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

2. Roadmap OverviewIntroduction 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 deterministic system sensitive to initial conditions sensitive to initial conditions unpredictable behavior unpredictable behavior values depend on attractors 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 visual analysis capacity, correlation, information dimension capacity, correlation, information dimension approximate entropy approximate entropy Lyapunov exponents 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 { S i }, calculate the approximate entropy for { S i } 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 { W i } on { S i }. If the entropy of the entire signal is being calculated consider { W i } = { S i }.

16. Algorithm Step 1: Truncate the peaks of { W i }. 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 { W i } 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 { W i } vs. { W i+τ }, where τ is the time lag, in an E = 2 space. Step 4: Calculate the Euclidean distance D i between each pair of points in the phase space. Count C i (R) the number of pairs in which D i < R, for each i.

18. Algorithm Step 5: Calculate the mean of C i (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 { W i } is calculated as Apn(2) - Apn(3).

19. Noise

20. HRV (young subject)

21. HRV (older subject)

22. Stock Signal

23. 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