IOP PUBLISHING PLASMA PHYSICS AND CONTROLLED FUSION

Plasma Phys. Control. Fusion 55 (2013) 085015 (7pp) doi:10.1088/0741-3335/55/8/085015

Permutation entropy analysis oftemperature fluctuations from a basicelectron heat transport experimentJ E Maggs and G J Morales

Department of Physics and Astronomy, University of California, Los Angeles, Los Angeles,CA 90095, USA

Received 7 February 2013, in final form 19 May 2013Published 10 June 2013Online at stacks.iop.org/PPCF/55/085015

AbstractThe permutation entropy concept of Bandt and Pompe (2002 Phys. Rev. Lett. 88 174102) isused to analyze the fluctuations in ion saturation current that spontaneously arise in a basicexperimental study (Pace et al 2008 Phys. Plasmas 15 122304) of electron heat transport in amagnetized plasma. From the behavior of the Shannon entropy and the Jensen–Shannoncomplexity it is found that the underlying dynamics are chaotic rather than stochastic. Apartitioning and scrambling technique is used to demonstrate that the exponential character ofthe associated power spectrum arises from individual Lorentzian pulses observed in thetime series.

(Some figures may appear in colour only in the online journal)

1. Introduction

The purpose of this paper is to conclusively identify the natureof the underlying dynamics that cause anomalous transport ina basic experiment of electron heat transport in a magnetizedplasma and to suggest that the techniques described here canbe usefully employed to examine signals from other plasmaexperiments. The details of the experimental arrangementand the major findings have been well documented [1–3], sothat only a cursory description is necessary for the presentinvestigation. The experiment typically uses a small (3 mmdiameter), single-crystal LaB6 cathode to inject a low-voltageelectron beam into a strongly magnetized, cold, afterglow-plasma. The low-voltage beam acts as an ideal heat source andproduces a long (∼8 m), narrow (∼5 mm in radius) filamentof elevated electron temperature that is disconnected fromthe walls of the device, and is surrounded by, an essentiallyinfinite, plasma of much lower temperature. The existence ofa transition from a period of classical transport, i.e. transportdue to Coulomb collisions, to one of anomalous transport hasbeen established through detailed measurements. During theperiod of classical transport, drift-Alfven waves grow linearly,driven by the temperature gradient in the filament edge [4].The transition in transport is characterized by a change inthe character of the frequency power spectrum of temperaturefluctuations, from a line spectrum in the classical transportcase, to a broadband spectrum when anomalous transport is

present. The broadband frequency spectrum is exponentialin nature.

Two major questions motivated by these experimentalfindings are: what is the reason for the characteristicexponential frequency dependence of the fluctuation spectrumduring the anomalous transport period, and what is thenature of the underlying dynamics that result in anomalousheat transport? Two prominent theoretical approaches thatare candidates for answering these questions are based onfundamentally different concepts: chaotic behavior [5, 6]and stochastic processes [7–9]. Therefore, it is importantin answering these questions to utilize methods of signalanalysis that are capable of distinguishing between thesetwo dynamical systems. This investigation uses two suchmethods: partitioning and temporal scrambling of time series,and, permutation entropy analysis. The temporal scramblingdemonstrates that individual Lorentzian pulses are responsiblefor the exponential character of the power spectrum, whilethe permutation entropy analysis, based on the concepts ofBandt and Pompe [10] and displayed in the entropy-complexityplane [11], identifies that the dynamics in the temperaturefilament are chaotic.

The paper is organized as follows. Section 2 presentsthe signal scrambling analysis leading to the identificationof Lorentzian pulses as the origin of the exponential powerspectrum. Section 3 provides a brief introduction to thepermutation entropy method. Section 4 applies the entropy and

0741-3335/13/085015+07$33.00 1 © 2013 IOP Publishing Ltd Printed in the UK & the USA

Plasma Phys. Control. Fusion 55 (2013) 085015 J E Maggs and G J Morales

Frequency (kHz)

Log

Pow

er (

arb.

uni

ts)

s

Figure 1. The ensemble-averaged fluctuation power spectrum of theunscrambled signals. The peaks below 50 kHz arise from driftwaves and a thermal diffusion wave. The time constant associatedwith the exponential fit to the power spectrum (i.e. exp(−2ωτ )) isτ = 5.0 µs.

complexity measures to experimental time series. Section 5provides conclusions.

2. Power spectrum

An ensemble of one hundred signals, each consisting of5.25 ms of data, is used in the analysis of the power spectrum.The temporal records are 8192 points in length collected with atime increment of 0.64 µs. The individual signals correspondto ion saturation current, Isat, collected by a small Langmuirprobe placed within the temperature filament. The signalsin this data set, collected when the probe is near the centerof the filament, display mainly negative pulses, presumablyrepresenting decreased temperature. The ensemble-averagedpower spectrum of the signals is shown in figure 1 in a log-linear format. The power spectrum is exponential over therange from 5–110 kHz with prominent lines at frequenciesbetween 5–50 kHz. The lowest frequency line, at about 8 kHz,corresponds to a resonant thermal diffusion wave [12] whilethe other peaks are drift-Alfven waves [4]. The slope of theexponential portion of the spectrum (represented by the red,dashed, straight line) corresponds to a temporal Lorentzianpulse whose width, τ , is 5 µs. Lorentzian pulses have thefunctional form

L(t) = A

1 + [(t − t0)/τ ]2 , (1)

where t0 is the time of peak arrival, τ is the pulse width and A

is the peak amplitude. The power spectrum of the Lorentzianpulse in equation (1) is proportional to exp(−2ωτ).

The proximate cause of the exponential power spectra isthe occurrence, in the Isat time signals, of Lorentzian-shapedpulses of the type shown in equation (1). This assertion is testedby investigating the effects of partitioning and scramblingthe ensemble of signals [13]. The scrambled signals areconstructed from the ensemble by breaking the original signal

2.2 2.3 2.4 2.5 2.60.00

0.05

0.10

0.15

0.20

Time (ms)

16 pieces

64 pieces

256 pieces

1024 pieces

Lorentzianpulse

Figure 2. A portion of one of the time signals from the temperaturefilament experiment used in this analysis. A Lorentzian pulse (redcurve) with a width corresponding to the slope of the power spectrais superposed on an isolated pulse appearing in the data and whosepeak is identified with the vertical black line. Scrambled signals areconstructed from 16 (red), 64 (orange), 256 (green) and 1024 (blue)pieces. The temporal length of these pieces relative to the typicalLorentzian pulse is indicated by the arrows of various colors.

into 2n equal-length pieces and then randomly rearranging thepieces. In the example presented here, each member of theoriginal ensemble of one hundred signals is broken into 16,64, 256 and 1024 pieces and then randomly scrambled.

A portion of one representative time trace taken fromthe original ensemble is shown in figure 2 together with asuperposed Lorentzian pulse (red solid curve), whose width,τ = 5.0 µs, corresponds to the time constant deduced fromthe power spectrum fit, exp(−2ωτ ), shown previously as thedashed red line in figure 1. Also illustrated in figure 2, relativeto the representative Lorentzian pulse, are the sizes of the timeintervals used in the partitioning and scrambling process. Thetemporal length of each segment in the partition with 64 pieces(82 µs) is over 16 times the typical Lorentzian pulse width. Incontrast, the length of each segment in the 256-piece partition(20 µs) is only about four times the typical pulse width, andthe interval with 1024 pieces is equal to the pulse width. It isexpected that scrambling the signal with partitions of 16 and64 pieces will leave many of the pulses intact, while the 1024piece partition should destroy most of the pulses.

The effect of the scrambling process on the ensemblepower spectra is shown in figure 3 in a log-linear format. Forease of viewing, each power spectrum displayed is multipliedby a factor of 102n (n = 0, 1, 2, 3, 4) to separate it from theone below it. The power spectrum for the unscrambled signal(bottom trace) is the same as displayed in figure 1. The curveappears ‘smoother’ here because fewer points are used in thedisplayed trace in order to reduce the file size of the figure.A linear fit to the ‘exponential region’ and the correspondingpulse width to which the slope corresponds are shown as dashedlines (color coded) for each partition of the scrambled signal.The partitions of 16 and 64 pieces do not affect the linear(i.e. exponential) portion of the power spectrum very much.The extent of the linear region decreases somewhat and the

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Plasma Phys. Control. Fusion 55 (2013) 085015 J E Maggs and G J Morales

Figure 3. The ensemble averaged power spectra of the scrambledsignals. Dashed lines are the best fit to exponential dependence withτ corresponding to the equivalent width of a Lorentzian pulse.When the Lorentzian pulses are destroyed by the temporalscrambling the exponential feature disappears.

slope decreases slightly for the 64-piece partition. However,the 64-piece partition procedure does greatly reduce the ‘line’features in the spectrum that arise from the presence of driftwaves. The 256-piece partition still exhibits a linear behavior,albeit over a much reduced range of frequencies and the slopeis shallower, corresponding to a narrower pulse. However,the effect of the 256-piece partition is very evident at lowfrequencies where the power spectrum is now flat. Finally, the1024-piece partition power spectrum is flat over the range from5 to 100 kHz and is similar to ‘white’ noise. The systematicchange in the power spectra as the signals are partitioned intosmaller and smaller pieces until the pulses are destroyed clearlydemonstrates that the Lorentzian pulses are the cause of theexponential part of the spectrum.

3. Entropy and complexity measures

The statistical character of a time signal can be determined byobtaining its permutation entropy and statistical complexity,which can be computed from a probability distributionintroduced by Bandt and Pompe [10]. This quantity representsthe probability of occurrence of the various (Nt − d + 1)realizations of the d! amplitude orderings of the d-tuplesthat appear in a signal consisting of Nt discrete elements eachhaving an amplitude Aj with 0 � j � Nt −1. As a concreteexample for a case with d = 5, a signal of Nt = 20 000elements has 19996 distinct 5-tuples (tj , tj+1, tj+2, tj+3, tj+4).Within each 5-tuplet, an ordering of the amplitude of the signalrecorded, e.g. (Aj , Aj+1, Aj+2, Aj+3, Aj+4), is just one of apossible set of 120 (i.e. 5!) permutations. The Bandt–Pompeprobability distribution of a particular time signal is determinedby computing the frequency of occurrence of each of thepossible permutations of the amplitude ordering observed inthe signal. The number of possible amplitude permutationsof a d-tuplet increases very rapidly with the value of d (e.g.10! = 3 628 800), accordingly, this number can easily exceed

the total number of data points typically stored in an actualexperimental situation. Thus, in practical implementations ofthe concept, the value of d is less than 7, and for the resultspresented here the value, d = 5, is used. Once the Bandt–Pompe probability distributions are computed for an ensembleof signals, the statistical nature of the signal can be evaluated bydetermining its location in the entropy-complexity plane [11].There are a variety of entropy definitions in common usage(Shannon, Tsallis and Renyi, for example) and even moremeasures of statistical complexities, as presented by Martinet al [14]. In the subsequent discussions, following Rossoet al [11], Shannon’s formulation is used to evaluate entropyand the Jensen–Shannon divergence serves as a measure ofstatistical complexity.

The nature of the dynamics resulting in a particularexperimental time signal can be determined from its location inthe entropy-complexity plane, or, using the notation of Rossoet al [11] , the so called CH-plane, when the normalizedShannon entropy, H, and Jensen–Shannon complexity, CS

J , areused as measures. For a set of probabilities, P , of dimension N

where pj denotes the probability of occurrence for one of theN possible states, pj � 0; j = 1, 2, . . . , N and

∑N1 pj = 1,

the Shannon entropy, S and normalized Shannon entropy, H ,are defined as [14]

S (P ) = −∑N

1pj ln

(pj

);H (P ) = S (P )/max (S) = S (P )/ln (N). (2)

Note that the maximum Shannon entropy is obtained whenall states have equal probability, pj = 1/N; j = 1, 2, �, N .This maximum entropy state is denoted as Pe. The Jensen–Shannon complexity is defined as [14]

CSJ = −2

S(

P +Pe2

) − 12 S (P ) − 1

2 S (Pe)

N+1N

ln (N + 1) − 2 ln (2 N) + ln (N)H (P ) .

(3)

To illustrate the different regions where chaotic and stochasticsignals appear in the CH plane, figure 4 shows two well-knownexamples of such dynamical behaviors, the chaotic logisticmap [15] and the stochastic fractional Brownian motion (fBm)[16]. The green curve marked ‘fBm’ in figure 4 is the locus ofpoints of fractional Brownian motion, whose increments arefractional Gaussian noise with Hurst exponent, He, ranging, insteps of 0.05, from 0.025 to 0.975, (0.025 � He � 0.975). Thedensely spaced collection of red dots represents the locations,in the CH-plane, of the logistic map, xn+1 = rcxn(1.0 − xn)

for those values of rc in the range, 3.58 < rc < 4.0, for whichthe logistic map exhibits chaotic behavior [15].

Also shown in figure 4 are two curves labeled ‘maximumcomplexity’ and ‘minimum complexity’. The genesis of thesecurves is discussed in detail by Martin et al [14], and onlya brief description is given here. For a chosen embeddingdimension, d, there are N = d! possible states in the Bandt–Pompe probability space, i.e. the Bandt–Pompe probabilityspace has dimension N . Maximum Shannon entropy isachieved when all states are equally populated, pj = 1/N forj = 1, 2, 3, ..., N . Minimum complexity, for an entropy lessthan maximum, corresponds to one state having probability,

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Plasma Phys. Control. Fusion 55 (2013) 085015 J E Maggs and G J Morales

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

H

fBm

MAXIMUM COMPLEXITY

MINIMUM COMPLEXITY

d = 5

Chaotic Behavior

Stochastic Behavior

Logistic MapCJ

S

Figure 4. Chaotic and stochastic dynamical processes occupydifferent regions of the entropy-complexity plane. Chaoticprocesses, as represented by the logistic map (densely spaced reddots), reside in the lilac-shaded region, and stochastic processes,represented by fractional Brownian motion (fBm) shown by thegreen curve, reside in the un-shaded region. The embedding spaceused to evaluate the Bandt–Pompe probability has dimension d = 5.

pk , with 1/N < pk � 1, and all other states having equalprobabilities, pj = (1 − pk)/(N − 1); j = 1, 2, 3, . . .,N , j �= k. As a concrete example, for d = 5, and thusN = 120, pk might be chosen to have 60 values between 1/120and 1. The minimum complexity is then computed on a spaceof dimension (60, 120). A realization of a time signal withminimum complexity is a slowly changing signal with shortbursts of white noise.

In contrast, maximum complexity is computed from thecollection of N − 1 probability distributions associated withprobability spaces having dimensions ranging from 2 to N .These N − 1 probability distributions have one state withprobability pk(i), where 0 � pk(i) � 1/(N − i + 1) andall other (N − i) states have equal probabilities, pj(i) =(1 − pk(i))/(N − i); j (i) = 1, . . ., N − i + 1, j (i) �= k(i)

and where i = 1, ..., N − 1. As a concrete example ofone of the probability distributions in the N − 1 collectionof distributions used in determining the maximum complexitycurve, consider the case, i = N − 5. Then the probabilityspace dimension is 6 and pk(N−5) can range from 0 to 1/6. Ifpk(N−5) = 0, the remaining 5 states have equal probabilities,pj(N−5) = 1/5, if pk(N−5) = 1/12, pj(N−5) = 11/60, andso forth. The maximum probability curve corresponds to thelocus of maximum complexities for the collection of entropy-complexity curves computed from this set of probabilitydistributions. If the various pk(i) vectors are chosen to havedimension 60, as in the minimum complexity example, themaximum probability is computed over a (119, 60, 120)dimensional space.

4. Application to time signals

Experimental time signals gathered from plasma probes aretypically obtained from analog to digital converters in which avoltage value is recorded at discrete time steps. The time step,

�t , is chosen small enough to capture the dynamics of interestto the experimenter. The time step determines the Nyquistfrequency fN = 1/2�t , and dynamical process that occur atfrequencies higher than fN are not resolved, but may affectthe power spectra through the phenomena of aliasing. It isassumed here that care is taken by the experimenter to avoidthe effects of aliasing, so that only the effects of noise needbe considered. Noise in the time signals increases the entropywhen using the Bandt–Pompe probability. Permutations ofthe basic amplitude ordering of a d-tuplet, [1, 2, 3, . . ., d]that are not realized by a dynamic process of interest can,however, be populated by noise. Maximum entropy occurswhen all possible amplitude permutations occur equally in atime signal. Thus it is desirable to reduce the effects of noiseon experimental signals to better ascertain the true nature of thedynamical process of interest. Of course, this is problematicin some cases, because the dynamical process under studymay be stochastic and strongly resemble noise. A commontechnique for reducing noise is to digitally filter the timesignals. Digital filtering alters the Fourier amplitude of thesignals, and, if the frequency range of the noise in the signalscan be identified, digital filtering can effectively eliminatenoise. Another technique for reducing the effects of noiseis wavelet de-noising [17] in which the wavelet coefficients,instead of the Fourier amplitudes, of the signals are alteredin order to reduce the contributions of noise to the signals.Both filtering and wavelet de-noising were explored, but thesetechniques are found to greatly alter the locations in the CH-plane of both chaotic and stochastic signals. A techniquefor reducing the effects of noise that does not much alterthe location of stochastic processes in the CH-plane is sub-sampling, or the embedding delay [18].

The Bandt–Pompe probability computed for a specificcollection of d-tuplets garnered from a time signal involvesthe concept of the d-tuplets representing structures in anembedding space of dimension d [14, 18]. In its primal form,the size, in time, of the structures investigated is d�t , where,�t is the sampling time. As mentioned earlier, since thenumber of possible structures grows as d!, it is not practical toinvestigate larger structures in time by increasing the size of thed-tuplet. Rather, larger structures in time can be investigatedby the technique of sub-sampling or embedding delay. In thesub-sampled signal the interval between successive data pointsis m �t rather than �t where m is taken to be a positive integerin order to avoid interpolation. The sub-sampling techniquereduces the Nyquist frequency, and, of course, the number ofpoints in the signal, N , is reduced to N /m, but the total timelength of the signal is unaltered. Thus, sub-sampling limitsthe upper frequency range of the dynamics under investigationwithout changing the low frequency information. In contrast,the lower range of frequencies available for investigation canbe limited by shortening the length of the time records used inthe analysis.

Often in the analysis of experimental data, certain features,or frequency intervals, of the power spectra are of particularinterest. For example, in the temperature filament dataconsidered here, the range of frequencies over which the datadisplays an exponential behavior is of particular interest, as

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Plasma Phys. Control. Fusion 55 (2013) 085015 J E Maggs and G J Morales

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Frequency (f/fN)

Data

Lorenz

Gissinger

Hindmarsh

Log

Pow

er (

arb.

uni

ts)

Subsample

(a) (b)

(c) (d)

(e) (f )

(g) (h)

0 0.2 0.4 0.6 0.8 1

MinimumComplexity

MaximumComplexity

d = 5

H

s_H

G

s_G

s_L

L

s_D

D

fBm

H

G

L

D

0.0

0.2

0.4

0.6

H

CJS

s_fBm

Figure 5. The effects of the sub-sampling operation are illustrated. (b),(d),(f ) and (h) are sub-sampled spectra extracted from time signalswhose spectra are shown in the adjacent left panels. Sub-sampling reduces the Nyquist frequency fN without changing the spectral shape.The corresponding locations, in the CH plane, of the sub-sampled time series for the Hindemarsh (s H), Gissinger (s G), Lorenz (s L)models and data (s D) all move toward the region of moderate entropy and maximum complexity, while sub-sampled fractional Brownianmotion (s fBm) is relatively invariant.

illustrated in figure 1. In employing the concept of Bandt–Pompe probability, the sub-sampling technique can be usedto limit the range of frequencies in the power spectrum and,thereby, limit the study of the dynamics to those time scalesof interest. This procedure is analogous to that employed inanalysis of stochastic time signals [19] in which the spectralrange of the power spectrum is limited to only those frequenciesover which the power spectrum exhibits a power law behavior.

Figure 5 illustrates the effects of sub-sampling on thepower spectra of chaotic signals obtained from numericalsolutions of some well-known nonlinear dynamics models:Gissinger (G) [20], Hindemarsh (H) [21] and Lorenz (L) [22].The effects on the data (D) used to generate figure 1 arealso indicated. All power spectra are shown as functions offrequency, normalized to the Nyquist frequency, fN, whosenumerical value changes after sub-sampling. The powerspectra of the original signals recorded are shown in the left-side panels ((a), (c), (e), (g)). The corresponding powerspectra of the sub-sampled signals are shown in the adjacent,left-side panels ((b), (d), (f ), (h)). Panel (h) is essentiallythe same as figure 1. The sub-sampling interval is separatelychosen for each class of signal so that the Nyquist frequencyis reduced to where the spectral range of interest spans themajority of the normalized frequency range. The signalsfrom nonlinear dynamics models (G, H and L) are calculatedusing small time steps to ensure accuracy in the Runga–Kutta

integration, so that the Nyquist frequency of the calculatedsignals is much higher than needed to capture the dynamics.Thus, the sub-sampling intervals for these signals is large, 20for the Gissinger model, 80 for the Hindemarsh model and18 for the Lorenz model. In contrast, the data was only sub-sampled on an interval of 4 to limit the spectral range of interest.The right side of figure 5 shows the locations, in the CH-plane, of the original signals (G, H, L and D) together withthe sub-sampled signals (s G, s H, s L and s D). The sub-sampled signals move toward moderate entropy and highercomplexity. For comparison, the dashed curves display thecorresponding effect of sub-sampling, with an interval of 8,on the stochastic signal associated with fractional Brownianmotion (fBm) shown earlier in figure 4. It is seen that the locusof fractional Brownian motion (fBm) is not much changed bythe sub-sampling operation (s fBm). The power spectra ofthe fractional Brownian motion signals are not shown becausethey are power laws (by construction) over the entire frequencyrange and remain so under the sub-sampling operation.

Figure 6 shows a comparison between data obtainedfrom the temperature filament experiment and the output of anumerical model of chaotic advection [23, 24] that incorporatesthe essential features of the experimental arrangement.Figure 6(a) shows the location in the CH-plane of numericaloutput from the model and data from the experiment. Spatialcontours of the temperature in the plane across the confinement

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Plasma Phys. Control. Fusion 55 (2013) 085015 J E Maggs and G J Morales

Figure 6. (a) Locations in the CH plane of temperature filament data (crosses), and numerical output of chaotic advection model (triangles).The shaded region, as in figure 4, delineates the region of chaotic behavior. The stochastic process of fractional Brownian motion (fBm) isalso shown for comparison. (b) Instantaneous spatial contours (x, y) of normalized electron temperature from the chaotic advection modelat a particular time. Time signals of temperature are obtained from the spatial locations indicated by ‘black’ dots to generate locations in CHplane. (c) Same as in (b), but for experimental data.

magnetic field (x, y) obtained from the numerical solution ofthe chaotic advection model are shown in (b) and measureddata (different than used in figure 1) from the temperaturefilament experiment in (c). Time signals are taken from sixteenlocations along a circle with radius 0.6 cm as indicated by the‘black’ dots in (b) and (c). Nyquist frequencies of both themodel and the experimental signals are adjusted by choosinga sub-sampling interval that results in the exponential part ofthe power spectra extending over at least half the frequencyrange. The sub-sampled time signals from both the model andexperiment have 4000 points. The Bandt–Pompe probabilitiesof time signals from the model and experiment are computedfor an embedding space with dimension, d = 5, so thateach sub-sampled time signal contains 3996 5-tuplets. Fora uniform distribution, the average number of realizations, pertuplet, is roughly 33.

The locations of the 16 model points are shown as trianglesin figure 6(a) and the experimental data are shown as crosses.Both the model output and experimental data are within theshaded region that demarks ‘chaotic behavior’. The dynamicsof the model have been identified by orbit sampling techniquesto be chaotic, so it is reassuring that the sampled time seriesfalls in the ‘chaotic behavior’ region of the CH-plane. Thevery close proximity of the experimental data to the chaoticadvection model data bolsters the assertion made previously

by the authors [23] that the dynamics associated with transportin the temperature filament is chaotic.

Figure 7 shows the changes in the location of the datasignals used to generate figure 3 under the partitioning andscrambling operation. The signals are first sub-sampled, withsub-sample interval m = 4, so that the spectral range islimited to that shown in figure 1. A red triangle marked(u) represents the ensemble averaged position of the sub-sampled, unscrambled signals. Other red triangles markedwith 16, 64, 256 and 1024 are ensemble-averaged positions ofthe partitioned and scrambled, sub-sampled signals, with thenumber of pieces in the partition indicated. Consistent with thebehavior of the power spectra shown in figure 3, the partitionedand scrambled signals become less chaotic in nature and movetoward the white noise region of the CH plane as the numberof partitions grows.

5. Conclusions

The method of ‘partitioning and scrambling’ has been shownto destroy the exponential character of the power spectrumof fluctuations in ion saturation current measured in a basicexperiment of electron heat transport. But importantly, thedestruction occurs only when the temporal scrambling intervalis chosen small enough to disrupt the individual Lorentzian

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Plasma Phys. Control. Fusion 55 (2013) 085015 J E Maggs and G J Morales

0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

MinimumComplexity

MaximumComplexity

d = 5

white noise

u16 64

256

1024

H

CJS

fBm

Figure 7. The locations, in the CH-plane, of the scrambled signalsused to generate the power spectra displayed in figure 3. As thenumber of scrambled partitions is increased, the position in the CHplane moves toward maximum entropy and minimum complexity,approaching the location of ‘white noise’.

pulses observed in the time signals. The exponential characterof the spectrum remains basically intact until the width ofthe scrambling window is below 20 µs (the 256 and 1024piece partitions), which is approximately the auto-correlationtime for a single pulse. In addition, the dynamical nature ofthe signals, as indicated by their location in the C–H planedoes not change appreciably until the scrambling partition issmaller than 20 µs in length. Thus, as long as the scramblingprocess does not significantly affect the individual pulses,both the statistical nature, as measured by the permutationentropy, and the power spectrum remain relatively unchanged.In previous work [3] it was demonstrated that the measuredvalue of the Lorentzian pulse width, τ , obtained from fittingindividual pulse events in the time series, is consistent with theindependent determination of τ from the slope of the powerspectrum in a log-linear plot. Combining this information withthe behavior of the spectrum when the scrambling techniqueis applied, we conclude that the Lorentzian pulses observed inthe temperature filament data are responsible for the observedexponential character of the power spectrum.

It has been illustrated how the Bandt–Pompe probabilitycan be used, in conjunction with the CH-plane, to clearlyseparate chaotic from stochastic behavior in well-knowndynamical models that exhibit only one behavior or the other.Furthermore, the effect of ‘time sub-sampling’ or ‘embeddingdelay’ on the location of these systems in the CH-plane hasbeen established. The chaotic systems systematically moveinto the region of moderate entropy and high complexity,while the stochastic systems remain basically unchanged. Thisimplies that sub-sampling can extract chaotic signals thatare contaminated by extraneous noise that is not an intrinsicproperty of the underlying dynamics. When the method isapplied to the data obtained from the heat transport experiment,it is found that the location in the CH-plane moves from aregion on the edge of stochasticity to the middle of the regionoccupied by chaotic systems. This behavior, independently ofthe previous findings about the exponential origin of the power

spectrum, ascertains that the underlying dynamic processes inthe temperature filament experiment are chaotic. Since it hasbeen well established [25–35] in a broad range of systemsthat a signature of deterministic chaos is the generation ofexponential power spectra, the two independently establishedfindings of this study are entirely consistent with previousknowledge common to the fluid turbulence and nonlineardynamics communities. The study contributes yet a furtherexample, from plasma physics, of this connection.

The relatively simple systems considered in this studyhave illustrated the usefulness of combining the Pompe–Bandtprobability with a display in the entropy-complexity plane toidentify chaotic and stochastic behavior. This signal analysistechnique has the potential to uncover new features in a widerange of fusion and basic plasma experiments.

Acknowledgments

The work of JEM and GJM is performed under the auspices ofthe BaPSF at UCLA, which is jointly supported by a DOE-NSFcooperative agreement, and by DOE grant SC0004663.

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