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Low-Dimensional Chaotic Signal Characterization Using Approximate Entropy

Low-Dimensional Chaotic Signal Characterization Using Approximate Entropy. Soundararajan Ezekiel Matthew Lang Computer Science Department Indiana University of Pennsylvania. Roadmap. Overview Introduction Basics and Background Methodology Experimental Results Conclusion. Overview.

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Low-Dimensional Chaotic Signal Characterization Using Approximate Entropy

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  1. Low-Dimensional Chaotic Signal Characterization Using Approximate Entropy Soundararajan Ezekiel Matthew Lang Computer Science Department Indiana University of Pennsylvania

  2. Roadmap • Overview • Introduction • Basics and Background • Methodology • Experimental Results • Conclusion

  3. Overview • Many signals appear to be random • May be chaotic or fractal in nature • Wary of noisy systems • Analysis of chaotic properties is in order • Our method - approximate entropy

  4. Introduction • Chaotic behavior is a lack of periodicity • Historically, non-periodicity implied randomness • Today, we know this behavior may be chaotic or fractal in nature • Power of fractal and chaos analysis

  5. Introduction • Chaotic systems have four essential characteristics: • deterministic system • sensitive to initial conditions • unpredictable behavior • values depend on attractors

  6. Introduction • Attractor's dimension is useful and good starting point • Even an incomplete description is useful

  7. Basics and Background • Fractal analysis • Fractal dimension defined for set whose Hausdorff-Besicovitch dimension exceeds its topological dimensions. • Also can be described by self-similarity property • Goal: find self-similar features and characterize data set

  8. Basics and Background • Chaotic analysis • Output of system mimics random behavior • Goal: determine mathematical form of process • Performed by transforming data to a phase space

  9. Basics and Background • Definitions • Phase Space: n dimensional space, n is number of dynamical variables • Attractor: finite set formed by values of variables • Strange Attractors: an attractor that is fractal in nature

  10. Basics and Background • Analysis of phase space • Determine topological properties • visual analysis • capacity, correlation, information dimension • approximate entropy • Lyapunov exponents

  11. Basics and Background • Fractal dimension of the attractor • Related to number of independent variables needed to generate time series • number of independent variables is smallest integer greater than fractal dimension of attractor

  12. Basics and Background • Box Dimension • Estimator for fractal dimension • Measure of the geometric aspect of the signal on the attractor • Count of boxes covering attractor

  13. Basics and Background • Information dimension • Similar to box dimension • Accounts for frequency of visitation • Based on point weighting - measures rate of change of information content

  14. Methodology • Approximate Entropy is based on information dimension • Embedded in lower dimensions • Computation is similar to that of correlation dimension

  15. Algorithm • Given a signal {Si}, calculate the approximate entropy for {Si} by the following steps. Note that the approximate entropy may be calculated for the entire signal, or the entropy spectrum may be calculated for windows {Wi} on {Si}. If the entropy of the entire signal is being calculated consider {Wi} = {Si}.

  16. Algorithm • Step 1: Truncate the peaks of {Wi}. During the digitization of analog signals, some unnecessary values may be generated by the monitoring equipment. • Step 2: Calculate the mean and standard deviation (Sd) for {Wi} and compute the tolerance limit R equal to 0.3 * Sd to reduces the noise effect.

  17. Algorithm • Step 3: Construct the phase space by plotting {Wi} vs. {Wi+τ}, where τ is the time lag, in an E = 2 space. • Step 4: Calculate the Euclidean distance Di between each pair of points in the phase space. Count Ci(R) the number of pairs in which Di<R, for each i.

  18. Algorithm • Step 5: Calculate the mean of Ci(R) then the log (mean) is the approximate entropy Apn(E) for Euclidean dimension E = 2. • Step 6: Repeat Steps 2-5 for E = 3. • Step 7: The approximate entropy for {Wi} is calculated as Apn(2) - Apn(3).

  19. Noise

  20. HRV (young subject)

  21. HRV (older subject)

  22. Stock Signal

  23. Seismic Signal

  24. Seismic Signal

  25. Conclusion • High approximate entropy - randomness • Low approximate entropy - periodic • Approximate entropy can be used to evaluate the predictability of a signal • Low predictability - random

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