Efficiently Evaluating Order Preserving Similarity Queries over Historical Market-Basket Data

1. Efficiently Evaluating Order Preserving Similarity Queries over Historical Market-Basket Data Reza Sherkat and Davood Rafiei Department of Computing Science University of Alberta Canada Travel assistance provided by the Mary Louise Imrie Graduate Student Award ICDE06

2. Overview • Introduction • Histories and Time-series • Similarity model for histories • Problem Definition • Proposed Approach • Results Highlight • Conclusions ICDE06

3. Querying Histories: Introduction • Querying multiple snapshots of data • Temporal selection, projection, and join queries • Finding similar time-series • Finding companies having similar stocks • Is it possible to define a notion of similarity for objects based on the similarity of their histories? ICDE06

4. the history of a web-page : bag of word Histories History: A sequence of time-stamped observations • Time-series: observations are real-values • Observations can be more general the history of a patient ICDE06

5. Similarity Model for Histories History for 3 patients • Similarity of two histories depends on: • Pair-wise similarity of their observations ICDE06

6. Similarity Model for Histories History for 3 patients • Similarity of two histories depends on: • Pair-wise similarity of their observations • The order that similar observations are recorded • Constraints on time-stamps of observations ICDE06

7. Problem Definition Given a history as a query: • Evaluate k-NN and Range queries efficiently. • For each history in the result, find its common signature with the query - where the similarity comes from? ICDE06

8. Similarity Measure for Histories Alignment of histories: • An approach to line-up subsequences of two histories • Denoted by a sequence of matches: • is an observation in A (B) or a gap ( ). • is the score of a match. • Alignment score measures the quality of an alignment. ICDE06

9. The best alignment of two histories: Alignments of Histories Alignment score can be the sum of the score of matches in the alignment. ICDE06

10. Alignments of Histories Alignment score can be the sum of the score of matches in the alignment. The best alignment of two histories: What is the best alignment of length 3? ICDE06

11. Alignments of Histories Alignment score can be the sum of the score of matches in the alignment. The best alignment of two histories: What is the best alignment of length 3? If the match could not be considered, what would be the best alignment of length 2? ICDE06

12. Constraints on the Alignments of Histories • The number of matches in the alignment. • l-alignment: alignment with l matches • The r-neighborhood constraint • For each match • r ,l : parameters of the similarity query. ICDE06

13. Principle of Optimality p(A) p(B) s(A) s(B) : optimal alignment of p(A) and p(B) : optimal alignment of s(A) and s(B) : optimal alignment of A and B : concatenation operator The principle of optimality holds if: ICDE06

14. b , , b , b , , b K K + 1 1 j j n • Optimal l-alignment of suffixes can formed by: • Concatenating with optimal (l-1)-alignment of suffixes • Matching with gap, and considering l-alignment of suffixes • Matching with gap, and considering l-alignment of suffixes Score of Optimal l-alignment ICDE06

15. can be used to find common signature of histories: • A sequence of observations that appear in the same order in • two histories. • Generalizes the notion of longest common subsequence. Similarity Measure for Histories : the score of optimal l-alignment of two histories. ICDE06

16. Similarity Queries over Collection of Histories • Straightforward (not practical) approach: naïve scan • Indexing techniques are proposed for metric spaces, but is not metric: • when the distance between observations is not metric. • when an r-neighberhood constraint is specified. • We propose upper bounds to prune history search space. ICDE06

17. A General Upper Bound for the Similarity Measure Intuition: The score of an optimal relaxed l-alignment is not less than the score of optimal l-alignment. • For each observation, find an optimal match. • Aggregate the scores for top l optimal matches to find an upper bound for . This upper bound can prune some extra computations, but still all histories will be accessed to evaluate a query. ICDE06

18. This upper bound can be evaluated efficiently by exploiting an inverted index if is Cosine or Extended Jaccard Coefficient. An Index-based Upper Bound for the Similarity Measure • Intuitions: • Observations are sparse in real life applications. • The score of an optimal relaxed match is not less • than the score of an optimal match. • The score of an optimal relaxed alignment is not • less than the score of optimal relaxed l-alignment. ICDE06

19. Experiments • Experiments performed on AMD/XP 2600 512 Mb RAM • Datasets: • DBLP • Synth1: Our synthetic data • Synth2: Modified IBM synthetic data generator • Investigated: • Effectiveness of similarity measure • Efficiency of our approach • Pruning power, Running time, Saleability ICDE06

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21. … … … … : Poisson distribution V(i+1): bit string following V(i) in a pre-determined order [Cho et al. VLDB 2000] V(1) V( i+1 ) V( n ) V( i ) • Synth2 dataset contains: • 20,000 histories • for each history is selected randomly from {1,…,10} • Length of histories: {32,…,64} Effectiveness of our Similarity Measure observation: document modeled as bit string First observation: randomly selected … … ICDE06

22. Effectiveness of our Similarity Measure (cnt.) Mean deviation of from for k-NN queries: * For 2,000 randomly generated queries ICDE06

23. Pruning Power vs. k Fraction of database examined 0 20 40 60 80 100 1 10 100 1024 No. of neighbours in k-NN query (LOG scale) ICDE06

24. Running Time vs. k Time (msec) 0 100 200 300 400 500 600 1 10 100 1024 Dataset: Synth2, 8,000 Histories, 1,000 items No. of neighbours in k-NN query (LOG scale) ICDE06

25. Scalability for 1-NN queries Time (msec) 8,000 16,000 32,000 64,000 No. of histories in the collection ICDE06

26. Running time vs. Sparseness of Observations Time (msec) 256 512 1,024 2,048 4,096 8,092 No. of items (LOG scale) ICDE06

27. Conclusions • Introduced a domain-independent framework to formulate and evaluate similarity queries over historical data. • Generalized few concepts, including edit distance and longest common subsequence to histories. • Developed upper bounds to efficiently evaluate queries. One of our upper bounds can directly take advantage of an index even though it is not metric. • Our experiments confirm the effectiveness and efficiency of our approach. ICDE06

28. Thank you for your attention! ICDE06

29. Related Works • Detecting, representing, querying histories • [Chawathe 1998], [Chien 2001] • Similarity-based sequence matching • [Altschul 1990], [Pearson 1990], [Bieganski 1994] • Finding similar sequence of events • [Wang 2003] • Finding similar time series • [Agrawal 1995], [Rafiei 1997], [Keogh 2002], [Vlachos 2002, 2003], ... ICDE06