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Dot Plots For Time Series Analysis

Dragomir Yankov, Eamonn Keogh, Stefano LonardiDept. of Computer Science & Eng.University of California Riverside

Ada Waichee FuDept. of Computer Science & Eng.

The Chinese University of Hong Kong

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Sequence analysis with dot plots

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• Introduced by Gibbs & McIntyre (1970)

• Observed patterns– Matches (homologies)– Reverses– Gaps (differences or

mutations)

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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

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Related work

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Recurrence plots (Eckman et al (1987))

Problem with recurrence plots

Matches are locally (point) based ratherthan subsequence based

)( ixr- Provide intuitive 2D view of multidimensional dynamical systems

- Matrix is computed over the heaviside function

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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)

• Presegmenting the series to achieve time warping invariance

It satisfies the initial requirements of robustness to outliers and invariance to time and value shifts

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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 trivial match 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

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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

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Random Projection – symbolization

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- Applies PAA (Piecewise Aggregate Approximation)

Input TS:

PAA TS:

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wpppP ...21

- Assigns letters to the PAA segments

Utilizes the Symbolic Aggregate Approximation (SAX) scheme:

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Random Projection–motif finding- The symbolic representations of the plotted time series are stored into tables

- d random dimensions are masked and the strings are divided into separate bins

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Random Projection–motif finding

- Updating the dot plot collision matrix

- The update is performed for m iterations.

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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

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Experimental evaluation

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

Dot Plots for anomaly detection

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Experimental evaluation

Recurrent data with fixed state length

Dot Plots for anomaly detection

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Experimental evaluation

Dot Plots for pattern detection

Stock marketdata

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Experimental evaluation

Dot Plots for pattern detection

Audio data

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Experimental evaluation

Dot Plots for pattern detection

Discrete data: for some tasks obtaining a real value representation is beneficial

MUMer

Random Projection

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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

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Dynamic sliding window

Comparison of the dynamic and fixed sliding windows

The dynamic sliding window preserves moreinformation about the frequency variability

Synthetic dataset Tide data set

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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