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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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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

  • 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


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


Introduction

  • Chaotic systems have four essential characteristics:

    • deterministic system

    • sensitive to initial conditions

    • unpredictable behavior

    • values depend on attractors


Introduction

  • Attractor's dimension is useful and good starting point

  • Even an incomplete description is useful


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


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


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


Basics and Background

  • Analysis of phase space

  • Determine topological properties

    • visual analysis

    • capacity, correlation, information dimension

    • approximate entropy

    • Lyapunov exponents


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


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


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


Methodology

  • Approximate Entropy is based on information dimension

  • Embedded in lower dimensions

  • Computation is similar to that of correlation dimension


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}.


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.


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.


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).


Noise


HRV (young subject)


HRV (older subject)


Stock Signal


Seismic Signal


Seismic Signal


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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