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Dot Plots For Time Series Analysis

Dot Plots For Time Series Analysis. Dragomir Yankov, Eamonn Keogh, Stefano Lonardi Dept. of Computer Science & Eng. University of California Riverside Ada Waichee Fu Dept. of Computer Science & Eng. The Chinese University of Hong Kong. Sequence analysis with dot plots.

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Dot Plots For Time Series Analysis

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  1. Dot Plots For Time Series Analysis Dragomir Yankov, Eamonn Keogh, Stefano Lonardi Dept. of Computer Science & Eng. University of California Riverside Ada Waichee Fu Dept. of Computer Science & Eng. The Chinese University of Hong Kong

  2. Sequence analysis with dot plots • Introduced by Gibbs & McIntyre (1970) • Observed patterns • Matches (homologies) • Reverses • Gaps (differences or mutations)

  3. Dot Plots For Time Series Analysis • Problem statement: How can we meaningfully adapt the DP analysis for real value data • The DP method would ideally be: • Robust to noise • Invariant to value and time shifts • Invariant to certain amount of time warping • Efficiently computable

  4. Related work Recurrence plots (Eckman et al (1987)) • Provide intuitive 2D view of multidimensional dynamical systems • Matrix is computed over the heaviside function Problem with recurrence plots Matches are locally (point) based rather than subsequence based

  5. The proposed solution • Reducing the dot plot procedure to the motif finding problem • Applying the Random Projection algorithm for finding motifs in time series data (Chiu et al 2003) It satisfies the initial requirements of robustness to outliers and invariance to time and value shifts • Presegmenting the series to achieve time warping invariance

  6. Dot plots and motif finding • Def: match, trivial match, motif - D(P,Q) <= R, we say that Q is a match of P - D(P,Q) <= R,D(P,Q1)<= R, we say that Q1 is a trivialmatch of P - A non trivial match is a motif • Def: Time series dot plot – a plot that contains a point at position (i,j) iff TS1(i) and TS2(j) represent the same motif

  7. The Random Projection algorithm • Based on PROJECTION (Buhler & Tompa 2002) • Algorithm outline • Split the TS into subsequences and symbolize them • Separate the symbolic sequences into classes of equivalence using PROJECTION • Mark as motifs sequences from the same class of equivalence

  8. Input TS: PAA TS: - Assigns letters to the PAA segments Random Projection – symbolization Utilizes the Symbolic Aggregate Approximation (SAX) scheme: • Applies PAA (Piecewise Aggregate Approximation)

  9. Random Projection–motif finding • d random dimensions are masked and the strings are divided into separate bins - The symbolic representations of the plotted time series are stored into tables

  10. Random Projection–motif finding - Updating the dot plot collision matrix - The update is performed for m iterations.

  11. Random Projection for streaming • Complexity: space – O(|M|), time – O(m|M|) • For practical data sets M is “very sparse” • For time series data small values of m (order of 10) generate highly descriptive plots • Random Projection as online algorithm • Good time performance • Updatability

  12. Experimental evaluation Dot Plots for anomaly detection Recurrent data with variable state length • The anomaly is of the same type: A • Small time warpings (shifts) are detected: B • Larger time warpings are omitted: C

  13. Experimental evaluation Dot Plots for anomaly detection Recurrent data with fixed state length

  14. Experimental evaluation Dot Plots for pattern detection Stock market data

  15. Experimental evaluation Dot Plots for pattern detection Audio data

  16. Experimental evaluation Dot Plots for pattern detection MUMer Random Projection Discrete data: for some tasks obtaining a real value representation is beneficial

  17. Dynamic sliding window • The fixed window does not perform well when: • The size of the recurrent states varies • We do not “guess” correctly the size of the states • Solution: use time series segmentation heuristics and a dynamic sliding window

  18. Dynamic sliding window Comparison of the dynamic and fixed sliding windows Tide data set Synthetic dataset The dynamic sliding window preserves more information about the frequency variability

  19. Conclusion • This work studies the problem of building dot plots for real value time series data • It demonstrates its equivalence to the motif finding problem • Introduced is an efficient and robust approach for building the dot plots • The performance of the tool is evaluated empirically on a number of data sets with different characteristics • Finally, a dynamic sliding window technique is proposed, which improves the quality and the descriptiveness of the plots

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