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### 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

- Introduced by Gibbs & McIntyre (1970)

- Observed patterns
- Matches (homologies)
- Reverses
- Gaps (differences or mutations)

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

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

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

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

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

PAA TS:

- Assigns letters to the PAA segments

Random Projection – symbolizationUtilizes the Symbolic Aggregate Approximation (SAX) scheme:

- Applies PAA (Piecewise Aggregate Approximation)

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

Random Projection–motif finding

- Updating the dot plot collision matrix

- The update is performed for m iterations.

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

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

Experimental evaluation

Dot Plots for

pattern detection

MUMer

Random

Projection

Discrete data: for some tasks obtaining a real value

representation is beneficial

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

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

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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