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APS/123-QED

Surrogate Test to Distinguish between Chaotic and Pseudoperiodic Time Series

Xiaodong Luo,∗ Tomomichi Nakamura, and Michael SmallDepartment of Electronic and Information Engineering,

Hong Kong Polytechnic University, Hung Hom, Hong Kong.

(Dated: February 8, 2008)

In this communication a new algorithm is proposed to produce surrogates for pseudoperiodic timeseries. By imposing a few constraints on the noise components of pseudoperiodic data sets, we devisean effective method to generate surrogates. Unlike other algorithms, this method properly copes withpseudoperiodic orbits contaminated with linear colored observational noise. We will demonstrate theability of this algorithm to distinguish chaotic orbits from pseudoperiodic orbits through simulationdata sets from the Rossler system. As an example of application of this algorithm, we will alsoemploy it to investigate a human electrocardiogram (ECG) record.

PACS numbers: 05.45.-a

I. INTRODUCTION

Surrogate tests [1] are examples of Monte Carlo hy-pothesis tests [4]. Taking the surrogate test of nonlin-earity in a time series [1] as an example, we first needto adopt a null hypothesis, which usually supposes thetime series is generated by a linear stochastic processand potentially filtered by a nonlinear filter [5]. Basedon this null hypothesis, a large number of data sets (sur-rogates) are to be produced from the original time series,which keeps the linearity of the original time series butdestroys all other structures. We then calculate somenonlinear statistics (discriminating statistics), for exam-ple, correlation dimension, of both the original time seriesand the surrogates. If the discriminating statistic of theoriginal time series deviates from those of the surrogates,we can reject the null hypothesis we proposed and claimthat the original time series is deterministic with certainconfidence level (depending on how many surrogates wehave generated, to be shown later). In general, to applythe surrogate technique to test if a time series possessesthe property P we are interested, we first select a nullhypothesis, which assumes the time series instead has aproperty Q opposite to P . We then devise a correspond-ing algorithm to produce surrogates from the observeddata set. In principle, these surrogates shall preserve thepotential property Q while destroying all others. Thenext step is to choose a suitable discriminating statistic,which shall be an invariant measure for both the surro-gates and the original time series if the null hypothesisis true. Hence if the discriminating statistic of the origi-nal time series distinctly deviates from the distribution ofthe discriminating statistic of the surrogates, the null hy-pothesis is unlikely to be true, or in other words, the timeseries is much more likely to possess the property P thanQ. In this way, we can assess the statistical significanceof our calculations through surrogate test technique evenwhen we have only a very limited amount of observa-

∗Electronic address: enxdluo@eie.polyu.edu.hk

tions. Such assessments are important because in manypractical situations statistical fluctuations are inevitabledue to the presence of noise, hence the surrogate test isa proper tool to evaluate the reliability of our results ina statistical sense.

In this communication, we are focused on discussingthe algorithm to generate surrogates for pseudoperiodictime series. By pseudoperiodic time series we mean arepresentative of a periodic orbit perturbed by dynami-cal noise, or contaminated by observational noise, or withthe combination of the both noises, whose states withinone cycle are largely independent of those within previouscycles given a cycle length. Note that, in our discussionswe will always assume we have detected that the time se-ries are produced from nonlinear deterministic systems,but they are also possibly contaminated by some noises.As we know, if an irregular time series comes from anonlinear deterministic system, it shall be either chaoticor pseudoperiodic in most cases. In some situations, itmight be important for us to discriminate between pseu-doperiodicity and chaos. However, chaotic and pseudope-riodic time series often look similar, we might not be ableto distinguish them from each other only through visualinspections, quantitative techniques are needed insteadat this time. One choice is to apply the direct test tech-niques. For instance, we can calculate some characteristicstatistics of the time series, such as the Lyapunov expo-nent and the correlation dimension. However, a directtest usually will not give out the confidence level. If wefind the Lyapunov exponent of a time series is, for ex-ample, 0.01, it may be difficult for us to tell whether thetime series is chaotic or the time series is pseudoperiodic,but the presence of noise causes the Lyapunov exponentto be slightly larger than zero. As an alternative choice,we suggest one utilizes the surrogate test rather than thedirect test, which can provide us the confidence level bycalculating a large number of surrogates. Through thesurrogate tests, if we could exclude the possibility thatthe time series is pseudoperiodic, then the time series ismore likely to be chaotic. This is the essential idea toapply our algorithm to distinguish chaos from pseudope-riodicity, as to be shown in section III.

2

First let us briefly review some of the algorithms togenerate surrogates for pseudoperiodic time series. Ini-tially, to generate surrogates for pseudoperiodic time se-ries, Theiler [2] proposed the cycle shuffling algorithm.The idea is to divide the whole data set into some seg-ments and let each segment contain exactly an integernumber of cycles. The surrogates are obtained by ran-domly shuffling these segments, which will preserve theintracycle dynamics but destroy the intercycle ones byrandomizing the temporal sequence of the individual cy-cles. The difficulty in applying this algorithm is thatit requires preknowledge of the precise periodicity, other-wise shuffling the individual cycles might lead to spuriousresults [9].

Recently, with the development of the cyclic theory ofchaos [6], many authors have shown interest in searchingunstable periodic orbits (UPOs) in noisy data sets fromchaotic dynamical systems. The algorithms proposed in[7] essentially deal with the unstable fixed points of theUPOs. But as observed, the presence of noise will re-duce the statistical significance of these algorithms. Oneremedy is to introduce the surrogate test for reliability as-sessments, e.g., Dolan et.al [7] claimed that the randomlyshuffling surrogate algorithm [1] together with the simplerecurrence method [7] correctly tests the appropriate nullhypothesis. Essentially, this approach is very similar tothe cycle shuffling algorithm described previously. Thesimple recurrence algorithm is equivalent to applying aPoincare map on the continuous dynamical systems andthen studying only the data points falling on the cross-section plane, hence one does not need to consider theintracycle dynamics and no knowledge of the periodicityis required, while randomly shuffling these data pointsexactly aims to randomize the temporal sequence of thecycles. However, one potential problem of this algorithmis that it might generate spuriously high statistical sig-nificance due to the correlation between the cycles [8].

Later, Small et.al [9] proposed the pseudoperiodic sur-rogate (PPS) algorithm from another viewpoint. Theyfirst apply the time delay embedding reconstruction [16]to the original data set, then utilize a method basedon local linear modelling techniques to produce surro-gate data which approximate the behavior of the under-lying dynamical system. As the authors pointed out,this algorithm works well even with very large dynami-cal noise, but it may incorrectly reject the null hypothesisif the intercycles of the pseudoperiodic orbit have a lin-ear stochastic dependence induced by colored additiveobservational noise [10].

In this communication we propose a new surrogate al-gorithm for continuous dynamical systems, which prop-erly copes with linear stochastic dependence between thecycles of the pseudoperiodic orbits. The null hypothesisto be tested is that the stationary data set is pseudope-riodic with noise components which are (approximately)identically distributed and uncorrelated for sufficientlylarge temporal translations. Note the constraints of thenoise components in our null hypothesis are stronger than

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FIG. 1: (a) Pseudoperiodic time series contaminated by obser-vational noise; (b) State space xi+n vs. xi of the pseudoperiodictime series from the Rossler system with n = 16; (c) Surrogatetest for the pseudoperiodic time series based on our algorithm.The abscissa is the indices of 100 surrogates and the ordinateis the corresponding correlation dimensions. The middle lineis the mean correlation dimension of the original time seriescalculated 100 times using the GKA, the upper and lower linesdenote the correlation dimensions twice the standard deviationaway from the mean value and the asterisks indicate the corre-lation dimensions of 100 surrogates.

that of Theiler’s algorithm, which requires the noise dis-tribution only periodically depends on the phase of thesignal. However, under our hypothesis, we can producethe surrogates in a simple way through the algorithm tobe described below. In addition, a large scope of noiseprocesses often encountered in practical situations, in-cluding (but not limited to) linear colored additive obser-vational noise described by the ARMA(p, q) model [15],match the above constraints.

The remainder of this communication is organized asfollows. In Sec. II we will introduce the new algorithm togenerate pseudoperiodic surrogates, while in Sec. III wewill apply this algorithm to simulation data sets from theRossler system, which demonstrates the ability of the sur-rogate test based on this algorithm to distinguish chaoticorbits from pseudoperiodic ones. As one of the applica-tions, we will use this surrogate technique to investigatewhether a human electrocardiogram (ECG) record is pos-sibly presentative of a chaotic dynamical system. Finally,in Sec. IV, we will have a summary of the whole com-munication.

II. A NEW ALGORITHM TO GENERATE

PSEUDOPERIODIC SURROGATES

Let {xi}Ni=1

be a data set with N observations (theform {xi} is adopted instead for convenience when caus-ing no confusion), where xi means the observation mea-sured at time ti = i · ∆ts with ∆ts denoting the sam-

pling time. We assume {xi}Ni=1

is stationary and can bedecomposed into the deterministic components and thenoise components, which are approximately independentof each other. Similar to the surrogate test idea of timeshifting to desynchronize two data sets [12], we assumethe noise components (approximately) follow an identi-cal distribution and are uncorrelated for sufficiently largetemporal translations (or time shifts). According to the

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FIG. 2: (a) Pseudoperiodic time series with both observationalnoise and dynamical noise; (b) State space xi+n vs. xi of thepseudoperiodic time series from the Rossler system with n = 16;(c) Surrogate test for the pseudoperiodic time series based onour algorithm. The meaning of the lines and the asterisks isthe same as that in panel (c) of Fig. 1.

null hypothesis we proposed in the previous section, if thedeterministic components are periodic, then we can writea data point xi as xi = pi + ni, where pi and ni denotethe periodic component and the noise component respec-tively. In many cases, we can set E(pi) = E(ni) = 0where E is the expectation operator. Since {pi} areroughly independent of {ni}, we have the autocovariancevar(xi) = var(pi) + var(ni). Let

yτi = αxi +βxi+τ = (αpi +βpi+τ )+(αni +βni+τ ) (1)

with i = 1, 2, ..., N − τ , where coefficients α and βsatisfy α2 + β2 = 1 and parameter τ is the temporal

translation between subsets {xi}N−τi=1

and {xi+τ}N−τi=1

,then the autocovariance function var(yτ

i ) = var(αpi +βpi+τ ) + var(αni + βni+τ ). Now let us consider thenoise components. If τ is sufficiently large, under ourhypothesis, ni and ni+τ are uncorrelated. We also notethat {ni} and {ni+τ} are drawn from (approximately)the same distribution, we have var(αni + βni+τ ) =var(ni). For the deterministic component, if we re-quire the translation τ to satisfy cov(pi, pi+τ ) = 0, thenvar(αpi+βpi+τ ) = var(pi). Hence by choosing a suitabletemporal translation, the noise levels of {yτ

i }, defined by

(var(αni + βni+τ )/var(yτi ))

1/2, will be the same as that

of {xi}Ni=1

, i.e., (var(ni)/var(xi))1/2

. The reason to pre-serve the noise level is that, the presence of noise willaffect the calculation of the correlation dimension, hencewe would like to let the surrogates and the original timeseries (roughly) have the same noise level in order to makethe results more conceivable.

The above deduction leads to the central idea of oursurrogate algorithm. From Eq. (1), we note that if{pi} is periodic, the nonconstant deterministic compo-nents {αpi + βpi+τ} shall also be periodic. In addition,

{xi}Ni=1

and {yτi } shall have the same noise level if a

suitable translation τ is selected. Therefore by random-izing the coefficient α or β, we can generate many data

sets {yτi } as the surrogates of {xi}N

i=1. Note that {pi}

and {αpi + βpi+τ} have the same degree-of-freedom, ifboth of them are periodic, their correlation dimensions[13] will theoretically be the same. Now let us consider

the noise components. Although the noise components{αni + βni+τ} may have a different distribution fromthat of {ni}, the noise level is preserved after the trans-form in Eq. (1). As Diks [20] has reported, the Gaussiankernel algorithm (GKA) can reasonably estimate the cor-relation dimensions of noisy data sets with different noisedistributions. This implies that, under the same noise

level, the correlation dimensions of {xi}Ni=1

and {yτi },

calculated by the GKA, shall statistically be the same

if {xi}Ni=1

and {yτi } are both pseudoperiodic (and sat-

isfy the constraints we imposed). In contrast, if {pi} ischaotic, its linear combination, {αpi + βpi+τ}, may havea new dynamical structure with a different correlation di-mension from that of {pi}, hence by adopting the corre-lation dimension as the discriminating statistic we mightdetect this difference.

We shall also note that, for an unstable periodic orbit,even a small dynamical noise might drive the resultant or-bit far away from the original position after a sufficientlylong time, and the pseudoperiodicity might be broken. Insuch situations, our algorithm might fail to work. Nev-ertheless, we suggest to apply our algorithm as the firststep in pseudoperiodicity test, which is computationallyfast and in principle deals well with a large scope of ob-servational noise (comparatively, the PPS algorithm willsometimes fail for colored observational noise). If we canreject the null hypothesis proposed previously, the timeseries in test is possibly chaotic or pseudoperiodic per-turbed by dynamical noise. Then we can adopt the PPSalgorithm for further tests, which works well even undera large amount of dynamical noise. If the correspondingnull hypothesis, i.e., the time series is pseudoperiodic per-turbed by dynamical noise, can be rejected again, thenwe may claim the time series is very likely to be chaotic.

We now consider several computational issues in our al-gorithms. As described in Eq. (1), to generate the surro-

gates {yτi }, we select two subsets of {xi}N

i=1, {xi}N−τ

i=1and

{xi+τ}N−τi=1

, multiply them by the coefficients α and β re-spectively and then add them together. We shall empha-size that choosing the temporal translation τ is a crucialissue for our algorithm. From one aspect, we require thetranslation τ to satisfy the condition cov(pi, pi+τ ) = 0.The reason is that we want to keep the noise level forthe original time series and the surrogates. In addition,we want the deterministic components {αpi} to be or-thogonal to {βpi+τ} for arbitrary coefficients α and β,otherwise the projection of {αpi} onto {βpi+τ} mightcounteract {βpi+τ} under some situations, for example,if pi ≈ −pi+τ and α = β, the deterministic components{αpi + βpi+τ} will almost vanish while the noise compo-nents {αni + βni+τ} remain. Hence the correlation di-mensions calculated are actually those of the noise com-ponents instead of the deterministic components, whichwill certainly cause the false rejection of the null hy-pothesis. From another aspect, we require τ to be suffi-ciently large to guarantee the decorrelation between the

noise components. However, we expect {xi}N−τi=1

and

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FIG. 3: (a) Chaotic time series contaminated by observationalnoise ; (b) State space xi+n vs. xi of the chaotic time seriesfrom the Rossler system with n = 16; (c) Surrogate test for thechaotic time series based on our new algorithm. The meaningof the lines and the curve is the same as that in panel (c) ofFig. 1; (d) Surrogate test for the chaotic time series based onthe PPS algorithm. The meaning of the lines and the asterisksis the same as that in panel (c) of Fig. 1.

{xi+τ}N−τi=1

shall have at least some overlaps to make use

of the information of the whole data set {xi}Ni=1

, whichmeans τ shall not exceed N/2. In addition, it is recom-mended the length of a data set shall not be too short inorder to appropriately calculate its correlation dimension[14], which also implies τ shall not be too large.

From Eq. (1) we see that the surrogates are gen-

erated from two segments {xi}N−τi=1

and {xi+τ}N−τi=1

ofthe original time series {xi}N

i=1. We want segments

{xi}N−τi=1

and {xi+τ}N−τi=1

to equivalently affect the gen-eration of the surrogates, therefore we would like to letmax(|α/β|) = max(|β/α|), min(|α/β|) = min(|β/α|) andPr(|α/β| > 1) ≃ Pr(|β/α| > 1), where max(·), min(·)and Pr(·) denote the maximal function, the minimal func-tion and the probability function respectively. But notethat the coefficient ratio α/β (or β/α) shall not be toolarge or too small, otherwise {yτ

i } will be very close

to {xi}N−τi=1

or {xi+τ}N−τi=1

, which will lead to approxi-

mately the same correlation dimensions of {xi}Ni=1

and

{yτi } regardless of the dynamical behavior of {xi}N

i=1,

and thus decrease the discriminating power of the cor-relation dimension. In our calculations we let α be uni-formly drawn from the interval [−0.8,−0.6] ∪ [0.6, 0.8]

and β =√

1 − α2, which satisfies our requirements andprovides moderate values for the ratio α/β.

III. SURROGATE TEST TO DISTINGUISH

BETWEEN CHAOTIC AND PSEUDOPERIODIC

TIME SERIES

In this section, through four examples from the Rosslersystem, we demonstrate the ability of surrogate testbased on our algorithm to discriminate chaotic orbitsfrom pseudoperiodic ones. As an application, we will alsoemploy the surrogate technique to investigate whethera recorded human electrocardiogram (ECG) data set ispossibly chaotic.

A. EXAMPLES

The equations of the Rossler system are given by

x = −y − z,y = x + ay,z = b + z(x − c).

(2)

with the initial conditions x(0) = y(0) = z(0) = 0.1. Wechoose parameters b = 2, c = 4 and the sampling time∆ts = 0.1 time units. For each example, the system isto be integrated 10, 000 times and the first 1, 000 datapoints are discarded to avoid including transient states.

In the first example, we set parameter a = 0.39095.The Rossler system exhibits limit cycle behavior of pe-riod 6. To obtain pseudoperiodic time series, we intro-duce 5% observational noise into the periodic time series.Although Gaussian white observational noise is the mostcommon choice in this situation, in order to demonstratethe ability of our surrogate algorithm to deal with col-ored noise, we will instead adopt the noise generated fromthe AR(1) process [15] ξi+1 = 0.8ξi + ǫi with the vari-able ǫ following the normal Gaussian distribution N(0, 1),which is the more difficult case due to the correlation be-tween noise components. However, one shall note that,Gaussian white noise and other colored noises satisfyingthe constraints in our null hypothesis, for example, thosemodelled by the ARMA(p, q) processes, in principle canbe dealt with in the same way. For convenience of obser-vation and comparison, we plot the time series and thecorresponding attractor in two dimensional state space(or embedding space) in panels (a) and (b) of Fig. 1respectively.

To produce surrogate data, first we shall choose a suit-able temporal translation. Since it is impractical to sep-arate noise from signal completely, in general it is diffi-cult to estimate the correlation decay time between noisecomponents. Fortunately, to decorrelate noise compo-nents, all temporal translations are equivalent as long asthey are large enough. In addition, in many real sit-uations, it is often possible to observe the backgroundnoise and thus estimate the decay time. In our exam-ple, we think the AR(1) noise to be uncorrelated whenthe temporal translation is larger than 50 (in units ofthe sampling time ∆ts). As another requirement, tem-poral translation satisfying cov(pi, pi+τ ) = 0 is desired.

5

In practice, of course, this requirement is generally im-practical due to the digitization and quantization in sam-pling process. Recall the discussion in the previous sec-tion, by letting E(pi) = 0 and α2 + β2 = 1, we havevar(αpi + βpi+τ ) = var(pi) + 2αβ · cov(pi, pi+τ ). Func-tion cov(pi, pi+τ ) 6= 0 means we do not preserve the noiselevel. However, under the null hypothesis of pseudope-riodicity, there shall always be some temporal transla-tions to make cov(pi, pi+τ ) ∼ 0, hence the noise levelwill not deviate from the original one too much. Be-sides, according to Eq. (1), we generate the surro-gates by uniformly drawing coefficient α from interval[−0.8,−0.6]∪ [0.6, 0.8] (β =

√1 − α2 is always kept pos-

itive), the noise level of the surrogates will fluctuatearound that of the original one due to the alternativesigns of product αβ. Therefore, cov(pi, pi+τ ) 6= 0 willonly cause some fluctuations when to calculate the cor-relation dimension because of the fluctuations of noiselevel, however, generally such fluctuations will not affectour conclusion if we can select a temporal translationτ to let cov(pi, pi+τ ) ∼ 0. Since we have assumed thenoise components are roughly independent of the deter-ministic components, then cov(xi, xi+τ ) = cov(pi, pi+τ )for a large enough temporal translation (to decorrelatenoise components), therefore in all of the examples, inorder to let cov(pi, pi+τ ) ∼ 0, we can equivalently requirecov(xi, xi+τ ) ∼ 0. In the first example, there are manytemporal translations satisfying the two constraints dis-cussed above, i.e., τ > 50 and cov(xi, xi+τ ) ∼ 0. Topick a value from all these candidates, we first selectan interval [100, 150], then search the temporal trans-lation which makes the absolute value |cov(xi, xi+τ )| bethe minimum (most close to zero) among all translations100 6 τ 6 150. One shall note that the choice of theinterval [100, 150] is arbitrary, except that we have tomake sure that the lower bound of the interval is largerthan 50, and there exists temporal translations to letcov(xi, xi+τ ) ∼ 0 within the interval. After selecting thetemporal translation, by randomizing the coefficient α wewill generate 100 surrogates according to Eq. (1).

In order to calculate the correlation dimension, weadopt the time delay embedding reconstruction [16] torecover the underlying system from the scalar time se-ries. Two parameters, i.e., embedding dimension andtime delay, shall be properly chosen to apply this tech-nique. Throughout this communication, we will use thefalse nearest neighbour criterion [17] to determine theglobal optimal embedding dimension. Using the programin TISEAN package [18], the embedding dimension m ofthe original time series is selected at 4, which is the mini-mal value to make the fraction of false nearest neighboursbe zero. To choose a suitable time delay, we will use thealgorithm of redundancy and irrelevance tradeoff expo-nent (RITE) proposed in [22]. This algorithm selects thetime delay by searching the optimal tradeoff between re-dundancy (due to too small time delay) and irrelevance(due to too large time delay). As demonstrated, theRITE algorithm can select equivalently suitable time de-

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FIG. 4: (a) Chaotic time series with both dynamical and ob-servational noises; (b) State space xi+n vs. xi of the chaotictime series from the Rossler system with n = 16; (c) Surrogatetest for the chaotic time series based on our new algorithm.The meaning of the lines and the asterisks is the same as thatin panel (c) of Fig. 1; (d) Surrogate test for the chaotic timeseries based on the PPS algorithm. The meaning of the linesand the asterisks is the same as that in panel (c) of Fig. 1.

lays compared to the average mutual information (AMI)criterion [19], however, its implementation is much sim-pler and the computational cost is fairly low. Thereforein case of large data sets, adopting the RITE algorithmcan facilitate our calculations. In the first example wegenerate 100 surrogates, and for each surrogate we keepthe embedding dimension m = 4 and use the RITE al-gorithm to choose the suitable time delay for time delayreconstruction.

We will follow Diks’s method [20] to calculate the cor-relation dimension, which is more robust against noiseby extending the hard kernel function (or the Heavi-side function) [13] in calculation of correlation integralto the general kernel functions. In his discussions, Diksadopted the Gaussian kernel function, hence this methodis called Gaussian kernel algorithm (GKA). Here we willuse the GKA implemented in [21] to calculate the corre-lation dimensions, which further enhances the computa-tional speed. Note that to speed up the calculation, only2000 data points are used as the reference points for theGKA. There are some statistical fluctuations even for thesame data set when calculating its correlation dimension,therefore for the original time series, we will calculate 100times to estimate the mean correlation dimension and thestandard deviation. As shown in panel (c) of Fig. 1, thereare three lines parallel to the abscissa. The middle linedenote the estimation of the mean correlation dimensionof the original time series, while the upper and lower linesindicate the positions twice the standard deviation away

6

from the mean value. For the surrogates, however, wewill calculate their correlation dimensions only once tosave time. The results are illustrated as the asterisks inpanel (c) of Fig. 1.

After the calculation of the correlation dimensions, weneed to inspect whether the result is consistent with ournull hypothesis. Here we use the ranking criterion [3]to determine whether the null hypothesis shall be re-jected or not. The idea of this criterion is that, sup-pose the discriminating statistic of the original data setis Q0, and those of NS surrogates are {Q1, Q2, ..., QNS

}.Rank the statistics {Q0, Q1, ..., QNS

} in the increasingorder and denote the rank of Q0 by r0, if the data setis consistent with the hypothesis (i.e., no evidence toreject), r0 can have an equal possibility be any integervalue between 1 and NS + 1. However, if the hypothesisis false, Q0 might deviate from the surrogate distribu-tion {Q1, Q2, ..., QNS

}, i.e, Q0 will be the smallest orlargest amongst {Q0, Q1, ..., QNS

}, hence we can rejectthe null hypothesis if r0 = 1 or NS + 1, the probabilityof a false rejection is 1/ (NS + 1) for one-sided tests and2/ (NS + 1) for two-sided tests.

For the first example, from panel (c) of Fig. 1 we cansee that, the mean correlation dimension of the originaltime series falls within the dimension distribution of thesurrogates, therefore we cannot reject the null hypothe-sis as we expect, which means the original time series ispossibly pseudoperiodic [23].

Now let us examine the other examples. In the secondexample, we still set parameter a = 0.39095 in Eq. (2).However, to obtain the pseudoperiodic time series, wefirst generate a data set by adding Gaussian white noisewith the standard deviation of 0.15% to the x compo-nent at each integration step, which simulates the systemperturbed by additive dynamical noise, and then intro-duce 5% observational AR(1) noise into the previouslyobtained data set. The global optimal embedding dimen-sion is chosen at m = 4. Note in all of the four examples,we will generate 100 surrogates, and parameter choicesfor surrogate generation will be the same, i.e., we let thetemporal translation be selected from [100, 150] and coef-ficient α be uniformly drawn from [−0.8,−0.6]∪ [0.6, 0.8]

(β =√

1 − α2). For the second example, the correlationdimensions of the original time series and the surrogatesare shown in panel (c) of Fig. 2. Under the rankingcriterion, once again we cannot reject our null hypothe-sis. Therefore the time series is possibly pseudoperiodic,which is consistent with our knowledge.

In the third example, we change parameter a of Eq.(2) to be 0.395. The Rossler system exhibits chaotic be-havior. We integrate Eq. (2) to obtain a time seriesand then introduce 5% observational AR(1) noise. Theoptimal embedding dimension m is selected at m = 5.From panel (c) of Fig. 3, we find that the mean correla-tion dimension of the original time series deviates fromthe distribution of the surrogate dimensions. Using theranking criterion, we can reject our null hypothesis. Inorder to exclude the possibility that the time series is

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0 20 40 60 80 1001.01.11.21.31.41.51.61.71.8

Indices of 100 generated surrogates

Cor

rela

tion

dim

ensi

on

(d) Surrogate test results

0 2000 4000 6000 8000−2000

0

2000

4000

6000

8000

i

x i

(c) One generated surrogate

FIG. 5: (a) Time series of a human electrocardiogram (ECG)record; (b) State space xi+n vs. xi of the ECG record withn = 23; (c) A surrogate data generated from our algorithmwith coefficient α = β = 1/

√

2 (cf. Eq. (1)); (d) Surrogate testfor the ECG record based on our algorithm. The meaning ofthe lines and the asterisks is the same as that in panel (c) ofFig. 1.

generated from an unstable period orbit perturbed bydynamical noise, we also apply the PPS algorithm forfurther test. From the PPS algorithm we generate 100surrogates, and then use the GKA to calculate their cor-relation dimensions. The results are shown in panel (d) ofFig. 3, as we can see, the mean correlation dimension ofthe original time series also falls outside the distributionof the surrogate dimensions, therefore we can reject thenull hypothesis again. After the two surrogate tests forpseudoperiodicity, we can claim the time series is chaoticwith a confidence level up to 96% (98% × 98%) for two-sided test.

The final example to be demonstrated is a chaotic timeseries also from the Rossler system. To generate the timeseries, we keep parameter a = 0.395. Similar to the wayin the second example, we add Gaussian white noise withthe standard deviation of 0.15% to the x component ateach integration step as the dynamical noise, and thenintroduce 5% observational AR(1) noise into the previ-ously obtained data set. The global optimal embeddingdimension is found to be m = 4. The results of surrogatetests based on the new algorithm and the PPS algorithmare shown in panel (c) and (d) of Fig. 4 respectively, fromwhich we can see that, surrogate tests based on both al-gorithms can detect the chaos in the time series. Againwe can claim the time series is chaotic with a confidencelevel up to 96% for two-sided test.

We have also investigated examples under different ob-servational noise levels (but keep the same dynamicalnoise if they have). For example, if we reduce the AR(1)observational noise levels to 3% in the above four exam-ples, we can obtain the same results as we have reported.

7

If we increase the observational noise levels to 10%, forthe pseudoperiodic time series we can still obtain the ex-pected results, i.e., we cannot reject our null hypothesis.However, for the chaotic time series, we will falsely ac-cept our null hypothesis due to the correlation dimensionof the original time series marginally falling within thedimension distribution of the surrogates. The reason offalse acceptance might be that, under large noise levels,the correlation dimension is not sensitive enough to de-tect the structure changes of the chaotic time series. Forsuch cases, we will have to look for more powerful dis-criminating statistics [24].

B. AN APPLICATION

As an example of application, we employ the surrogatetest based on our algorithm to investigate whether a hu-man electrocardiogram (ECG) record (with 8975 datapoints) is likely to be chaotic. The ECG record wasobtained by measuring from a resting healthy subject(22 years old) in a shielded room at the sampling rateof 1 KHz. The ECG record indicated in panel (a) ofFig. 5 looks very regular and even possibly periodic,but we need a quantitative approach to verify the pe-riodicity. Here we choose the surrogate test technique.Using the false nearest neighbour criterion, the globaloptimal embedding dimension is chosen at m = 5. Thebackground noise is mainly from the measurement in-struments, usually it is a blend of white and correlatednoise. By observing the linear second order correlationfunction of the ECG data, we let the temporal translationbe within the interval [100, 150] (large enough to decor-relate the noise components), where there exists an in-teger temporal translation to make the correlation func-tion almost be zero. Then by uniformly drawing valuesfrom [−0.8,−0.6] ∪ [0.6, 0.8] for coefficient α in Eq. (1)

(β =√

1 − α2 ), we generate 100 surrogates. For demon-stration, we plot in panel (c) one surrogate generated

from Eq. (1) with coefficient α = β = 1/√

2. We can seethat the surrogate is different from the original ECG datain that there appear more spikes in the surrogate. How-ever, as we can also find, the surrogate indicates the sim-ilar regularity to that in the original data, which impliesthat the surrogate preserves the potential periodicity inthe original data as we expect (although in a differentpattern). With regards to the surrogate test, our cal-culation of the correlation dimensions is also based onthe GKA. The results are indicated in panel (d) of Fig.5, from which we can see that the mean correlation di-mension of the ECG data falls within the distribution of

the correlation dimensions of the surrogates, therefore wecannot reject our null hypothesis. Hence the ECG recordis possibly periodic. Moreover, there is no evidence ofchaos.

IV. CONCLUSION

To summarize, by imposing a few constraints on thenoise process, we devise a simple but effective way toproduce surrogates for pseudoperiodic orbits. The mainidea of this algorithm is that a linear combination of anytwo segments of the same periodic orbit will generate an-other periodic orbit. By properly choosing the temporaltranslation between the two segments, under the samenoise level we can obtain statistically the same correla-tion dimensions of the pseudoperiodic orbit and its sur-rogates. Choosing the temporal translation is a crucialissue for our algorithm, which in principle shall guaran-tee the decorrelation between the noise components andpreserve the noise level. Another important issue is toselect a proper discriminating statistic which helps de-termine whether to reject the null hypothesis. The cor-relation dimension is a suitable discriminating statistic inthis case. It is possible there are other suitable discrimi-nating statistics, we will leave the problem of finding suchstatistics for future study.

The surrogate test technique based on our algorithmcan be utilized to distinguish between chaotic and pseu-doperiodic time series. Initially, the PPS algorithm wasproposed to generate surrogates for a pseudoperiodic or-bit driven by dynamical noise, but sometimes surrogatetests based on this algorithm will falsely reject the nullhypothesis if the time series is also contaminated by col-ored observational noise. As a complement to the PPSalgorithm, our algorithm deals well with observationalnoise, but it might fail for large dynamical noise. How-ever, due to the convenience in computation, we suggestto adopt surrogate test based on our algorithm as thefirst step for pseudoperiodicity detection. If we can re-ject the null hypothesis of our algorithm, then we shalluse the PPS algorithm for further tests. If we can re-ject the null hypotheses of both the algorithms, then thetime series under investigation is very likely to be chaotic.In this communication, the concrete procedures of surro-gate test for pseudoperiodicity are demonstrated throughfour simulation examples. As an application in practice,we also employ the surrogate technique based on our al-gorithm to investigate whether a human ECG record ispossible to be chaotic.

This research is supported by a Hong Kong UniversityGrants Council Competitive Earmarked Research Grant(CERG) number PolyU 5235/03E.

[1] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J.D.Farmer, Physica D 58, 77 (1992).

[2] J. Theiler, Phys. Lett. A 196, 335 (1995).

[3] J. Theiler and D. Prichard, Fields Inst. Commun. 11, 99(1997).

[4] A. Galka, Topics in Nonlinear Time Series Analysis with

8

Implications for EEG Analysis (World Scientific, 2000).[5] Here we would like to elucidate that, the irregularity of a

time series is usually brought by stochasticity or nonlinear-ity (often chaos), therefore in the example of nonlinearitydetection, if we can exclude (most of) the probability thatthe time series is generated by a stochastic process, thenit is very likely that the time series is generated from anonlinear deterministic system.

[6] D. Auerbach, P. Cvitanovic, J.P. Eckmann, and G. Gu-naratne, Phys. Rev. Lett. 58, 2387 (1987); P. Cvitanovic,Phys. Rev. Lett. 61, 2729 (1988).

[7] D. Pierson and F. Moss, Phys. Rev. Lett. 75, 2124 (1995);X. Pei and F. Moss, Nature (Lodon) 379, 618 (1996); P.So, E. Ott, S.J. Schiff, D.T. Kaplan, T. Sauer, and C.Grebogi, Phys. Rev. Lett. 76, 4705 (1996); P. Schmelcherand F.K. Diakonos, Phys. Rev. Lett. 78, 4733 (1997); K.Dolan, A. Witt, M.L. Spano, A. Neiman, and F. Moss,Phys. Rev. E 59, 5235 (1999).

[8] D. Petracchi, Chaos, Solitons & Fractals 8, 327 (1997);[9] M. Small, D. Yu, and R.G. Harrison, Phys. Rev. Lett. 87,

188101 (2001).[10] For the definitions of the terminologies such as additive

observational noise, please refer to [11] for more details.[11] J. Argyris, I. Andreadis, G. Pavlos, and M. Athanasiou,

Chaos, Solitons & Fractals 9, 343 (1998).[12] R. Quian Quiroga, A. Kraskov, T. Kreuz, and P. Grass-

berger, Phys. Rev. E 65, 041903 (2002).[13] P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346

(1983).[14] A. Jedynak, M. Bach, and J. Timmer, Phys. Rev. E 50,

1770 (1994).[15] G.E.P. Box, G.M. Jenkins, and G.C. Reinsel, Time Series

Analysis: Forecasting and Control, 3rd Ed. (Prentice-Hall,1994).

[16] F. Takens, in D. Rand and L.S. Young, editors, Dynam-

ical Systems and Turbulence (Springer, 1981).[17] M. B. Kennel, R. Brown, and H. D. I. Abarbanel, Phys.

Rev. A 45, 3403 (1992).[18] R. Hegger, H. Kantz, and T. Schreiber, CHAOS 9, 413

(1999).[19] A. M. Fraser and H. L. Wwinney, Phys. Rev. A 33, 1134

(1986).[20] C. Diks, Phys. Rev. E 53, 4263 (1996).[21] D.J. Yu, M. Small, R.G. Harrison, and C. Diks, Phys.

Rev. E 61, 3750 (2000).[22] X. Luo and M. Small (submitted, available from the

URL: http://arxiv.org/abs/nlin.CD/0312023/).[23] We cannot claim the time series is pseudoperiodic defi-

nitely, because if we cannot reject a null hypothesis, therecould be many interpretations other than that the data intest is consistent with our null hypothsis. For more details,see, for example, Ref. [4].

[24] T. Schreiber and A. Schmitz, Phys. Rev. E 55, 5443(1997).