Indexing of Time Series by Major Minima and Maxima

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Indexing of Time Seriesby Major Minima and Maxima
Eugene Fink Kevin B. Pratt Harith S. Gandhi
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Time series
A time series is a sequence of real values
measured at equal intervals.
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Results

• Compression of a time series by extracting
  its major minima and maxima

• Indexing of compressed time series

• Retrieval of series similar to a given pattern

• Experiments with stock and weather series

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Outline

• Compression

• Indexing

• Retrieval

• Experiments

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Compression
We select major minima and maxima, along with the
start point and end point, and discard the other
points.
We use a positive parameter R to control the
compression rate.
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Major minima

• A point am in a1..n is a major minimum if
  there are i and j, where i lt m lt j, such that
• am is a minimum among ai..j, and
• ai am ? R and aj am ? R.

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Major maxima

• A point am in a1..n is a major maximum if
  there are i and j, where i lt m lt j, such that
• am is a maximum among ai..j, and
• am ai ? R and am aj ? R.

am
? R
? R
aj
ai
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Compression procedure
The procedure performs one pass through a given
series.
It takes linear time and constant memory.
It can compress a live serieswithout storing it
in memory.
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Outline

• Compression

• Indexing

• Retrieval

• Experiments

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Indexing of series
We index series in a database by their major
inclines, which are upward and downward segments
of the series.
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Major inclines

• A segment a1..j is a major upward incline if
• ai is a major minimum
• aj is a major maximum
• for every m ? i..j, ai lt am lt aj.

The definition of a major downward inclineis
symmetric.
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Identification of inclines
The procedure performs two passes through a list
of major minima and maxima.
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Identification of inclines
The procedure performs two passes through a list
of major minima and maxima.
Its time is linear in the number of inclines.
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Indexing of inclines
We index major inclines of series in a database
by their lengths and heights.
We use a range tree, which supports indexing of
points by two coordinates.
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Outline

• Compression

• Indexing

• Retrieval

• Experiments

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Retrieval
The procedure inputs a pattern series
andsearches for similar segments in a database.
Pattern
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Retrieval
The procedure inputs a pattern series
andsearches for similar segments in a database.

• Main steps
• Find the patterns inclines with the greatest
  height

• Retrieve all segments that have similar
  inclines

• Compare each of these segments with the
  pattern

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Highest inclines
First, the retrieval procedure identifies the
important inclines in the pattern.

, and selects the highest inclines.
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Candidate segments
Second, the procedure retrieves segments with
similar inclines from the database.

• An incline is considered similar if
• its height is between height / C and height
  C
• its length is between length / D and length
  D.

We use the range tree toretrieve similar
inclines.
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Similarity test
Third, the procedure compares the retrieved
segments with the pattern.

,using a given similarity test.
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Outline

• Compression

• Indexing

• Retrieval

• Experiments

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Experiments
We have tested a Visual-Basic implemen- tation on
a 2.4-GHz Pentium computer.

• Data sets
• Stock prices 98 series, 60,000 points

• Air and sea temperatures 136 series, 450,000
  points

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Stock prices (60,000 points) Search for 100-point
patterns
The x-axes show the ranks of matches retrieved by
the developed procedure, and the y-axes are the
ranks assigned by a slow exhaustive search.
210
perfect ranking
0
0
200
fast rankingC D 5 time 0.05 sec
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Stock prices (60,000 points) Search for 500-point
patterns
The x-axes show the ranks of matches retrieved by
the developed procedure, and the y-axes are the
ranks assigned by a slow exhaustive search.
400
328
202
perfect ranking
perfect ranking
perfect ranking
0
0
0
200
167
0
0
0
200
fast rankingC D 5 time 0.31 sec
fast rankingC D 2 time 0.12 sec
fast rankingC D 1.5 time 0.09 sec
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Temperatures (450,000 points) Search for
200-point patterns
The x-axes show the ranks of matches retrieved by
the developed procedure, and the y-axes are the
ranks assigned by a slow exhaustive search.
400
400
202
perfect ranking
perfect ranking
perfect ranking
0
0
0
82
0
151
0
0
200
fast rankingC D 5 time 1.18 sec
fast rankingC D 2 time 0.27 sec
fast rankingC D 1.5 time 0.14 sec
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Conclusions
Main results Compression and indexing of time
series by major minima and maxima.
Current work Hierarchical indexing by importance
levels of minima and maxima.
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