latter is a viable candidate, then it can be obtained by merging w 1 with < {1} {2 6} {5}>
17 GSP Example Frequent 3-sequences < {1} {2} {3} > < {1} {2 5} > < {1} {5} {3} > < {2} {3} {4} > < {2 5} {3} > < {3} {4} {5} > < {5} {3 4} > Candidate Generation < {1} {2} {3} {4} > < {1} {2 5} {3} > < {1} {5} {3 4} > < {2} {3} {4} {5} > < {2 5} {3 4} > Candidate Pruning < {1} {2 5} {3} >
18 Finding Length-1 Sequential Patterns Examine GSP using an example Initial candidates: all singleton sequences <a>, <b>, <c>, <d>, <e>, <f>, <g>, <h> Scan database once, count support for candidates min_sup =2 Seq. ID Sequence <(bd)cb(ac)> <(bf)(ce)b(fg)> <(ah)(bf)abf> <(be)(ce)d> <a(bd)bcb(ade)> Cand Sup <a> 3 <b> 5 <c> 4 <d> 3 <e> 3 <f> 2 <g> 1 <h> 1
19 GSP: Generating Length-2 Candidates 51 length-2 Candidates <a> <b> <c> <d> <e> <f> <a> <aa> <ab> <ac> <ad> <ae> <af> <b> <ba> <bb> <bc> <bd> <be> <bf> <c> <ca> <cb> <cc> <cd> <ce> <cf> <d> <da> <db> <dc> <dd> <de> <df> <e> <ea> <eb> <ec> <ed> <ee> <ef> <f> <fa> <fb> <fc> <fd> <fe> <ff> <a> <b> <c> <d> <e> <f> <a> <(ab)> <(ac)> <(ad)> <(ae)> <(af)> <b> <(bc)> <(bd)> <(be)> <(bf)> <c> <(cd)> <(ce)> <(cf)> <d> <(de)> <(df)> <e> <(ef)> <f> Without Apriori property, 8*8+8*7/2=92 candidates Apriori prunes 44.57% candidates
20 THE GSP Mining Process 5 th scan: 1 cand. 1 length-5 seq. pat. 4 th scan: 8 cand. 6 length-4 seq. pat. 3 rd scan: 47 cand. 19 length-3 seq. pat. 20 cand. not in DB at all 2 nd scan: 51 cand. 19 length-2 seq. pat. 10 cand. not in DB at all 1 st scan: 8 cand. 6 length-1 seq. pat. <(bd)cba> <abba> <(bd)bc> <abb> <aab> <aba> <baa> <bab> <aa> <ab> <af> <ba> <bb> <ff> <(ab)> <(ef)> <a> <b> <c> <d> <e> <f> <g> <h> Seq. ID Cand. cannot pass sup. threshold Cand. not in DB at all Sequence min_sup = <(bd)cb(ac)> <(bf)(ce)b(fg)> <(ah)(bf)abf> <(be)(ce)d> <a(bd)bcb(ade)>
21 Candidate Generate-and-test: Drawbacks A huge set of candidate sequences generated Especially 2-item candidate sequence Multiple Scans of database needed The length of each candidate grows by one at each database scan Inefficient for mining long sequential patterns A long pattern grow up from short patterns The number of short patterns is exponential to the length of mined patterns
22 Timing Constraints (I) {A B} {C} {D E} <= x g >n g x g : max-gap n g : min-gap <= m s m s : maximum span x g = 2, n g = 0, m s = 4 Data sequence Subsequence Contain? < {2,4} {3,5,6} {4,7} {4,5} {8} > < {6} {5} > Yes < {1} {2} {3} {4} {5}> < {1} {4} > No < {1} {2,3} {3,4} {4,5}> < {2} {3} {5} > Yes < {1,2} {3} {2,3} {3,4} {2,4} {4,5}> < {1,2} {5} > No
23 Mining Sequential Patterns with Timing Constraints Approach 1: Mine sequential patterns without timing constraints Postprocess the discovered patterns Approach 2: Modify GSP to directly prune candidates that violate timing constraints Question: Does Apriori principle still hold?
24 Apriori Principle for Sequence Data Object Timestamp Events A 1 1,2,4 A 2 2,3 A 3 5 B 1 1,2 B 2 2,3,4 C 1 1, 2 C 2 2,3,4 C 3 2,4,5 D 1 2 D 2 3, 4 D 3 4, 5 E 1 1, 3 E 2 2, 4, 5 Suppose: x g = 1 (max-gap) n g = 0 (min-gap) m s = 5 (maximum span) minsup = 60% <{2} {5}> support = 40% but <{2} {3} {5}> support = 60% Problem exists because of max-gap constraint No such problem if max-gap is infinite
25 Contiguous Subsequences s is a contiguous subsequence of w = <e 1 >< e 2 > < e k > if any of the following conditions hold: 1. s is obtained from w by deleting an item from either e 1 or e k 2. s is obtained from w by deleting an item from any element e i that contains more than 2 items 3. s is a contiguous subsequence of s and s is a contiguous subsequence of w (recursive definition) Examples: s = < {1} {2} > is a contiguous subsequence of < {1} {2 3}>, < {1 2} {2} {3}>, and < {3 4} {1 2} {2 3} {4} > is not a contiguous subsequence of < {1} {3} {2}> and < {2} {1} {3} {2}>
26 Modified Candidate Pruning Step Without maxgap constraint: A candidate k-sequence is pruned if at least one of its (k-1)-subsequences is infrequent With maxgap constraint: A candidate k-sequence is pruned if at least one of its contiguous (k-1)-subsequences is infrequent
27 Timing Constraints (II) {A B} {C} {D E} <= x g >n g <= ws <= m s x g : max-gap n g : min-gap ws: window size m s : maximum span x g = 2, n g = 0, ws = 1, m s = 5 Data sequence Subsequence Contain? < {2,4} {3,5,6} {4,7} {4,6} {8} > < {3} {5} > No < {1} {2} {3} {4} {5}> < {1,2} {3} > Yes < {1,2} {2,3} {3,4} {4,5}> < {1,2} {3,4} > Yes
28 Modified Support Counting Step Given a candidate pattern: <{a, c}> Any data sequences that contain < {a c} >, < {a} {c} > ( where time({c}) time({a}) ws) < {c} {a} > (where time({a}) time({c}) ws) will contribute to the support count of candidate pattern
29 Other Formulation In some domains, we may have only one very long time series Example: monitoring network traffic events for attacks monitoring telecommunication alarm signals Goal is to find frequent sequences of events in the time series This problem is also known as frequent episode mining
30 BIDE+ Algorithm BIDE (BI-Directional Extension): Saves space Speed up Finds frequent closed sequences Closed itemset: An itemset is closed if none of its immediate supersets has the same support as the itemset
31 Frequent Closed Sequences Sequence identifier Sequence 1 CAABC 2 ABCB 3 CABC 4 ABBCA Frequent sequences A:4, AA:2, AB:4, ABB:2, ABC:4, AC:4, B:4, BB:2,BC:4, C:4, CA:3, CAB:2, CABC:2, CAC:2, CB:3, CBC:2,CC:2 Frequent closed sequences AA:2, ABB:2, ABC:4, CA:3, CABC:2,CB:3
32 Frequent Closed Sequences
33 BIDE: space usage Most of the frequent closed pattern mining algorithms need to maintain the set of already mined frequent closed patterns (or just candidates) in memory subpattern checking super-pattern checking
34 Compact Efficient BIDE: Mining Frequent Closed Sequences Costly in both runtime and space usage BI-Directional Extension mining frequent closed sequences without candidate maintenance
35 BIDE: Closure checking scheme if an n-sequence, S=e 1 e 2... e n, is non-closed e is a forward-extension event S = e 1 e 2... e n e and sup SDB (S ) = sup SDB (S) e is a backward-extension event i (1 i < n), S = e 1 e 2... e i e e i+1...e n and sup SDB (S ) = sup SDB (S) S = e e 1 e 2... e n and sup SDB (S ) = sup SDB (S)
36 BIDE: Theorem, Lemma Theorem: BI-Directional Extension closure checking If there exists no forward-extension event nor backward extension event w.r.t. a prefix sequence S p, S p must be a closed sequence; otherwise, S p must be non-closed Lemma: Forward-extension event checking For a pre-fix sequence S p, its complete set of forwardextension events is equivalent to the set of its locally frequent items whose supports are equal to SUP SDB (S p )
37 BIDE: Definitions Projected sequence of a prefix sequence the projected sequence of prefix sequence AB in sequence ABBCA is BCA Projected database of a prefix sequence the projected database of prefix sequence AB in our running example is {C, CB, C, BCA} Sequence identifier Sequence 1 CAABC 2 ABCB 3 CABC 4 ABBCA
38 BIDE: Lemma Backward-extension event checking First, we assume S p =AC:4, it is easy to find that item B appears in each of the 2nd maximum periods of S p. As a result AC:4 is not closed. let S p =ABC:4, we cannot find any backward-extension item for it. Also there is no forward-extension item for it, therefore ABC:4 is a frequent closed sequence Sequence identifier Sequence 1 CAABC 2 ABCB 3 CABC 4 ABBCA
39 BIDE: Definition First instance of a prefix sequence the first instance of the prefix sequence AB in sequence CAABC is CAAB Last instance of a prefix sequence the subsequence from the beginning of S to the last appearance of item e i in S the last instance of the prefix sequence AB in sequence ABBCA is ABB Given an input sequence S which contains a prefix i- sequence e 1 e 2... e i The i-th last-in-last appearance w.r.t. a pre-fix sequence it is the last appearance of e i in the last instance of the prefix S p in S S=CAABC and S p =AB, the 1st last-in-last appearance w.r.t. prefix S p in S is the second A in S
40 BIDE: Definition Given an input sequence S which contains a prefix i- sequence e 1 e 2... e i The i-th maximum period of a prefix sequence if 1 < i n, it is the piece of sequence between the end of the first instance of prefix e 1 e 2... e i 1 in S and the i-th last-in-last appearance w.r.t. prefix S p if i = 1, it is the piece of sequence in S locating before the 1st last-in-last appearance w.r.t. prefix S p S=ABCB and the prefix sequence S p =AB» 2nd maximum period of prefix S p in S is BC,» 1st maximum period of prefix S p in S is φ
41 BIDE: Example S=CAABCBC and S p =AB First instance of a prefix sequence CAAB Last instance of a prefix sequence CAABCB The i-th last-in-last appearance w.r.t. a pre-fix sequence 1st nd A 2nd nd B The i-th maximum period of a prefix sequence 1st ---- CA 2nd ---- ABC
42 BIDE: Definition Given an input sequence S which contains a prefix i-sequence e 1 e 2... e i The i-th last-in-first appearance w.r.t. a prefix sequence it is the last appearance of e i in the first instance of the prefix S p in S S=CAABC and S p =CA» 2nd last-in-first appearance w.r.t. prefix S p in S is the first A in S
43 BIDE: Definition Given an input sequence S which contains a prefix i-sequence e 1 e 2... e i The i-th semi-maximum period of a prefix sequence if 1 < i n, it is the piece of sequence between the end of the first instance of prefix e 1 e 2... e i 1 in S and the i-th last-in-first appearance w.r.t. prefix S p if i = 1, it is the piece of sequence in S locating before the 1st last-in-first appearance w.r.t. pre-fix S p S=ABCB and the prefix sequence S p =AC» 2nd semi-maximum period of prefix AC in S is B» 1st semi-maximum period of prefix AC in S is φ
44 BIDE: Theorem BackScan search space pruning Let the pre-fix sequence be an n-sequence, S p =e 1 e 2... e n. If i (1 i n) and there exists an item e which appears in each of the i-th semi-maximum periods of the prefix S p in SDB, we can safely stop growing prefix S p Sequence identifier Sequence 1 CAABC 2 ABCB 3 CABC 4 ABBCA
45 BIDE: ScanSkip The ScanSkip optimization technique closure checking scheme needs to scan backward a set of maximum-periods w.r.t. a certain prefix S p =ABC : 4 3rd maximum periods is {φ, φ, φ, B} skip last three 2nd maximum periods is {A, φ, φ, B} skip last two 1st maximum periods is {CA, φ, C, φ} skip last two Sequence identifier Sequence 1 CAABC 2 ABCB 3 CABC 4 ABBCA
46 BIDE Algorithm scans the database once to find the frequent 1- sequences projected database for each frequent 1- sequence BackScan pruning method backward-extension-item & forward-extension-item output S p as a frequent closed sequence grow S p to get a new prefix projected database for the new pre-fix BackScan and backward extension use the ScanSkip to speed up the mining process
47 BIDE: Frequent Closed Sequences
48
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