Important Extrema of Time Series Eugene Fink Harith

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 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 . . .