Download Stream Sequential Pattern Mining with Precise Error Bounds

Stream Sequential Pattern Mining
with Precise Error Bounds
Luiz F. Mendes；Bolin Ding；
Jiawei Han
ICDM 2008
Outlines
• Introduction
• Problem Definition
• SS-BE Method (Stream Sequence miner using Bounded Error)
• SS-MB Method (Stream Sequence miner using Memory Bounds)
• Experimental Results
• Conclusions
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Introduction
• A data stream is an unbounded sequence in which new
elements are generated continuously.
• Memory usage is restricted.
(meaning that we cannot store all the stream data in memory)
• Two methods : SS-BE and SS-MB.
• To break the data stream into fixed-sized batches and perform
sequential pattern mining on each batch.
• A lexicographic tree structure.
• Using different pruning strategies that restrict the memory
usage.
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Problem Definition
• A data stream of sequences is an arbitrarily large list of
sequences.
• A sequence s contains another sequence s’ if s’ is a
subsequence of s.
• count(s) : the number of sequences that contain s.
• supp(s) : count(s) divided by the total number of sequences
seen.
• If supp(s) ≥ σ, where σ is a user-supplied minimum support
threshold, then we say that s is a frequent sequence, or
a sequential pattern.
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Cont.
• Example.
• Suppose the length of our data stream is only 3 sequences :
S1 = <a, b, c>, S2 = <a, c>, and S3 = <b, c>.
Let us assume we are given that σ = 0.5. (supp(s) ≥ σ)
• .sequential pattern count(s) supp(s)
<a>
2
2/3=0.6 ≥ 0.5
<b>
2
2/3=0.6 ≥ 0.5
<c>
3
3/3=1 ≥ 0.5
<a, c>
2
2/3=0.6 ≥ 0.5
<b, c>
2
2/3=0.6 ≥ 0.5
<a, b>
1
1/3=0.3 < 0.5
<a, b, c>
1
1/3=0.3 < 0.5
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Cont.
• SS-BE and SS-ME use a lexicographic tree T0 to store the
subsequences seen in the data stream.
• Example. (Cont.)
• Sequential patterns are : <a>:2, <b>:2, <c>:3, <a, c>:2, <b, c>:2.
• T0 :
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SS-BE Method
• Example.
• . Input values
(1) a stream of sequences D = S1, S2,…
(2) minimum support threshold σ
0.75
(3) significance threshold ϵ
0.5
(4) batch length L
4
(5) batch support threshold α
0.4
(6) pruning period δ
2
• Batch B1 : <a, b, c>, <a, c>, <a, b> and <b, c>.
• Batch B2 : <a, b, c, d>, <c, a, b>, <d, a, b> and <a, e, b>.
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SS-BE Method (Cont.)
• Batch B1 : <a, b, c>, <a, c>, <a, b> and <b, c>,
with minimum support α = 0.4.
• Sequential patterns are : <a>:3, <b>:3, <c>:3, <a, b>:2, <a, c>:2,
and <b, c>:2.
• After batch 1 :
batchCount of 1
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SS-BE Method (Cont.)
• Batch B2 : <a, b, c, d>, <c, a, b>, <d, a, b> and <a, e, b>,
with minimum support α = 0.4.
• Sequential patterns are : <a>:4, <b>:4, <c>:2, <d>:2, and <a, b>:4.
• After batch 2 :
2
2
batchCount of 1
2
2
1
1
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SS-BE Method (Cont.)
• B = 2 (B : the number of batches elapsed since the last pruning
before it was inserted in the tree.)
• B’ = B − batchCount, batchCount = 1；2, B’ = 1；0
• Pruning :
=> count + B’ ≤ 4
• After pruning :
2 + 1 ≤ 4 =>pruning
5+0≤4
• Output all sequences, having count ci,
count ≥ ( σ − ϵ )N = (0.75−0.5)8 = 2.
• The output sequences and counts are:
<a>:7, <b>:7, <c>:5, and <a, b>:6. (false positive : <c>)
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SS-ME Method
• Example.
• . Input values
(1) a stream of sequences D = S1, S2,…
(2) minimum support threshold σ
0.75
(3) significance threshold ϵ
0.5
(4) batch length L
4
(5) maximum number of nodes in the tree m
7
• Batch B1 : <a, b, c>, <a, c>, <a, b> and <b, c>.
• Batch B2 : <a, b, c, d>, <c, a, b>, <d, a, b> and <a, e, b>.
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SS-ME Method (Cont.)
• Batch B1 : <a, b, c>, <a, c>, <a, b> and <b, c>,
with minimum support ϵ = 0.5.
• Sequential patterns are : <a>:3, <b>:3, <c>:3, <a, b>:2, <a, c>:2,
and <b, c>:2.
• After batch 1 :
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SS-ME Method (Cont.)
• Batch B2 : <a, b, c, d>, <c, a, b>, <d, a, b> and <a, e, b>,
with minimum support ϵ = 0.5.
• Sequential patterns are : <a>:4, <b>:4, <c>:2, <d>:2, and <a, b>:4.
• After batch 2 :
count as 2 + min = 2
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• A variable min : to keep track of the largest count of any node
that has been removed from our tree. (initially set to 0)
SS-ME Method (Cont.)
• m=7
• The sequence <b, c> is removed, and min = 2.
pruning
• After batch
2 ::
• Output all sequences, having count ci,
count > ( σ − ϵ )N = (0.75−0.5)8 = 2.
• The output sequences and counts are:
<a>:7, <b>:7, <c>:5, and <a, b>:6.
• If min = 2 ≤ ( σ − ϵ )N = 2, then the algorithm guarantees that
there are no false negatives. (false positive : <c>)
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Experimental Results
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Cont.
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Conclusions
• In this paper we propose two effective methods for mining
sequential patterns from data streams : SS-BE and SS-MB.
• The running time of each algorithm scales linearly as the
number of sequences grows.
• The maximum memory usage is restricted in both cases
through the pruning strategies adopted.
• These properties make both methods effective choices for
stream sequential pattern mining.
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