CEMiner – An Efficient Algorithm for Mining Closed Patterns from Time Interval-based DataYi-Cheng Chen, Wen-Chih Peng

and Suh-Yin Lee

ICDM 2011

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Outlines

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Motivation Preliminaries Endpoint representation CEMiner algorithm Experimental result Conclusion

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Motivation

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Existing studies only focus on mining closed sequential patterns from time point-based data.

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Cont.

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In this paper, we discuss and design an efficient method to discover closed temporal patterns from interval-based data.

Three contributions ： We simplify the processing of complex relations. i.e., only “before”, “after” and “equal.”

Endpoint representation

A novel algorithm, CEMiner (Closed Endpoint Temporal Miner).

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Preliminaries

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Definition 1. Event interval and event sequence

E = {e1 , e2 ,…, ek } be the set of event symbols :

{A, B, C, D, E } The triplet (ei , si , fi ) is an event interval : (A , 2 , 7) An event sequence is a series of event

interval triplets : <(A, 2 , 7), (B , 5 , 10), …, (E , 18 , 20)>.

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Cont.

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Definition 2. Temporal database Database DB = {r1 , r2 , …, rm }, each record ri , consists of a sequence-id, SID and an

event. DB is called a temporal database.

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Endpoint representation

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When describing relationships among more than three events, Allen’s temporal logics may suffer several problems.

A suitable representation is very important for describing a temporal pattern.

A new expression, endpoint representation is proposed to address the ambiguous and scalable problem.

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Cont.

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Definition 3. Endpoint sequence

event sequence q = <( A , 2 , 7 ), ( B , 5 , 10 ), ( C , 5 , 12 ), ( D , 16 , 22 ), ( E , 18 , 20 )>

Tq = { 2 ,7 ,5 ,10 ,5 ,12 ,16 ,22 ,18 ,20 } endpoint sequence : qe =

<2 ,5 ,5 ,7 ,10 ,12 ,16 ,18 ,20 ,22> endpoint representation : <>

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Cont.

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The endpoint representation has several benefits : Scalability

Nonambiguity

Simplicity

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CEMiner algorithm

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CEMiner (standing for Closed Endpoint temporal Miner) utilizes the arrangement of endpoints to accomplish the closed temporal pattern mining. Closure Checking

subsequence & supersequence Ex. Given two sequences = <A, B, C>, = <𝛽 A, D, B, C, E>, we say is a subsequence of , and is 𝛽 𝛽

a supersequence of.

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Cont.

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Definition 4. Closed temporal pattern CTP = {( ∈ TP ) ˄ ( ∄ ∈ TP ) such that ( ⊆ 𝛼 𝛽 𝛼

β) ∧ ( support ( ) = support ( ) )}𝛼 𝛽 Given two sequence and 𝛼 𝛽 If is a closed temporal pattern, 𝛼

𝛼 is a temporal pattern and there doesn’t exist a supersequence and support ( ) 𝛽 𝛼

= support ( ). 𝛽

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Cont.

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Ex. min_sup = 2 The endpoint sequence = <> is a temporal

pattern but not a closed temporal pattern. Because <> ⊂ <> and both support = 2.

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Cont.

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Closure Checking To verify a new closed temporal pattern p, we

require checking whether p is a sub-sequence or super-sequence of an existing temporal pattern p’ and the projected database of p and p’ is equal.

This paper borrow BI-Directional Extension [WH04] to check patterns’ closure. Forward-extension Backward-extension

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Cont.

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Definition 5. Forward-extension and backward-extension

If = <> is non-closed, there must exist at least one endpoint x, which can be used to extend to a new endpoint sequence ’, support () = support (’).

can be extended in five ways: (1)’= 〈〉 (2)’= 〈〉 ’ 𝛼 a forward-extension sequence (3)’= 〈〉 (4)’= 〈〉 (5)’= 〈〉 ’ backward-extension sequence

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Cont.

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If there exists no forward-extension endpoint nor backward-extension , must be a closed 𝛼endpoint sequence.

The CEMiner checks closure in two directions as follows, Forward directional checking Backward directional checking

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Cont.

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Definition First instance of a prefix sequence

Ex. The first instance of the prefix sequence AB in

sequence CAABC is CAAB.

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Cont.

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Definition 6. The i-th last-in-first appearance Ex. 〈 ABAB(AB)(AB) 〉 p = 〈〉1. The last-in-first appearance w.r.t. prefix p in ? (1) 1≤ i < n, n=4, i=2 first instance : 〈 ABAB(AB)(AB) 〉 2. The last-in-first appearance w.r.t. prefix p in? (2) i = n, i = n = 4 first instance : 〈 ABAB(AB)(AB) 〉

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Cont.

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Definition 7. The i-th semi-maximum period Ex. 〈 ABAB(AB)(AB) 〉 p = 〈〉 1. semi-maximum period of prefix p in (1) i =1 , before the last-in-first appearance : 〈 ABAB(AB)(AB) 〉 2. semi-maximum period of prefix p in (2) 1< i ≤ n, n=4, i=2 a. end of the first instance of 〈〉 : 〈 AB 〉 b. the 2-th last-in-first appearance w.r.t p: B 〈 ABAB(AB)(AB) 〉

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Cont.

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EbackScan search Let an endpoint sequence, if there exists i, 1 ≤ i ≤ n

and there exists an endpoint x which appears in each of the i-th semi-maximum periods of the prefixin database.

We can derive a new endpoint sequenceand we can stop growing the endpoint sequence .

Ex. Prefix sequence p = <A, C> B is the 2nd semi-max. period of the prefix p in

database We can derive a new prefix sequence p’ = <A, B, C>

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CEMiner Algorithm

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We use three pruning strategies to reduce the searching space efficiently and effectively.

(1) pre-pruning (2) post-pruning (3) pair-pruning

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CEMiner Algo.

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CEMiner Algo.

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CEMiner Algo.

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CEMiner Algo.

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CEMiner Algo.

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Pair-pruning: If the endpoint is a

starting endpoint, we can omit the closure checking.

Because the starting endpoint and finishing endpoint always occur in pairs in an endpoint sequence.

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CEMiner Algo.

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Ex. Prefix p =<> Endpoint B+ is a

backward-extension endpoint of p.

So we can stop growing p.

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CEMiner Algo.

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CEMiner Algo.

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CEMiner Algo.

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Pre-pruning: If y is finishing

endpoint and it has corresponding starting endpoint in .

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CEMiner Algo.

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Post-pruning: A finish point is

called significant, if it has a corresponding starting endpoint in projected postfix or in .

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Cont.

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Experimental result

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Conclusion

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We develop an efficient algorithm, CEMiner, to discover closed temporal patterns without candidate generation, based on proposed endpoint representation.

The algorithm further employs three pruning methods to reduce the search space effectively.