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

Low-Dimensional Chaotic Low-Dimensional Chaotic Signal Characterization Signal Characterization

Using Approximate EntropyUsing Approximate Entropy

Soundararajan EzekielSoundararajan EzekielMatthew LangMatthew Lang

Computer Science DepartmentComputer Science DepartmentIndiana University of PennsylvaniaIndiana University of Pennsylvania

RoadmapRoadmap

OverviewOverview

IntroductionIntroduction

Basics and BackgroundBasics and Background

MethodologyMethodology

Experimental ResultsExperimental Results

ConclusionConclusion

OverviewOverview

Many signals appear to be randomMany signals appear to be random

May be chaotic or fractal in natureMay be chaotic or fractal in nature

Wary of noisy systemsWary of noisy systems

Analysis of chaotic properties is in orderAnalysis of chaotic properties is in order

Our method - approximate entropyOur method - approximate entropy


Chaotic behavior is a lack of periodicityChaotic behavior is a lack of periodicity

Historically, non-periodicity implied Historically, non-periodicity implied randomnessrandomness

Today, we know this behavior may be Today, we know this behavior may be chaotic or fractal in naturechaotic or fractal in nature

Power of fractal and chaos analysisPower of fractal and chaos analysis


Chaotic systems have four essential Chaotic systems have four essential characteristics:characteristics: deterministic systemdeterministic system sensitive to initial conditionssensitive to initial conditions unpredictable behaviorunpredictable behavior values depend on attractorsvalues depend on attractors


Attractor's dimension is useful and good Attractor's dimension is useful and good starting pointstarting point

Even an incomplete description is usefulEven an incomplete description is useful


Fractal analysisFractal analysis

Fractal dimension defined for set whose Fractal dimension defined for set whose Hausdorff-Besicovitch dimension exceeds Hausdorff-Besicovitch dimension exceeds its topological dimensions.its topological dimensions.

Also can be described by self-similarity Also can be described by self-similarity propertyproperty

Goal: find self-similar features and Goal: find self-similar features and characterize data setcharacterize data set


Chaotic analysisChaotic analysis

Output of system mimics random behaviorOutput of system mimics random behavior

Goal: determine mathematical form of Goal: determine mathematical form of processprocess

Performed by transforming data to a Performed by transforming data to a phase spacephase space


DefinitionsDefinitions

Phase Space: n dimensional space, n is Phase Space: n dimensional space, n is number of dynamical variablesnumber of dynamical variables

Attractor: finite set formed by values of Attractor: finite set formed by values of variablesvariables

Strange Attractors: an attractor that is Strange Attractors: an attractor that is fractal in naturefractal in nature


Analysis of phase spaceAnalysis of phase space

Determine topological propertiesDetermine topological properties visual analysisvisual analysis capacity, correlation, information dimensioncapacity, correlation, information dimension approximate entropyapproximate entropy Lyapunov exponentsLyapunov exponents


Fractal dimension of the attractorFractal dimension of the attractor

Related to number of independent Related to number of independent variables needed to generate time seriesvariables needed to generate time series

number of independent variables is number of independent variables is smallest integer greater than fractal smallest integer greater than fractal dimension of attractordimension of attractor


Box DimensionBox Dimension

Estimator for fractal dimensionEstimator for fractal dimension

Measure of the geometric aspect of the Measure of the geometric aspect of the signal on the attractorsignal on the attractor

Count of boxes covering attractorCount of boxes covering attractor


Information dimensionInformation dimension

Similar to box dimensionSimilar to box dimension

Accounts for frequency of visitationAccounts for frequency of visitation

Based on point weighting - measures rate Based on point weighting - measures rate of change of information contentof change of information content

MethodologyMethodology

Approximate Entropy is based on Approximate Entropy is based on information dimensioninformation dimension

Embedded in lower dimensionsEmbedded in lower dimensions

Computation is similar to that of correlation Computation is similar to that of correlation dimensiondimension

AlgorithmAlgorithm

Given a signal {Given a signal {SSii}, calculate the }, calculate the

approximate entropy for {approximate entropy for {SSii} by the } by the

following steps. Note that the approximate following steps. Note that the approximate entropy may be calculated for the entire entropy may be calculated for the entire signal, or the entropy spectrum may be signal, or the entropy spectrum may be calculated for windows {calculated for windows {WWii} on {} on {SSii}. If the }. If the

entropy of the entire signal is being entropy of the entire signal is being calculated consider {calculated consider {WWii} = {} = {SSii}.}.

AlgorithmAlgorithm

Step 1:Step 1: Truncate the peaks of { Truncate the peaks of {WWii}. During }. During

the digitization of analog signals, some the digitization of analog signals, some unnecessary values may be generated by unnecessary values may be generated by the monitoring equipment. the monitoring equipment.

Step 2:Step 2: Calculate the mean and standard Calculate the mean and standard deviation (deviation (SdSd) for {) for {WWii} and compute the } and compute the

tolerance limit tolerance limit RR equal to equal to 0.3 * Sd0.3 * Sd to to reduces the noise effect. reduces the noise effect.

AlgorithmAlgorithm

Step 3: Step 3: Construct the phase space by Construct the phase space by plotting {plotting {WWii} vs. {} vs. {WWi+i+ττ}, where }, where ττ is the time is the time

lag, in an lag, in an EE = 2 = 2 space. space.

Step 4:Step 4: Calculate the Euclidean distance Calculate the Euclidean distance DDii between each pair of points in the between each pair of points in the

phase space. Count phase space. Count CCii(R)(R) the number of the number of

pairs in which pairs in which DDii << RR, for each , for each ii. .

AlgorithmAlgorithm

Step 5:Step 5: Calculate the mean of Calculate the mean of CCii(R)(R) then then

the log (mean) is the approximate entropy the log (mean) is the approximate entropy Apn(E)Apn(E) for Euclidean dimension for Euclidean dimension EE = 2 = 2..

Step 6:Step 6: Repeat Steps 2-5 for Repeat Steps 2-5 for EE = 3 = 3. .

Step 7: Step 7: The approximate entropy for {The approximate entropy for {WWii} }

is calculated as is calculated as Apn(Apn(22) - Apn() - Apn(33)). .

NoiseNoise

HRV (young subject)HRV (young subject)

HRV (older subject)HRV (older subject)

Stock SignalStock Signal

Seismic SignalSeismic Signal

Seismic SignalSeismic Signal

ConclusionConclusion

High approximate entropy - randomnessHigh approximate entropy - randomness

Low approximate entropy - periodicLow approximate entropy - periodic

Approximate entropy can be used to Approximate entropy can be used to evaluate the predictability of a signalevaluate the predictability of a signal

Low predictability - randomLow predictability - random

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