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. 1, 5, 5, 9, 9, 6, 3, 4, 5, 4, 3, 6, 9, 5, 1, 3, 7, 5, 6, 7 Example:

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.

Mountain analogy A major peak is the highest point of some mountain, which is much higher than the foot of the mountain. major minor

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. segment importance

Strict, left, and right extrema If a mountain top is a single point, it is called a strict maximum. strict If a mountain top is a plateau, its endpoints are left and right maxima. leftright

Algorithm 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) Fast identification of major extrema.

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. initial compressed

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 importance place in the series

Indexing of extrema We index extrema of a series by their importance and place in the series importance We use a range tree, which supports indexing of points by two coordinates. place in the series

Retrieval importance place in the series segment We can quickly look up a compressed version of any given segment, and then retrieve more and more of its details.

Retrieval importance place in the series segment We can quickly look up a compressed version of any given segment, and then retrieve more and more of its details.

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. PatternSeries

Extensions Generalized vertical distance between points of a series... Important extrema in the first and second derivatives of a series