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Advanced Pattern Mining 02
COP 6726: New Directions in Database Systems Advanced Pattern Mining 02
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Formal Definition of a Subsequence
A sequence <a1 a2 … an> is contained in another sequence <b1 b2 … bm> (m ≥ n) if there exist integers i1 < i2 < … < in such that a1 bi1 , a2 bi1, …, an bin 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)
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Sequential Pattern Mining: Definition
Given: a database of sequences a user-specified minimum support threshold, min-sup Task: Find all subsequences with support ≥ minsup
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Sub-sequence Mining
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Sequence Mining Let ∑ denote an alphabet, defined as a finite set of characters or symbols. A sequence (or a string) is defined as an ordered list of symbols, i.e., s = s1s2…sk, where si ∈ ∑ is a symbol at position i. Let s = s1s2…sn and r = r1r2…rm be two sequence over ∑. r is a subsequence of s denoted r ⊆ s, if there exists a non-to-one mapping ϕ : [1, m] [1, n], such that r[i] = s[ϕ(i)] and for any two position i, j in r, i < j ϕ(i) < ϕ(j). r is a consecutive subsequence (or substring) of s, if r[1 : m] = s[j : j+m], with 1 ≤ j ≤ n −m +1.
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Sequence Mining Given a sequence data, find frequent subsequences that satisfy the minimum support constraint. A AA AG AAG G GA GG GAA GAG GAAG T Id Sequence s1 CAGAAGT s2 TGACAG s3 GAAGT Minimum Support = 3
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Generalized Sequence Pattern (GSP) Mining
Id Sequence s1 CAGAAGT s2 TGACAG s3 GAAGT A 3 C 2 G T Minimum Support = 3
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Spade: Vertical Sequence Mining
For each symbol c ∈ ∑, we keep a set of tuples of the form < i, pos(c)>, where pos(c) is the set of positions in the sequence si. It maintains the list of positions for the occurrences of the last symbol. In this example, A occurs in s1 at positions 2,4, and 5. Id Sequence s1 CAGAAGT s2 TGACAG s3 GAAGT
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Spade: Vertical Sequence Mining
Id Sequence s1 CAGAAGT s2 TGACAG s3 GAAGT 4
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Projection-Based Sequence Mining (PrefixSpan)
The projected database with respect to ⍺, denoted D⍺ is obtained by finding the first occurrence (e.g., p) of A in si. Next, the suffix of ss starting at position p+1 is extracted from si. After that, any infrequent symbols are removed from suffix. Minimum Support = 3 ⍺ = G, i.e., DG Projection Id Sequence s1 CAGAAGT s2 TGACAG s3 GAAGT A 3 C 2 G T Id Sequence s1 CAGAAGT s2 TGACAG s3 GAAGT Id Sequence s1 AAGT s2 AAG s3 Given s1 = CAGAAGT, the projection of s1 with respect to G (i.e., DG) is AAGT. In this example, DG is {s1: AAGT, s2: AAG, s3: AAGT}
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PrefixSpan
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PrefixSpan
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Consecutive Subsequence (or substring) Mining
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Substring Mining via Suffix Trees
Id Sequence s1 CAGAAGT s2 TGACAG s3 GAAGT (id number, frequency occurrence)
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Substring Mining via Suffix Trees
Id Sequence s1 CAGAAGT
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Substring Mining via Suffix Trees
Id Sequence s1 CAGAAGT s2 TGACAG s3 GAAGT n N = support (i.e., frequency) If min-sup =3, Is GAA frequent? Is CAA frequent?
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Subgraph Pattern Mining
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Frequent Subgraph Mining
Extend association rule mining to finding frequent subgraphs Useful for Web Mining, computational chemistry, bioinformatics, spatial data sets, etc.
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Graph Definitions
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Computing the support of a subgraph.
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Frequent Subgraph Mining
Given a set of graph G and a support threshold, minsup, the goal of frequent subgraph mining is to find all subgraphs g such that s(g) ≥ minsup.
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Brute-force method
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Apriori-like method Transform each graph into a transaction-like format so that existing algorithms such as Aprior can be applied. During candidate generation, a pair of frequent (k-1)-subgraphs are merged to form a candidate k-subgraph.
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Take Home Message Sequence Mining Consecutive Sequence Mining
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