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

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

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. 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. a[j]a[j]a[i]a[i] a[m]a[m]  R R  R 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[j]a[j]a[i]a[i] a[m]a[m]  R R  R R

8. Compression procedure The procedure performs one pass through a given series. It can compress a live series without storing it in memory. It takes linear time and constant 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. 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]. a[i]a[i] a[j]a[j] The definition of a major downward incline is 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. 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. length height length height incline

15. Outline Compression Indexing Retrieval Experiments

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

17. Retrieval The procedure inputs a pattern series and searches 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. Highest inclines First, the retrieval procedure identifies the important inclines in the pattern., and selects the highest inclines. length 1 height length 2 12

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

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. 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. 0 0 210 fast ranking C = D = 5 time: 0.05 sec 200 perfect ranking 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. 0 0 331 fast ranking C = D = 2 time: 0.02 sec 200 perfect ranking 0 0 400 fast ranking C = D = 1.5 time: 0.01 sec 151 perfect ranking

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. 0 0 202 fast ranking C = D = 5 time: 0.31 sec 200 perfect ranking 0 0 328 fast ranking C = D = 2 time: 0.12 sec 200 perfect ranking 0 0 400 fast ranking C = D = 1.5 time: 0.09 sec 167 perfect ranking

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. 0 0 202 fast ranking C = D = 5 time: 1.18 sec 200 perfect ranking 0 0 400 fast ranking C = D = 2 time: 0.27 sec 151 perfect ranking 0 0 400 fast ranking C = D = 1.5 time: 0.14 sec 82 perfect ranking

26. 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 3 3 3 3 1 11 1 1 1