Ordinal spectrum: a frequency domain characterization of complex time series
OOrdinal spectrum: a frequency domain characterization of complex time series
Mario Chavez ∗ CNRS UMR-7225, Hˆopital de la Piti´e-Salpˆetri`ere. Paris, France
Johann H. Mart´ınez
Grupo Interdisciplinar de Sistemas Complejos (GISC). Madrid, Spain (Dated: September 8, 2020)Although classical spectral analysis is a natural approach to characterise linear systems, it cannotdescribe a chaotic dynamics. Here, we propose the ordinal spectrum , a method based on a spectraltransformation of symbolic sequences, to characterise the complexity of a time series. In contrastswith other nonlinear mapping functions (e.g. the state-space reconstruction) the proposed represen-tation is a natural approach to distinguish, in a frequency domain, a chaotic behavior. We test themethod in different synthetic and real-world data. Our results suggest that the proposed approachmay provide new insights into the non-linear oscillations observed in different real data.
Observed time series from a large number of physi-cal processes generally display erratic temporal behav-ior. In the last decades, various measures of complex-ity have been proposed to characterize these data anddistinguish regular (e.g., periodic), chaotic, and randomdynamics [1]. The spectral representation based on theFourier transform could be a natural approach to identifythe rich oscillatory dynamics in terms of frequency, mag-nitude and phase. Indeed, changes in power spectra ofsome dynamical systems as they bifurcate to chaos havebeen described [2–4]. Although several measures wereproposed to characterize these spectral changes [2, 5–7],a close relation between these measures and the attractortopology is unclear and it is not supported by numericalsimulations [2, 7].It is generally accepted that time series observed fromchaotic systems exhibit some characteristic signaturessuch as fractal geometry and broadband frequency con-tent. Indeed, time series of chaotic trajectories often dis-play an exponential decay in their power spectrum athigh frequencies, different from the classical power-law ofstochastic colored noises [5, 8, 9]. However, some studieshave proved that the spectra of colored noises cannot berelated to the dynamical route to chaos [10, 11]. In fact,the power spectrum estimated from chaotic sequencescan be replicated by a monotonic nonlinear transforma-tion of linearly filtered noise [12–14]. All this evidencesuggests that classical spectral analysis cannot provideadequate information for identifying chaotic systems.Although classical spectral analysis is adequate forcharacterizing linear systems, power spectra cannot re-flect the non-linear interactions between the Fourier com-ponents of a chaotic motion [3, 15, 16]. Bispectral tech-niques has been used to investigate such nonlinear inter-actions [15, 16], but some works have proved that higher-order statistics are required for a better characterizationof a chaotic dynamics [16, 17]. Recently, other nonlin-ear spectral methods have been proposed to distinguishdeterministic from stochastic dynamics in finite time se-ries. The so-called symbol spectrum test developed in Refs. [18, 19] does not take into account the temporaldynamics of the symbols, but the variability of their dis-tribution in the symbolic sequence. The spectrum pro-posed in Ref. [20] characterises, in the frequency domain,the recurrence of a reconstructed trajectory in the phasespace.Based on the concept of state-space reconstruction,some measures such as entropies, Lyapunov exponentsand fractal dimensions were shown to be effective to char-acterize and reconstruct the equations of motion whenthe data model has a deterministic dynamics [1, 21].Alternative nonlinear mapping functions were proposedto better capture the disorder degree of a time seriesthrough symbolization procedures [22, 23]. The methodknown as ordinal patterns (OP) is a transformation basedon order relations among values of the data, and it pro-vides a robust estimation of the probability distributionfunction associated with the time series [24]. This repre-sentation of the ordinal structure has provided a robusttool to discriminate, in the time domain, different dy-namical regimes in time series [25, 26]. Coarse-grainingapproaches [27, 28], or the use of ordinal structures fordifferent time delays [29] have been proposed to charac-terize complexity at different temporal scales.In this Letter we present the ordinal spectrum of timeseries. The method is based on the spectral trans-formation of a symbolic representation of data. Themethod is based on the ordinal patterns and it is there-fore fully data-driven. In contrast with other nonlinearapproaches, the spectral analysis proposed here providesa characterization of the data’s complexity in a frequencydomain. We assess the reliability of our method in dis-tinguishing periodic or random time series from chaoticdata. We evaluate the proposed method in different syn-thetic and real, linear, non-linear, stochastic and deter-ministic time series. Results depict a robust approach toidentify a chaotic dynamics in data.The main steps to estimate the ordinal spectrum froma time series X t are the following: a r X i v : . [ phy s i c s . d a t a - a n ] S e p i) Ordinal pattern representation of data . Symbolisa-tion procedures map a time series X t onto a discretizedsymbols sequence by extracting its amplitudes’ informa-tion [30]. Among several symbolisation proposals [31], weconsidered here the dynamical transformation of OP [24].This method maps a time series X t with t = 1 , . . . , T to a finite number of patterns that encode the rela-tive amplitudes observed in the D -dimensional vectors X t = { X t , X t + τ , . . . , X t +( D − τ } . The elements of thevector X t are mapped uniquely onto the permutation π = ( π , π , . . . , π D − ) of (0 , , . . . , D −
1) that fulfills X t + π τ (cid:54) X t + π τ , (cid:54) . . . (cid:54) X t + π D − τ .The set of all possible ordinal patterns derived froma time series, that represents the whole embedding statespace, is noted as Z , whose cardinality is D ! at most. Thewhole sequence of OP extracted from X t is known as thesymbolic representation S t of the time series. The higherthe order is, the more information is captured from thetime series. To sample the empirical distribution of ordi-nal patterns densely enough for a reliable estimation ofits probability distribution we follow the condition [30] T (cid:62) ( D + 1)!. The OP symbolisation has some practicaladvantages [25, 26]: a) it is computationally efficient, b) it is fully data-driven with no further assumptionsabout the data range to find appropriate partitions, c) it is invariant to any monotonous transformations and, d) a small D is generally useful in descriptive data anal-ysis [24, 30]. Furthermore, this representation is knownto be relatively robust against noise, and useful for timeseries with weak stationarity [25, 30, 32, 33].Contrary to phase state reconstruction, in ordinaltime-series analysis, the criteria to select an embeddingdimension D are computational cost and statistical sig-nificance in view of the amount of data available. Theselection of time delay embedding τ may, however, influ-ence the analysis of correlated data. Here, to minimizethe effects of this correlation we select the delay that cor-responds to either the first minimum or the zero crossingof the autocorrelation function ρ of the original time se-ries X t (the folding time ρ = 1 /e is used in case of amonotonic function ρ ). ii) Capturing information dynamics from the symbolicsequence . To characterize the time evolution of the ob-tained sequence S t we consider it to be a homogeneousergodic Markov chain with the finite state space Z = z , . . . , z n (all the possible distinct permutations). Let P m = { p mij } be the transition matrix that describes theprobability of leaving the symbol z i and entering the sym-bol z j at a distance m , i.e. p mij = p ( S t + m = z j | S t = z i ).If the chain is stationary we have p ( S t = z i ) = p ∗ i , where p ∗ i is the invariant or stationary distribution that satisfies p ∗ j = (cid:80) i p ∗ i p ij iii) Characterization of symbolic dynamics at differenttime lags . For a first-order Markov chain, any symbol insequence S t is independent of all the previous observa-tions. However, by construction, each ordinal pattern in S t depends on its predecessors, inducing thus a non-zerocorrelation between symbols S t and S t + m for m >
1. Thecorrelation between S t and S t + m , is termed the autoco-variance at lag m and, for a Markov chain that convergesto a unique stationary distribution, it is expected to de-crease as the lag is increased [34, 35].We notice that, in contrast with numeric signals, sym-bolic sets have no mathematical structure and algebraicoperations are usually meaningful. To solve this, dif-ferent rules have been proposed for mapping a symbolicsequence into a numerical domain [36, 37]. The numeri-cal algorithm used in the ordinal pattern transformationalso yields an enumeration of permutations, such thateach unique ordinal pattern can be associated to a non-negative integer, z i = i with i ∈ { , . . . , D ! } [38, 39]. In-terestingly, this enumeration procedure yields a naturalorder of the symbols such that the pattern with permu-tation representation (1 , . . . , D ) and that with represen-tation ( D, . . . ,
1) are at the maximal possible distancesince they represent completely opposite monotonic be-haviors [39].On the basis of this representation with ordered sym-bols or patterns, a rank variance and rank autocovarianceof the Markov chain can be obtained as follows [34, 35]:Var( S ) = (cid:88) i i p ( S t = z i ) − E { S t } Cov( m ) = Cov( S t , S t + m )= (cid:88) i (cid:88) j ijp ( S t = z i , S t + m = z j ) − E { S t } , (1)where E { S t } = (cid:80) i ip ( S t = z i ). We can also write p ( S t = z i , S t + m = z j ) = p ∗ i p mij , wich yields:Cov( m ) = (cid:88) i (cid:88) j ijp ∗ i p mij − (cid:32)(cid:88) i ip ∗ i (cid:33) (2) iv) Estimation of the ordinal spectrum . To explore thespectral properties of the ordinal patterns sequence, the ordinal spectrum (OS) can be obtained from the spec-tral representation of the autocovariance function definedabove: OS( f ) = N − (cid:88) m = − ( N − Cov( m ) exp( − i πf m ) (3)Periodicity in time series yields a periodic structure inthe symbolic sequences, and will be reflected in an or-dinal spectrum with clear peaks. Similarly, as in theircounterpart in classical signal processing, random sym-bolic sequence are decorrelated with a flat spectrum. Itis well known that the structure of symbolic sequencesdepends on the temporal correlations of original time se-ries [40]. Consequently, their symbolic autocovarianceand ordinal spectrum are expected to depend on the de-gree of such correlations. Although different estimatorscan be used to obtain the spectrum OS( f ) from the co-variance function Cov( S t , S t + m ), here we used the au-toregressive spectral estimate as it offers better spectralresolution and smaller variance than does Fourier-basedestimators [41, 42]. Normalized frequency (cycle/sample)
Normalized frequency (cycles/sample) O r d i n a l s p ec t r u m O r d i n a l s p ec t r u m (c) (d) (a) (b) -4 -5 FIG. 1. Ordinal spectra against normalised frequency (bluecurves) for data generated from the logistics map (embed-ding parameters D = 3 and τ = 1): with periodic behav-iors (a) r = 3 .
55; (b) r = 3 . r = 3 .
8; and (d) r = 4. Orange boxes in main subplots in-dicate the frequency regions where values of ordinal spectraare statistically different from those obtained from surrogatedata. Slashed curves only indicate the average spectra fromrandomized symbolic sequences, not from IAAFT surrogates.Insets depict the power spectra from original time series innormalised frequencies. v) Detection of relevant scales in the ordinal spectrum .Peaks in the ordinal spectrum could simply result fromlarge autocorrelation values at different time lags in theoriginal time series X t . To rule out this possibility, wecompare the ordinal spectrum with those obtained froman ensemble { X st } of surrogate time series that repli-cate the linear autocorrelation and amplitudes distribu-tion of the original time series. Here, we use the so-called Iterative Amplitude Adjusted Fourier Transform(IAAFT) [13, 14] that preserves autocorrelation func-tion and amplitude distribution of original data, whileall other higher-order statistics are destroyed. For each X st , we repeat the above steps i)-iv) to compute a setof { OS s ( f ) } spectra. If any value in the ordinal spec-trum OS( f ) is statistically distant from the distributionof { OS s ( f ) } we can reject the null hypothesis of a linearstochastic time series. In this study, all significance testsare set at p < .
05, Bonferroni-corrected for multiplecomparisons (over frequencies f ). Normalized frequency (cycles/sample)
Normalized frequency (cycles/sample) -1 O r d i n a l s p ec t r u m O r d i n a l s p ec t r u m (a) (b)(c) (d) -1 -1 -1 FIG. 2. Ordinal spectra against normalised frequency for datagenerated from the x -component of the R¨ossler system ( D = 4and τ = 30 samples) with: periodic behaviors (a) a = 0 . a = 0 .
41; and chaotic dynamics (c) a = 0 .
42; and (d) a = 0 .
54. Same stipulations as in the caption of Fig.1
To demonstrate our method, we apply it first to syn-thetic data generated by a logistic map, defined by theiterative equation x n +1 = rx n (1 − x n ) where r is thebifurcation parameter. This nonlinear map has severaltransitions in the dynamics occurring during r ∈ [3 , r ≈ . r display non-chaotic behavior [43].For this model, the length of each time series is set to T = 2000, after discarding the first 1000 points to avoidpossible transients.Main plots in Figs.1(a)-(b) show that, despite the largepeaks observed in the ordinal spectrum, the dynamicalproperties of the periodic process are not statistically dif-ferent from those replicated by the surrogate data andthus, the null hypothesis cannot be rejected at any fre-quency. As expected for chaotic sequences, results inFigs.1(c)-(d) indicate that the ordinal spectrum capturesin different frequency ranges a dynamical complexity dif-ferent from those of surrogates.We also test our method on data generated bythe R¨ossler system whose equations are given by[ ˙ x = − y − z, ˙ y = x + ay, ˙ z = 2 + z ( x − a ∈ [0 . , . a ≈ . a , some periodic windows can be still found. The lengthof each time series is set to T = 10 , and a transientcut-off of 1000 samples.Different plots in Fig.2 show that the proposed methodaccurately distinguish chaotic fluctuations from a peri-odic dynamics (event for the dynamics observed in thepocket of periodicity at r = 0 . T = 500 samples.For the chaotic R¨ossler system, the method requires thetime series to be larger than ten times the fundamentalperiod of the oscillation. For shorter sample sizes, themethod cannot reject the null hypothesis of a linearlyfiltered noise.In addition to testing across deterministic time series(chaotic or not), we evaluated the method on non-chaoticstochastic data. Here we firstly evaluated a Gaussiannoise with distribution N (0 ,
1) and a stochastic processwith a power law spectrum S ( f ) ∝ f − α where α = 1.For each stochastic process, the ordinal spectrum is cal-culated with 2000 samples. Normalized frequency (cycles/sample) O r d i n a l s p ec t r u m O r d i n a l s p ec t r u m (a) (b)(c) (d) Normalized frequency (cycles/sample) FIG. 3. Ordinal spectra for stochastic data generated from:(a) a Gaussian distribution ( D = 4 and τ = 1 samples); (b)Power law noise ( D = 4 and τ = 4 samples); (c) nonlinearsystem driven by non-Gaussian noise ( D = 4 and τ = 2 sam-ples); and (d) non-monotonic nonlinear transformation of alinearly filtered noise ( D = 4 and τ = 3 samples). Samestipulations as in the caption of Fig.1 Results in Figs. 3(a)-(b) suggest that, although theordinal spectra are affected by the temporal correlationsof the original time series, they cannot be distinguishedfrom those produced by the surrogate data. As expected,the corresponding statistical tests suggest that the nullhypothesis of a stochastic process cannot be rejected inboth cases (white decorrelated and correlated noises).We then evaluated the method on a nonlinear systemdriven by a non-Gaussian noise. We considered the sys-tem given by x t = 0 . x t − − . x t − +0 . y t − +0 . x t − +0 . y t − +0 . η (cid:48) t and y t = sin(4 πt )+sin(6 πt )+0 . η (cid:48)(cid:48) t , where noises { η (cid:48) t , η (cid:48)(cid:48) t } are iid drawn from the Laplaciandistribution p ( η ) = b exp (cid:16) −| η − µ | b (cid:17) , with µ = 0 and b = 1. To evaluate the performance of our method underthe null hypothesis of nonlinearly transformed stochas-tic processes, we applied a static non-monotonic nonlin-ear transformation, x t = tanh ( y t ), to the linear non-Gaussian process given by y t = 0 . y t − − . y t − . y t − + e t , where e t is obtained by squaring a uniform noise withamplitude distribution between − . . T = 2000,after discarding the first 1000 points.A clear distinction between chaos and stochastic be-havior can be difficult for data generated by nonlinearsystems driven by non-Gaussian noises [13, 14]. Simi-larly, it is well known that nonlinear transformations mayintroduce sufficient phase correlations in linearly filterednoises making difficult to identify the stochastic behav-ior. Results depicted in Figs. 3(c)-(d) indicate that theordinal spectrum, in combination with the IAAFT algo-rithm, correctly diagnoses the nonlinear and non Gaus-sian models as stochastic process, including the staticnonlinear non-monotonic transformation of a non Gaus-sian random process.Results clearly indicate that: i) large peaks in the non-linear spectrum cannot be considered as a proof of achaotic dynamics; and ii) whatever the underlying dy-namic is (periodic or chaotic), a random shuffling of theconstructed symbolic sequence yields a flat spectrum. Astatistical test based on randomly shuffled sequences istherefore unable to identify chaotic dynamics.Finally we demonstrate the potentials of our methodon real data of different nature: epidemiology (measlesand cholera time series), astrophysics (the sunspots num-ber series) and neuroscience (electroencephalographicdata from an epileptic patient). As data have differ-ent length we apply the ordinal symbolic transforma-tion in different dimensions, following the condition [26] T (cid:62) ( D + 1)!The pattern of measles epidemics in developed coun-tries is among the best documented population cycles inecology. Different studies have proposed evidence for lowdimensional dynamics in different epidemiological timeseries [45, 46]. The inset in Fig. 4(a) shows the monthlycases of measles in Copenhague, Denmark, between 1927and 1968 [46]. For the time series of measles consid-ered here, our method clearly rejects the null hypothesisof stochastic dynamics. In full agreement with previousworks, our results indicate that such measles’ dynamicscannot be analyzed by conventional linear models and alow dimensional complexity underlies the observed dy-namics.The interannual disease cycles observed in many infec-tious diseases result from the interplay between extrinsicand intrinsic factors [48]. These interactions with the dis-ease dynamics may produce oscillations of complex pat- Normalized frequency (cycles/sample)
Normalized frequency (cycles/sample) O r d i n a l s p ec t r u m (a) (b)
100 weeks2000 cases 100 months2000 deaths (c)
50 years100 sunspots
Normalized frequency (cycles/sample)
Normalized frequency (cycles/sample) (d) (e) μ V μ V Normalized frequency (cycles/sample)
FIG. 4. Ordinal spectra (blue curves) of different real data. Insets show the original time series, its temporal and amplitudescales. (a) Measles cases ( D = 3 and τ = 4 samples); (b) cholera data ( D = 4 and τ = 2 samples); (c) Sunspots data ( D = 4and τ = 30 samples); (d) EEG before and (e) during an epileptic seizure ( D = 4 and τ = 5 samples). Slashed curves indicatethe average spectra from randomly shuffled symbolic sequences. Orange boxes in main subplots denote the frequency regionswhere values of ordinal spectra are statistically different from those obtained from surrogate data. terns, including chaos [49]. The inset in Fig. 4(b) depictsthe monthly deaths from cholera in Dacca, East Bengalbetween 1891 and 1940 [50, 51]. Our analysis, based upona spectrum capturing the intrinsic nonlinear dynamics ofthe system, suggests a low dimensional dynamics in thetime series. This is in agreement with previous mathe-matical models of seasonally driven epidemics [49].Solar activity is driven by the emergence of mag-netic flux through the photosphere forming active re-gions which include sunspots. While the most char-acteristic feature of solar activity is its modulated 11-year cycle [52], some studies have suggested that theirregular behavior of the activity reflects the presenceof low-dimensional chaotic dynamics [52]. The inset inFig. 4(c) shows the monthly mean total sunspot number(the arithmetic mean of the daily total sunspot num-ber over all days of each calendar month) between 1749and 2020 [53]. Our method rejects the hypothesis of astochastic process, which agrees with previous findingssuggesting that sunspots fluctuations can be explainedby a nonlinear (chaotic) dynamics [52].As many other time series in biology and medicine,electroencephalographic (EEG) signals display strongnonlinearities during different cognitive or pathologicalstates [54]. In epilepsy, dynamical properties of EEGsignals can be a used as a marker of the epileptogeniczone [55]. Here, we applied our method to a scalpEEG recordings from a subject with intractable epilep-tic seizures. Data were recorded at 102.4 Hz with ascalp right central (C4) electrode (linked earlobes ref-erence) [56, 57]. Time series ploted in Figs. 4(d)-(e) cor-respond to data from interictal and ictal (seizure) period,respectively. These results confirm previous findings sug-gesting that interictal EEG dynamics can be associate toa stochastic process, whereas a low dimensional dynamicscharacterizes epileptic seizures [55].To conclude, this study proposes a nonlinear spectrumfor characterising complexity in the frequency domain.Our approach is able to distinguish chaotic fluctuationsfrom stochastic dynamics in finite time series. Basedon the ordinal patterns analysis, the proposed method compares the spectral information of the symbolic rep-resentation of X t and the counterpart of linearly filteredstochastic process (surrogate data). Our simulations sug-gest that it accounts for static nonlinear transformationsof linear data and accurately provides the expected re-sults, even under the null hypothesis of correlated noisesor nonlinearly transformed stochastic processes.In this work, we show that the detection of chaoticoscillations in time series can be successfully addressedusing a spectral analysis of ordinal symbolic representa-tion. Our findings depict a robust approach to identifya chaotic dynamics in a time series. The main advan-tage of our proposal relies on its simplicity, reliability,and computational efficiency. The method is fully data-driven and it does not require a priori knowledge aboutthe data sequence for its symbolic representation, whichis very useful in real-world data analysis. Although it isbased on the ordinal patterns representation, the methodcan be straightforwardly applied to other symbolic rep-resentations (based on ordered symbols or categories).Similarly, different spectral representations can be ob-tained from symbolic data.Our results confirm chaotic dynamics in sunspotstime series and suggest this property as a commonsignature in epidemiological data. Results also indicatethat the EEG recordings are characterized by a complexdynamics during epileptic seizures, whereas interictalactivity can be explained by a stochastic process. Theseresults suggest that nonlinear spectral methods mayprovide a more complete characterization of chaoticsequences, which could help to unfold their underlyingdynamics and provide a better landscape of the observedsystem. 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