Identifying Patterns in Time Series Data

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Identifying Patterns in Time Series Data
Daniel Lewis 04/06/06
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Time Series Data

• Definition
• An ordered set of m real-valued variables
• How can patterns that occur in time series data
  be located?

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Comparing Time Series

• Euclidean Distance

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Adaptive Piecewise Constant Approximation

• Segments of time series data are represented by 2
  values, the mean value of all points in the
  segment and the right endpoint of the segment
• Allows large queries to be quickly compared, but
  APCA representations must be created first.

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Adaptive Piecewise Constant Approximation (cont')
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APCA (cont')

• Data Compression allows for faster search
• Can be used for indexing large series
• Can handle large queries
• But what if we want to identify patterns in
  streaming time series data?

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Pattern Recognition

• For a given query q and a time series s, if the
  Euclidean distance between q and s lt r, the
  series match
• r user defined, application specific threshold

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Detecting Patterns in Streaming Data

• Brute Force Method
• P1 ,.., Pn Set of patterns of length k
• S Input Stream
• For every possible substring of length k in s,
  calculate the distance between the substring and
  all n patterns
• This method is obviously extremely costly in the
  case of a large pattern set and a large input
  stream
• O(n(S - k))

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Speeding Up Pattern Identification

• Early Abandoning
• If at any point error gt r 2, we can stop
  computation

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Wedge Creation

• Combine multiple patterns into a wedge
• Define Upper Limit
• Ui max( C1i , .. , Cki)
• Define Lower Limit
• Li min( C1i , .. , Cki)
• This produces a wedge such that

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Wedge Creation (cont')

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Wedge Comparison

• Distance Between a Query and a Wedge
• If distance gt r, then the distance between the
  query and all component patterns gt r, allowing
  you to eliminate multiple possible matches with a
  single comparison

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Hierarchical Wedges

• The usefulness of any wedge is determined by the
  similarity of the patterns used in its
  construction.
• More similar patterns create smaller, more useful
  wedges
• Patterns can be combined in a tree-like pattern
  to produce a hierarchy of wedges

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Hierarchical Wedges (cont')
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Atomic Wedgie

• Preparation
• All patterns are clustered by similarity
• The most similar patterns are combined into
  wedges
• The resulting wedges are combined to form larger,
  less specific wedges

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Atomic Wedgie (cont')

• Usage
• When streaming data arrives, each substring of
  length k is first compared to the largest wedge,
  if dist gt r, comparison stops, else, the distance
  is compared against the two component wedges,
  eliminating any branches where the distance
  exceeds r.
• Eventually, all branches are eliminated or a
  single (atomic) pattern is matched

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Atomic Wedgie Optimization

• Optimization
• Summation is order independent
• Large sections are less likely to increase error
  than small sections
• Thus, if error is summed starting with the
  smallest sections first, the requirements for
  early abandon are more likely to be met earlier

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Atomic Wedgie (cont')

• Advantages
• If wedges are well formed, large speed increases
  can occur
• A large number of similar possible patterns can
  be analyzed quickly
• Disadvantages
• If wedges are poorly formed, the time required
  will exceed the Brute Force Method
• Dissimilar patterns are not handled well

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Special Considerations

• The choice of r (similarity threshold) is of
  great importance
• if r is too large, a substring can match too many
  patterns to be useful
• if r is too small, too little matching may occur
• Good Choice
• r average distance from any pattern to its
  nearest neighbor

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Atomic Wedgie Results
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References

• Chakrabarti, K., Keogh, E., Mehrotra, S., and
  Pazzani, M., Locally Adaptive Dimensionality
  Reduction for Indexing Large Time Series
  Databases, ACM Transactions on Database Systems,
  Vol 27, 2002.
• Wei, L., Keogh, E., Van Herle, H., Mafra-Neto,
  A., Atomic Wedgie Efficient Query Filtering
  for Streaming Times Series, Data Mining, Fifth
  IEEE International Conference on 27-30 Nov.
  2005 Page(s)490 - 497

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Questions?