Important Extrema of Time Series

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Important Extremaof Time Series
Eugene Fink 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

• Concept of important extrema

• Fast identification of these extrema

• Applications to compressionand indexing of time
  series

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Summary
We have developed a technique for identifying
major minima and maxima in a time series.
,
and finding the importance of each minimum and
maximum.
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Mountain analogy
A major peak is the highest point of some
mountain, which is much higher than the foot of
the mountain.
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Importance of an extremum
A local maximum in a time series is the top of a
mountain, that is, the maximal value in some
segment of the series.
THE DEFINITION FOR MINIMA IS SYMMETRIC
The importance of a maximum is the mountain
height, that is, its vertical distance from the
foot of the mountain.
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Strict, left, and right extrema
If a mountain top is a single point,it is called
a strict maximum.
THE DEFINITION FOR MINIMA IS SYMMETRIC
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Algorithm
Fast identification of major extrema.

• Determines the importances of all extrema in one
  pass through a series

• Can process a live series in real time, without
  storing it in memory

• Complexity
• For an n-point series with m extrema
• Running time is O(n)
• Required memory is O(m)

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Demo
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Applications

• Compression of a time series by extracting its
  major extrema

• Indexing of a series and retrieval of segments
  similar to a given pattern

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Lossy compression
Select a given percentage of the most important
extrema, along with the two endpoints, and
discard all other points.
initial
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Lossy compression
Select a given percentage of the most important
extrema, along with the two endpoints, and
discard all other points.

• Advantages
• Very fast compression procedure
• Preserving major minima and maxima
• Real-time compression of live series

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Indexing of extrema
We index extrema of a series by importance and
place in the series.
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2
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Indexing of extrema
We index extrema of a series by their importance
and place in the series.
We use a range tree, which supports indexing of
points by two coordinates.
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importance
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2
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place in the series
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Retrieval
We can quickly look up a compressed version of
any given segment, and then retrieve more and
more of its details.
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6
importance
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2
0
place in the series
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Retrieval
We can quickly look up a compressed version of
any given segment, and then retrieve more and
more of its details.
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importance
4
2
0
place in the series
segment
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Retrieval
We can quickly look up a compressed version of
any given segment, and then retrieve more and
more of its details.
This procedure supports fast search for segments
similar to a given pattern.
Pattern
Series
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Extensions

• Generalized vertical distancebetween points of a
  series

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