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Indexing of Time Series by Major Minima and Maxima

Indexing of Time Series by Major Minima and Maxima. Eugene Fink Kevin B. Pratt Harith S. Gandhi. Example:. 0, 3, 1, 2, 0, 1, 1, 3, 0, 2, 1, 4, 0, 1, 0. 4. 3. 2. 1. 0. Time series. A time series is a sequence of real values measured at equal intervals. Results.

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Indexing of Time Series by Major Minima and Maxima

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  1. Indexing of Time Seriesby Major Minima and Maxima Eugene Fink Kevin B. Pratt Harith S. Gandhi

  2. Example: 0, 3, 1, 2, 0, 1, 1, 3, 0, 2, 1, 4, 0, 1, 0 4 3 2 1 0 Time series A time series is a sequence of real values measured at equal intervals.

  3. 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

  4. Outline • Compression • Indexing • Retrieval • Experiments

  5. 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.

  6. a[i] a[j]  R  R a[m] Major minima • A point a[m] in a[1..n] is a major minimum if there are i and j, where i < m < j, such that: • a[m] is a minimum among a[i..j], and • a[i] – a[m]  R and a[j] – a[m]  R.

  7. Major maxima • A point a[m] in a[1..n] is a major maximum if there are i and j, where i < m < j, such that: • a[m] is a maximum among a[i..j], and • a[m] – a[i]  R and a[m] – a[j]  R. a[m]  R  R a[i] a[j]

  8. 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.

  9. Outline • Compression • Indexing • Retrieval • Experiments

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

  11. a[j] a[i] Major inclines • A segment a[1..j] is a major upward incline if • a[i] is a major minimum; • a[j] is a major maximum; • for every m [i..j], a[i] < a[m] < a[j]. The definition of a major downward inclineis symmetric.

  12. Identification of inclines The procedure performs two passes through a list of major minima and maxima.

  13. 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.

  14. incline height height length length 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.

  15. Outline • Compression • Indexing • Retrieval • Experiments

  16. Example: Database 3 2 1 Retrieval The procedure inputs a pattern series andsearches for similar segments in a database. Pattern

  17. Retrieval The procedure inputs a pattern series andsearches for similar segments in a database. • Main steps: • Find the pattern’s inclines with the greatest height • Retrieve all segments that have similar inclines • Compare each of these segments with the pattern

  18. 1 2 height length1 length2 Highest inclines First, the retrieval procedure identifies the important inclines in the pattern. , and selects the highest inclines.

  19. height · C incline height / C length / C length · C Candidate segments Second, the procedure retrieves segments with similar inclines from the database. • An incline is considered similar if • its height is betweenheight / C and height· C; • its length is betweenlength / D and length· D. We use the range tree toretrieve similar inclines.

  20. Similarity test Third, the procedure compares the retrieved segments with the pattern. ,using a given similarity test.

  21. Outline • Compression • Indexing • Retrieval • Experiments

  22. 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

  23. 400 331 perfect ranking perfect ranking 0 0 200 151 0 0 fast rankingC = D = 2 time: 0.02 sec fast rankingC = D = 1.5 time: 0.01 sec 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

  24. 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

  25. 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

  26. 3 3 1 1 1 1 1 1 3 3 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. 4

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