SEG 4630 Tutorial 10

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SEG 4630 Tutorial 10

• Sequential Patterns
• Lijun Chang

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Sequential Patterns

• Elements Events
• Sequence an ordered list of elements
  (transactions)
• s lte1e2e3engt
• Element a collection of one or more events
  (items)
• e1 i1,i2,ik
• k-sequence a sequence with k events(items)
• E.g. a 8-sequence lt1,2 4,6 7 8,2,10gt

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Examples of Sequence Data
Sequence Database
Element (Transaction)
Event (Item)
235
61
1
Sequence
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Definition of a Subsequence

• A sequence lta1 a2 angt is contained in another
  sequence ltb1 b2 bmgt (m n) if there exist
  integers i1 lt i2 lt lt in such that a1 ? bi1 ,
  a2 ? bi1, , an ? bin

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The count of subsequences

• How many k-subsequences can be extracted from a
  given n-sequence?
• lta b c d e f g h igt n 9
• k4 Y _ _ Y Y _ _ _ Y
• lta d e igt

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Sequential Patterns Mining

• sequential pattern
• a frequent subsequence
• support is minsup
• Aprior principle for sequential data mining
• Any data sequence that contains a particular
  k-sequence must also contain all of its
  (k-1)-subsequences.
• Iteratively
• Generate new candidate k-sequences
• Prune candidates whose (k-1)-sequences are
  infrequent
• Count the supports of the candidates
• Eliminate infrequent candidate k-sequences

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Candidates Generation

• Sequence Merging (kgt2)
• To generate a candidate k-sequence
• Merging of two frequent (k-1)-sequence w1 and w2
• if the subsequence obtained by removing the first
  event in w1 is the same as the subsequence
  obtained by removing the last event in w2
• Resulting candidates -- extended w1 with the last
  event of w2.
• Situations on merging
• If the last two events in w2 belong to the same
  element, then the last event in w2 becomes part
  of the last element in w1
• e.g. Generate a 4-sequence from two 3-sequence
• w1 lt 1 5 3 gt dropping first event ? lt
  53 gt
• W2 lt 5 3, 4 gt dropping the last event ? lt
  53 gt and 4 is in same element 3,4
• lt1 5 3, 4gt
• Otherwise, the last event in w2 becomes a
  separate element appended to the end of w1

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Timing Constraints

• Maxspan
• The maximum allowed time difference between the
  earliest event and the latest event in the entire
  sequence.
• Mingap
• The minimum allowed time difference between two
  consecutive elements in a sequence
• Maxgap
• The maximum allowed time difference between two
  consecutive elements in a sequence
• Window Size
• The maximum allowed time difference of the latest
  and earliest occurrences of events in any element
  of a sequential pattern.
• ws0 ? all events in the same element of a
  pattern must occur simultaneously

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Problem on Maxgap

• Using of maxgap may produce contradictions to the
  apriori principle
• Suppose
• xg 1 (max-gap)
• ng 0 (min-gap)
• minsup 60
• lt2 5gt support 40
• lt2 3 5gt support 60
• Support increased when the number of events in a
  sequence increases!?

Aprior principle for sequential data Any data
sequence that contains a particular k-sequence
must also contain all of its (k-1)-subsequences.
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Modified Apriori Principle

• If a k-sequence is frequent, then all of its
  contiguous k-1-subsequences must also be
  frequent.
• Contiguous subsequence
• S is a contiguous subsequence of w if
• Deleting an event from either the first or the
  last element
• E.g. lt1 2gt is contiguous of lt123gt
• Deleting an event from any element having at
  least two events
• E.g. lt1 2gt is contiguous of lt1 2,3 3gt
• S is a contiguous subsequence of t which is a
  contiguous subsequence of w.

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Examples

• Q.13(P.480)
• Consider the following frequent 3-sequences
• lt 1, 2, 3 gt, lt 1, 23 gt, lt 12, 3 gt, lt
  1, 24 gt,
• lt 1, 34 gt, lt 1, 2, 4 gt, lt 2, 33 gt, lt
  2, 34 gt,
• lt 233 gt, and lt 234 gt.
• (a) the candidate 4-sequences produced by the
  candidate generation step of the GSP algorithm.
• lt 1, 2, 3 3 gt, lt 1, 2, 3 4 gt, lt 1, 2
  3 3 gt, lt 1, 2 3 4 gt,
• lt 1 2, 3 3 gt, lt 1 2, 3 4 gt.
• (b) The candidate 4-sequences pruned during the
  candidate pruning
• step of the GSP algorithm (assuming no timing
  constraints).
• lt 1, 2, 3 3 gt, lt 1, 2 3 3 gt, lt 1, 2
  3 4 gt, lt 1 2, 3 3 gt,
• lt 1 2, 3 4 gt.

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Examples

• Q.12(p.480)
• For each of the sequence w lte1, . . . , elastgt
  below,
• determine whether they are subsequences of the
  following data sequence
• s A,BC,DA,BC,DA,BC,D
• subjected to the following timing constraints
• mingap 0 (interval between last event in ei
  and first event in ei1 is gt 0)
• maxgap 2 (interval between first event in ei
  and last event in ei1 is 2)
• maxspan 6 (interval between first event in e1
  and last event in elast is 6)
• ws 1 (time between first and last events in ei
  is 1)

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More Examples

• s lt1 2 2,5 3 4gt
• w lt1,2 3,4gt

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More Examples

• s1 lt1,2,3,4gt
• w1 lt1,2 3,4gt ?
• s2 lt1,2,3,4 5gt
• w2 lt1,2 3,4,5gt ?
• s3 lt1 2 3 4gt
• w3 lt1,3 2,4gt ?