Important Extrema
of Time Series
Eugene Fink
Harith S. Gandhi
Time series
A time series is a sequence of real
values measured at equal intervals.
Example:
1, 5, 5, 9, 9, 6, 3, 4, 5, 4, 3, 6, 9, 5, 1, 3, 7, 5, 6, 7
10
8
6
4
2
0
Results
• Concept of important extrema
• Fast identification of these extrema
• Applications to compression
and indexing of time series
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.
8 6
8
2
2
2
6
6
6
Mountain analogy
major
major
minor
minor
major
A major peak is the highest point of
some mountain, which is much higher
than the foot of the mountain.
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 importance of a maximum is the
“mountain height,” that is, its vertical
distance from the foot of the mountain.
importance
segment
Strict, left, and right extrema
If a mountain top is a single point,
it is called a strict maximum.
If a mountain top is a plateau, its
endpoints are left and right maxima.
strict
left
right
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)
Demo
Applications
• Compression of a time series
by extracting its major extrema
• Indexing of a series and retrieval of
segments similar to a given pattern
Lossy compression
Select a given percentage of the most
important extrema, along with the two
endpoints, and discard all other points.
8 6
initial
8
compressed
6
6
6
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
Indexing of extrema
We index extrema of a series by
importance and place in the series.
8 6
8
2
2
2
6
6
importance
6
8
6
4
2
0
place in the series
Indexing of extrema
importance
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.
8
6
4
2
0
place in the series
Retrieval
importance
We can quickly look up a compressed
version of any given segment, and then
retrieve more and more of its details.
8
6
4
2
0
place in the series
segment
Retrieval
importance
We can quickly look up a compressed
version of any given segment, and then
retrieve more and more of its details.
8
6
4
2
0
place in the series
segment
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
Extensions
• Generalized vertical distance
between points of a series
• Important extrema in the first and
second derivatives of a series
...