Wavelet-based discrimination of isolated singularities masquerading as multifractals in detrended fluctuation analyses
Paweł Oświęcimka, Stanisław Drożdż, Mattia Frasca, Robert Gębarowski, Natsue Yoshimura, Luciano Zunino, Ludovico Minati
WWavelet-based discrimination of isolated singularitiesmasquerading as multifractals in detrended fluctuationanalyses
Pawe(cid:32)l O´swi¸ecimka a, ∗ , Stanis(cid:32)law Dro˙zd˙z a,b , Mattia Frasca c , RobertG¸ebarowski d , Natsue Yoshimura e,f , Luciano Zunino g,h , Ludovico Minati e,a,i a Complex Systems Theory Department, Institute of Nuclear Physics Polish Academy ofSciences, ul. Radzikowskiego 152, 31–342 Krak´ow, Poland b Faculty of Computer Science and Telecommunications, Cracow University ofTechnology, ul. Warszawska 24, 31–155 Krak´ow, Poland c Department of Electrical Electronic and Computer Engineering (DIEEI), University ofCatania, 95125 Catania, Italy d Faculty of Materials Engineering and Physics, Cracow University of Technology, 30-084Krak´ow, Poland e FIRST, Institute of Innovative Research, Tokyo Institute of Technology, Yokohama226-8503, Japan f PRESTO, JST, Saitama 332-0012, Japan g Centro de Investigaciones ´Opticas (CONICET La Plata - CIC), C.C. 3, 1897 Gonnet,Argentina h Departamento de Ciencias B´asicas, Facultad de Ingenier´ıa, Universidad Nacional de LaPlata (UNLP), 1900 La Plata, Argentina i Center for Mind/Brain Science (CIMeC), University of Trento, 38123 Trento, Italy
Abstract
The robustness of two widespread multifractal analysis methods, one basedon detrended fluctuation analysis and one on wavelet leaders, is discussedin the context of time-series containing non-uniform structures with onlyisolated singularities. Signals generated by simulated and experimentally-realized chaos generators, together with synthetic data addressing particularaspects, are taken into consideration. The results reveal essential limitationsaffecting the ability of both methods to correctly infer the non-multifractalnature of signals devoid of a cascade-like hierarchy of singularities. Namely,signals harboring only isolated singularities are found to artefactually give ∗ Corresponding author:
Email address: [email protected] (Pawe(cid:32)l O´swi¸ecimka)
Preprint submitted to Nonlinear Dynamics April 8, 2020 ise to broad multifractal spectra, resembling those expected in the presenceof a well-developed underlying multifractal structure. Hence, there is a realrisk of incorrectly inferring multifractality due to isolated singularities. Thecareful consideration of local scaling properties and the distribution of H¨olderexponent obtained, for example, through wavelet analysis, is indispensablefor rigorously assessing the presence or absence of multifractality.
Keywords:
Chaotic oscillator; Complexity; Dynamical system; H¨olderexponents; Multifractal analysis; Multifractal spectrum; Singularity;Time-series analysis
1. Introduction
The concept of multifractality, whereby not just one but an extendedspectrum of exponents is required to account for the dynamics of a system,represents one of the pillars of complex signal analysis [1–5]. The term wascoined in the context of fully-developed turbulence [6] and mathematicallyformalized, with the multifractal or singularity spectrum as principal char-acteristic, by Halsey et al. in the year 1986 [7]. Within the frameworkof multifractal formalism, a function is decomposed into subsets, which arecharacterized by a H¨older exponent α and a fractal dimension f ( α ) [7]. Theidentified set of H¨older exponents provides explicit information about theregularity of a time-series. When only one type of regularity is present in asignal, its statistical properties are quantified by a single H¨older exponent;this results in the convergence of the multifractal spectrum to a single point.A typical example of such a monofractal process is fractional Brownian mo-tion (fBm) with a homogenous organization of the fluctuations. On the otherhand, multifractal signals, related to intermittency phenomena and correla-tions heterogeneity, are characterized by an extended set of H¨older exponentsand developed multifractal spectrum resembling an inverted parabola [8]. Arepresentative example is the binomial cascade generated through the itera-tive and multiplicative procedure. Thus, the multifractal methodology offersan opportunity to distinguish between signals characterized by the same au-tocorrelation function or power spectrum, but having a different underlyingorganization. The pervasiveness of emergent fractal and multifractal struc-tures has made it possible to apply this methodology fruitfully across diverseareas such as physics [9, 10], biology [11–13], physiology and neuroscience[14, 15], chemistry [16, 17], economy [18–24], even linguistics [25, 26] and2usic [27, 28].The study of the multifractal properties of time-series can be approachedin two ways. The most common approach involves estimating the spectrumthrough global procedures that neglect the precise location of the singulari-ties. These procedures, as we show in our study, work well for fractal struc-ture with a well-developed hierarchy of singularities and densely roughness ofthe signals. An alternative one, based on assessing singularity locally, offersthe possibility to determine also the location of the H¨older exponents. Thisopens up the possibility of quantifying the singular behavior of the signalwhenever singularities appear isolated, e.g., when analyzed processes reveallocal smoothness or outliers due to measurement error. Although this ap-proach provides more information about the fluctuation organization, it isconsiderably more numerically unstable, and as such, rarely applied in prac-tical time-series analysis. However, as we clearly demonstrate in our study,this is possibly the only way of distinguishing artefactual (or apparent) fromgenuine multifractality.Several methods have been devised for numerically estimating the mul-tifractal spectrum of a process given a simulated or recorded signal (in thepresent context, equivalent to time-series) [29–31]. The prevalent algorithmsencompass the multifractal detrended fluctuation analysis (MFDFA) [32], thewavelet transform modulus maxima [33], and its development wavelet leaders(WL) [34]. Due to their numerical stability and usual accuracy, these twomethods are commonly recognized as the most reliable means of estimat-ing the multifractal spectrum [35–37]. Their accuracy has been repeatedlydemonstrated via synthetic time-series having known multifractal properties[35]. Moreover, the results obtained with these two methods can be usedas a cross-test of the validity of the assessed multifractality, since MFDFAand WL are based on different numerical methods of quantifying the multi-fractality. The former hinged around the scaling properties of the variance,whereas the latter is grounded on the wavelet transform for decomposingthe signal and characterize its self-similar properties. Thus, in our study,we applied both approaches as complementary methodologies. Albeit dis-tinct, these methods are all based on the well-known q -filtering technique,which decomposes a time-series into subsets predicated on the fluctuationamplitudes. As such, they similarly provide information about the averagelevel of fractality across all time-series segments. While their practical use-fulness is beyond question, the results should be interpreted cautiously dueto the inherent complexity of both the signals under analysis and of the al-3orithms themselves. Here, several compelling examples are given of howeven elementary systems can yield signals for which a naive interpretationof the multifractal spectrum, obtained via both the MFDFA and the WL,leads to completely flawed conclusions. To this end, consideration is given tosimulated and experimentally-recorded time-series from the Saito chaos gen-erator, a simple four-dimensional non-linear dynamical system with a stronghysteretic component, as well as to synthetic signals which by constructioncannot possess any multifractal structure. The results show that relying onlyon the multifractal spectrum width as an indicator of multifractality can beprofoundly misleading. At a minimum, other spectral attributes such as theasymmetry have to be taken into account, where the key is considering thelocal scaling properties and the distribution of H¨older exponents.This paper is organized as follows. In Sec. II, the notion of local andglobal H¨older exponents is introduced, together with the methods of MFDFAand WL. In Sec. III, a selection of examples of true multifractality is firstlyintroduced. Then, the case of a hysteretic oscillator is considered to exemplifythe consequences of isolated singularities on both MFDFA and WL, followedby the presentation of paradigmatic synthetic signals leading to similar issues.Finally, in Sec. IV, practical recommendations for the proper application ofthese analyses are offered.
2. Methodology
Analyzing the regularity of a signal provides essential insight into its sta-tistical properties and possible underlying geometrical structure. To char-acterize the local singular properties of a time-series, then, the point-wiseH¨older exponent α can be considered. Given a function f , for each x ∈ (cid:60) the same is defined as follows [38]: α ( x ) = sup { h : f ∈ C h ( x ) } , (1)where f belongs to the H¨older space C h ( x ) if and only if | x − x | ≤ (cid:15), | f ( x ) − f ( x ) | ≤ C | x − x | h . (2)Values of the H¨older exponent approaching zero indicate increasing irregular-ity of the function f ; conversely, larger values of α denote more regular fluc-tuations. The multifractal formalism is a statistical description of functionsthrough quantifying their distribution of the point-wise H¨older exponents,which is naturally extended to continuous signals and discretized time-series.4 .2. Multifractal detrended fluctuation analysis Multifractal detrended fluctuation analysis (MFDFA) [32] is a method fordetecting and quantifying the scaling properties of time-series which is widelyapplied across diverse areas of experimental and computational science [39–44]. It comprises multiple steps, which may be summarized as follows. Letus consider a time-series x i having length N , i = 1 , ...N and, as a first step,calculate its profile according to X ( j ) = j (cid:88) i =1 [ x i − (cid:104) x (cid:105) ] , (3)where (cid:104) x (cid:105) denotes the mean of time-series x i . Since fractality manifests aspatterns which are self-similar across different temporal scales, the profile hasto be analyzed over segments of different length. Thus, the time-series is nextsubdivided into N s non-overlapping segments ν of length s ( N s = int( N/s ))starting from the beginning. However, since the length is not necessarily aninteger multiple of the scale s , the procedure is also repeated starting from theend, yielding a total of 2 N s segments. To remove possible trends in the time-series, which can distort the results, in each segment ν a polynomial of order m ( P ( m ) ν ) is fit and subsequently subtracted from the data. The effectivenessof this detrending step strongly depends on the polynomial order, and itis generally agreed that small values of m provide the most reliable results[45]. Here, no statistically discernible differences were found between resultsobtained for m = 2 , m = 4are given.As a next step, the detrended variance is calculated within each segmentaccording to F ( ν, s ) = 1 s s (cid:88) k =1 ( X (( ν − s + k ) − P ( m ) ν ( k )) . (4)Then, in order to quantify the fractal properties of the signal with respectto the amplitude, the q -order filtering technique is applied, obtaining the q -order fluctuations function F q ( s ) = (cid:110) N s N s (cid:88) ν =1 [ F ( ν, s )] q/ (cid:111) /q , q ∈ (cid:60) \ { } , (5)5here q operates as a filter which discriminates fluctuations based on theiramplitude; more precisely, negative and positive settings of q respectivelyemphasize small and large changes. Fractality in a time-series manifestsitself as power-law behavior of F q ( s ) over different scales, that is, F q ( s ) ∼ s h ( q ) , (6)where h ( q ) denotes the generalized Hurst exponent. Hence, h ( q ) repre-sents the fractality of the fluctuations selected by a given setting of q . Formonofractal time-series, h ( q ) is constant and equals the Hurst exponent h ( q ) = H [46, 47]. This can be used to classify time-series with respect tolinear correlations. Namely, H > . H < . H = 0 . h ( q ) is a de-creasing function of q , and the Hurst exponent is retrieved at h ( q = 2) = H .Thus, for better visualizing the results and interpreting the spectrum of thegeneralized Hurst exponents, the same can be converted into the multifractalspectrum via the Legendre transform of the scaling function τ ( q ) = qh ( q ) − α = h ( q ) + qh (cid:48) ( q ) , f ( α ) = q [ α − h ( q )] + 1 , (7)where α is the H¨older exponent, and f ( α ) refers to the fractal dimension ofthe data supported by a particular α .The intensity of multifractality, and thus the degree of signal complexity,is often quantified through the width of the multifractal spectrum, that is,∆ α = α max − α min . The larger ∆ α , the more developed a multifractal struc-ture is deemed to be. Another important feature of the multifractal spectrumis its asymmetry. For the paradigmatic case of the binomial cascade, a math-ematical multifractal, f ( α ) resembles a symmetric inverted parabola [48].However, for real-world time-series, the spectrum is often asymmetric, hav-ing one side better developed than the other; this stems from a heterogeneousorganization of the signal fluctuations across scales. Hence, through quanti-fying the spectral asymmetry, one can retrieve critical information about thetemporal organization of a time-series. The asymmetry parameter is definedas [49] A α = (∆ α L − ∆ α R ) / (∆ α L + ∆ α R ) , (8)6here ∆ α L and ∆ α R stand, respectively, for the distances between the spec-tral maximum and the smallest and largest values of α . In turn, the de-gree of the asymmetry is quantified as | A α | , whereas the sign indicates theasymmetry direction. A positive value of A α hallmarks a leftwards-stretchedspectrum and denotes a well-developed fractal organization of the large fluc-tuations, while smaller ones are governed by simpler dynamics. Contrariwise,for negative A α the spectrum is stretched towards the right, denoting morecomplex behavior of the small fluctuations compared to the larger ones. Another class of techniques for estimating the multifractal characteristicsof a non-stationary time-series is based on the wavelet transform [50, 51].According to these techniques, a signal is decomposed into the elementaryspace-scale wavelet coefficients by means of a family of functions stemmingfrom a basic function, the so-called mother wavelet. By scaling and trans-lation of the mother wavelet ψ a,s ( x ) = s − / ψ ( x − as ), one can obtain a de-composition of the signal at each scale s corresponding to a frequency band,separately for all time-points a ( a, s ∈ (cid:60) , s > f ( x ) is defined as [52] W f ( a, s ) = 1 s − / (cid:90) + ∞−∞ f ( x ) ψ (cid:18) x − as (cid:19) dx . (9)Importantly, visualization of the resulting wavelet spectrum W f ( a, s ) on thescale-time plane promptly reveals the skeleton of the hierarchical structure ofthe process being analyzed. The choice of the mother wavelet is dictated byit being well-localized in both the time and frequency domains (derivatives ofa Gaussian function are often used as mother wavelets). A crucial propertyof the wavelet transform is its close relation with the H¨older exponent α [50],wherein W f ( x , s ) ∼ s α ( x ) , s → + . (10)Hence, a local singularity α ( x ) can be characterized by the scaling behaviorof the wavelet transform around the point x . Moreover, the maxima of thewavelet transform produce maxima lines in space-scale half-plane, which con-verge towards loci of singularity. Thus, by retrieving the power-law behaviorof the wavelet transform coefficients along these lines, one can estimate theH¨older exponents, and in turn, quantify the singularity strength [33]. Due tothe instability of the canonical wavelet-based multifractal methods whenever7 large number of coefficients are close to zero, and due to its insensitivityto oscillating singularities [53], the notion of wavelet leaders (WL) was intro-duced [34, 51]. For a discrete scale parameter s j = 2 − j and time a j,k = 2 − jk ( j, k ∈ Z ), the signal can be recovered via the formula [54, 55] f ( x ) = (cid:88) j,k ∈ Z c j,k ψ (2 − j x − k ) , (11)where the wavelet coefficient c j,k is given by c j,k = 2 − j (cid:90) (cid:60) f ( x ) ψ (2 − j x − k ) dx . (12)In this study, the Daubechies wavelet with 4 vanishing moments was used[56]. The wavelet leader of x at the level j denotes the largest waveletcoefficient among those existing in the spatial neighborhood of x at finerscales [57]. Formally, for the dyadic interval λ j,k = [2 j k, j ( k + 1)], it isdefined as L j ( x ) = sup λ (cid:48) ⊂ λ j,k ( x ) | c j,k ( λ (cid:48) ) | , (13)where 3 λ j,k ( x ) = λ j,k − ∪ λ j,k ∪ λ j,k +1 = [2 j ( k − , j ( k + 2))) and contains x . For a given scale 2 j , one can define structure functions S ( q, j ) based onthe q -th order average of the leaders S ( q, j ) = 2 j (cid:88) λ ∈ Λ j L qj , (14)where q is a real number, and Λ j is a set of dyadic intervals at scale j .Power-law behavior of the structure function in the limit of small scales S ( q, j ) ≈ C q jζ ( q ) , (2 j →
0) is a manifestation of scale invariance. Thus, ζ ( q )determines the scaling exponents, and can be numerically estimated by meansof a log-log regression. Since the ζ ( q ) function is necessarily concave [58],the Legendre transform can be used to estimate the multifractal spectrumaccording to the formula f ( α ) = inf q ∈(cid:60) ( qα − ζ ( q )) + 1 . (15) The multifractality of a time-series manifests itself through sets of non-trivial H¨older exponents, which quantify the local variation in its irregularity859]. These exponents may be collectively quantified by means of a “global”measure, obtained from the multifractal spectrum in (Eq. (7) or (Eq. 15) anddenoted as α G , or directly through the analysis of local scaling propertiesof the signal by means of Eq. (10) and denoted as α L . Ideally, the twoapproaches should give consistent results. However, as demonstrated below,the multifractal analysis of complex time-series has limitations which, undercertain circumstances, yield misleading signatures of multifractality. To assess the statistical validity of the results, additional analyses wereperformed on surrogate time-series for all scenarios under consideration. Twocommonly-used surrogate sets were generated. One relies on randomizationof the Fourier phases and, as such, preserves only the spectral amplitudeswhile obliterating all non-linear inter-dependencies [60]. The other involvesrandomly permuting the time-points, destroying all temporal correlationswhile preserving the value distribution [21].
3. Results
For comparison with the particular cases considered below, representativeinstances of real multifractals having diverse properties are firstly presented,based on analyses conducted over the range q ∈ [ − ,
4] [61]. To this end,two mathematical multifractals are considered, namely the binomial cascadeand the chaotic metronome derived from the Ikeda map [62], together withseveral real-world time-series: the inter-beat intervals extracted from elec-trocardiographic signals (103885 data points), the sentence length variabil-ity of the “Finnegans Wake” book by James Joyce, the logarithmic returnsof the American stock market index S&P500 (7440 data points), and thesunspot number variability (43495 data points) [11, 21, 26, 35, 49]. In allthese cases, the multifractal spectrum f ( α G ) assumes the shape of a wide in-verted parabola, spanning ∆ α G > .
2, indicating a multifractal organizationof the data (Fig. 1, left). Yet, the spectra develop different degrees of asym-metry. For the binomial cascade, the inter-beat intervals, and the sentencelength variability, the spectra appear almost symmetrical ( A α ≈ α L even broader than in the multifractal spectrum, incidentally revealing thehigher sensitivity of the wavelet transform on the local scaling propertiescompared to the global methodology, which mainly reflects the prevalentsingularities in the time-series. Figure 1: Examples of true multifractality. (a) Multifractal spectra and (b) relative fre-quency f r histograms of the H¨older exponents. The following cases are presented: binomialcascade, chaotic metronome derived from the Ikeda map, inter-beat intervals, sentencelength in “Finnegans Wake”, logarithmic returns of the S&P500 index, and sunspot num-ber variability. To illustrate the potential pitfalls inherent in drawing hasty conclusionssolely from global measures, let us now consider the case of the Saito chaos10enerator, which is a four-dimensional non-linear oscillator consisting of thefollowing dimensionless state equations [63]: ˙ x = − z − w ˙ y = γ (2 δy + z )˙ z = ρ ( x − y ) ,˙ w = ( x − h ( w )) /ε (16)wherein h ( w ) = w − (1 + η ) if w ≥ η − η − w if | w | < ηw + (1 + η ) if w ≤ − η . (17)Despite its simple form and low dimensionality, this system readily generatesrich dynamics spanning periodicity, quasi-periodicity, chaos, and eventuallyhyperchaos as a function of the parameters γ , δ , ε , η , and ρ . Here, it wasinitially deemed of interest from the perspective of its hypothetical ability togenerate signals having a truly multifractal structure; however, in the courseof numerical investigation, another feature was realized to be fundamentallyimportant for the purposes of the present work, namely, the presence of thehysteresis function h ( w ), which only enters the equation of the state variable w . As a consequence of it, even though all state variables conjointly par-ticipate in the temporal dynamics, x, y, z have rather smooth an activity,whereas, in the limit of ε →
0, the temporal evolution w is characterizedby sudden jumps. As shall become clear, it is these discontinuities, namelythe combination of slow and fast motions corresponding to the continuousmanifold and sudden jumps, which may lead to a mistaken inference of multi-fractality. Unless indicated otherwise, the parameters were set for operationin the hyperchaotic regime, that is, γ = 1, δ = 0 . ε = 0 . η = 1, and ρ = 14 [63, 64].Preliminary examination revealed differences between short- and long-range temporal correlations in the simulated time-series, giving rise to across-over in the fluctuation functions. In particular, the multiscale charac-teristics revealed a strong autocorrelation only over short time scales (i.e., s < .2.1. Numerical simulations Time-series having a length of 10 points were simulated given equations(16)-(17), applying the adaptive step-size Runge-Kutta (4,5) method andreturnig the results at a fixed step size of ∆ t = 0 . x, y, z are characterized by the markedlyirregular behavior characteristic of chaotic systems. However, the dynamicsof w are even more complex, featuring sharp upward and downward jumps.Even though the underlying system is the same, the multifractal propertiesof the signals, being influenced by the presence of singularities, could thenbe partially dependent on the variable under consideration. This observa-tion is confirmed by the corresponding fluctuation functions F q ( s ) (Fig. 2b).Therein, it is clearly visible that the functions obey power-law behavior,which is a signature of fractal organization: however, while for x, y and z the scaling is rather homogeneous, for w a pronounced heterogeneity is ap-parent. Moreover, for the latter the majority of fluctuation functions havea slope close to those found close for the extreme values of q , i.e., q = ± F q ( s ) are observed in the quasiperi-odic regime, with δ = 0 .
65 (Fig. 2c; for brevity, results are only shown for w ). Though the dynamics are profoundly different compared to the hyper-chaotic regime, the heterogeneity of the fluctuations functions remains mostpronounced for w , with the distribution of slopes nearly unchanged and char-acteristic of a bifractal structure (Fig. 2d).The multifractal analyses for the time-series of w generated as a functionof the control parameter δ are depicted in Fig. 3. The parameter was sweptin δ ∈ [0 . , H and multifractal spectrum width∆ α G as estimated through the MFDFA and WL algorithms are depicted inFig. 3a. It is evident that the multifractal characteristics are insensitive to thequalitative features of the system dynamics. The time-series remain stronglypersistent, with H ≈ .
5, and feature a wide spectrum with ∆ α G ≈ . igure 2: Dynamics of the simulated Saito chaos generator. (a) Time-series in the hy-perchaotic regime (all variables), and (b) corresponding fluctuation functions F q ( s ). (c)Time-series in the quasiperiodic regime ( w only), and (d) corresponding fluctuation func-tions. the multifractal spectra for the hyperchaotic and quasiperiodic regimes arecompared. Their shape is almost identical, with a strong left-sided asymme-try A α ≈ .
5: importantly, this coexists with an uneven distribution of thepoints along the spectrum, which concentrate mainly towards its ends. Here,analysis of the local scaling revealed fundamental subtleties of the data or-ganization. The relative frequency histogram f r of the H¨older exponents α L forms two separable peaks, whose locations coincide with high-concentrationpoints close to the minimal and maximal values of α G identified on the mul-tifractal spectrum.The locations of the singularities and their “strength” were recovered, asgiven by the H¨older exponents, through analysis of the local wavelet trans-form coefficients (Fig. 4). It is well-evident that for the time-series of the w variable, in both the quasiperiodic and hyperchaotic regimes (Fig. 4a), themaxima form separate lines on the space-scale half-plane (cf. Fig. 4b), whichdelineate isolated singularities (Fig. 4c). In the presence of a truly multifrac-tal geometry, the maxima would follow a tree-like structure, stemming fromthe self-similar organization of the fluctuations. By contrast, consideration ofthe locations and strength of the singularities reveals that two discrete types13 igure 3: Analysis of the simulated w time-series from the Saito chaos generator. (a)Width of the multifractal spectrum ∆ α and Hurst exponent given different settings ofthe control parameter δ . Horizontal dashed line: average spectra widths estimated forFourier-based surrogates; for randomly shuffled data, ∆ α G < . are present in these time-series, and related to volatile portions of the signal:one reflects instants wherein the hysteretic behavior is apparent ( α L ≈ . α L ≈ q -filteringmethod inherently yields a concave spectrum, and isolated peaks are impos-sible to obtain. The artifactual result, then, is purely the product of theaveraging procedures inherent in the MFDFA methodology: together withthe dense sampling of the q parameter, these generate a broad spectrum ofH¨older exponents, even when only isolated singularities are present in thesignal. Although it is markedly stretched towards the left-hand side, with ahigh concentration of points towards the two limit values of H¨older exponent,the estimated spectrum resembles an inverted parabola. Instead, the scalingproperties of the time-series generated by this system should be representedby a single exponent for the x, y, z variables, and by a bifractal organization14or the w variable. -3-1135 w Hyperchaotic ( =0.94) a) -3-113 w Quasiperiodic ( =0.65) b)
1 2 3 4567 l og ( s ) l og ( s ) c) t L t L Figure 4: Local scaling properties of the w time-series from the Saito chaos generator,simulated in the hyperchaotic (left) and quasiperiodic (right) regimes. (a) Time-coursesand (b) their wavelet transforms obtained via the fourth derivative of the Gaussian wavelet.Color coding denotes the magnitude of the wavelet coefficients W f ( x , s ) ranging fromdark blue (the smallest W f ( x , s )) to red (the largest one). (c) Corresponding time-localized H¨older exponents. To independently confirm that the results presented above stem faithfullyfrom the dynamics of this system, an experimental version of the same wasconveniently constructed using two operational amplifiers (type TL082) anda non-linearity based on two anti-parallel series Zener diodes (type BZT52-C5V1). The corresponding circuit diagram is given in Fig. 5a, where r = r = R = R = R = 10 kΩ, r o = 820 Ω, C = C = 3 . L = 3 . L = 32 mH (two inductors in series), and U Z = 5 . γ = C /C = 1, ε = L / ( r C ) = 0 . η = r /r = 1, and ρ = r C /L = 12 .
2. The signal corresponding to the variable w was digitizedfrom the physical circuit board (Fig. 5b) using a recording oscilloscope at arate of 1 MSa/s, tuning g − to obtain δ = { . , . , . , . , . , . , } .The corresponding time-series have been made publicly available [66].15 igure 5: Experimental implementation of the Saito chaos generator. (a) Circuit diagram,and (b) representative example of physical realization.Figure 6: (a) Experimental time-series of variable w recorded from the physical Saito chaosgenerator in the hyperchaotic regime and (b) corresponding fluctuation functions F q ( s ).(c) Time-series for the same recorded in the quasiperiodic regime and (d) correspondingfluctuation functions. Y-axis presented in arbitrary units.
16n agreement with the simulations, apparent multifractality only arisesfor the variable w . In Fig. 6, the time-series in the quasiperiodic and hyper-chaos regimes are shown alongside the corresponding fluctuation functions.The heterogeneity of the latter is equally apparent in both cases, suggestingthat the multifractal properties of the signals are only weakly dependent onthe dynamics. The multifractal spectra are shown in Fig. 7: they are wide(∆ α G >
2) but, in contrast to the simulations, more symmetric ( A α ≈ . The next example of dynamical system that we consider is the R¨osslersystem. Its dynamics are governed by the following system of three differen-tial equations [67] ˙ x = − y − z ˙ y = x + ay ˙ z = b + z ( x − c ) . (18)Similarly to the Saito generator, the R¨ossler system reveals rich dynam-ics spanning periodic and chaotic behaviors. Notably, its dynamical prop-erties depend on state equations without a hysteresis element. They arecontrolled by parameters that were set to a =0.3, b =0.2, c =5.7, knowinglyrealizing chaotic behavior with an intermediate level of folding. To ensurestatistical reliability, we generated time-series having a length of 10 points,a representative fragment of which is visible in Fig. 8a. The distribution ofthe fluctuation functions already suggests a bi-fractal organization (Fig. 8b)[68]. However, the multifractal spectra estimated through the MFDFA and17 igure 7: Analysis of the experimental recordings from the electronic Saito chaos generator.(a) Width of the multifractal spectrum and Hurst exponents estimated for the w variablegiven different settings of the control parameter δ . Average ∆ α G for the Fourier based-surrogates and randomly shuffled data, respectively, < .
65 and < .
1. (b) Multifractalspectra estimated for the quasiperiodic and hyperchaotic regimes. (c) Corresponding rel-ative frequency histograms for the H¨older exponents.
WL algorithms resemble typical multifractal characteristics, with a strongleft-sided asymmetry (Fig. 8c). This is in contrast with the local scalingproperties (Fig. 8d). The histogram of local H¨older exponents forms, simi-larly to the Saito generator case, two separate peaks that concentrate in thevicinity of the extreme α G values. Thus, the underlying structure reflectsisolated singularities rather than a unified multifractal organization. The results presented above suggest that singular behavior in the Saitochaos generator can be quantified through just two scaling exponents. More-over, the subsets corresponding to different singularities index separate com-ponents of the time-series, rather than constituting hierarchically-interwovenstructures, which are the hallmark of true multifractality. Thus, a naive in-terpretation of the spectrum width ∆ α G as a signature of multifractality canbe faulty. To highlight this issue even more clearly, we finally consider pro-cesses that are, by construction, not multifractal. Yet, the methods based on18 igure 8: Analysis of the simulated z time-series from the R¨ossler system. (a) Repre-sentative fragment of the times series. (b) Fluctuation functions F q ( s ) from MFDFAalgorithm. (c) Multifractal spectra estimated for the analysed signal. (d) Correspondingrelative frequency histogram of the H¨older exponents. q -filtering, namely the MFDFA and WL, yield misleadingly wide multifractalspectra in these cases.As an instructive example, results from the multifractal analysis of theL´evy process, which possesses a well-recognized bifractal structure, are firstlypresented. The multifractal spectrum of the L´evy time-series consists of twopoints, whose locations are directly related to the asymptotic behavior of thedistribution tail P ( x ) ∼ x − ( α Levy +1) and are given by [35, 69]: α = (cid:26) /α Levy ( q ≤ α Levy )0 ( q > α
Levy ) f ( α ) = (cid:26) q ≤ α Levy )0 ( q > α
Levy ) (19)where α Levy is the L´evy index and q is q -th moment of the fluctuation function F q ( s ). The multifractal analysis of the L´evy time-series having a length of50000 points (Fig. 9a) with α Levy = 1 . α G = 0 .
7) and strongly left-sidedasymmetrical ( A α ≈ . w ( t ) of two related sig-nals. One is a pseudo-periodic signal given by, e.g., u ( t ) = (cid:80) i sin 2 ωt/ ( p i / max p )where p i = { , , , , } . The other is a sequence of binary fluctuations v ( t ) = W [ u ( t ) , ξ ] generated by a hysteresis operator W acting on that signal,with v ∈ [ − ,
1] and hysteresis parameter ξ = 0 . u . Their linear com-bination, e.g., w ( t ) = v ( t ) + u ( t ) / max u is, by definition, not hierarchicallyinterwoven and does not obey different scaling exponents [70, 71], hence, themultifractal spectrum should consist of two separable points. To test thishypothesis, 10 time-series segments each having a lengths of 10 points weregenerated, and MFDFA was performed. The average of the multifractal spec-tra and the histogram of the H¨older exponents are depicted in Fig. 10c,d. Inthis case too, the multifractal spectrum appears well-developed (∆ α G = 1 . A α = 0 . w inthe Saito chaos generator; here, however, there was no underlying non-lineardynamical system.Finally, a pseudo-multifractal process, consisting of the superposition of afractal time-series with periodic components, is considered. At a first glance,this process resembles the multifractal time-series of sunspot number vari-ability (cf. Fig.1). However, as demonstrated below, careful inspection ofthe fractal characteristics illuminates its pseudo-multifractality. To this end,fractional Gaussian noise (fGn) [4] was generated: it represents a well-knownexample of a stochastic monofractal structure with possible long-range cor-relations quantified by the Hurst exponent, which has been applied to modelphenomena across various fields of science. Namely, simulations produceda fractional Gaussian noise with an arbitrarily-chosen Hurst exponent of H = 0 . igure 9: Examples of synthetic time-series leading to apparent multifractality. (a) L´evyprocess, (b) pseudo-periodic signal with sudden jumps (mimics w variable in the Saitochaos generator), (c) pseudo-multifractal process based on fractional Gaussian noise. cess was modulated by a cosinusoidal function F ( i ) = A + A cos(2 πi/T ) in i = 1 ...N , where A and T are the model parameters (cf. Fig. 9c). Then,the periodic function F ( i ) was added to this amplitude modulated noise. Inour simulations, N = 10 , A = 0 . T = 4000 were set. The results ofthe local scaling analysis, as well as the multifractal spectrum, are shown inFig. 10e,f. Analysis of the distribution of H¨older exponents confirms thatthe time-series is a composition of the two independent processes havingdifferent singular behaviors. The smaller values of α L concentrate around α L ≈ H = 0 .
8, corresponding to the fGn component, whereas the largerones are related to the periodic trend. A cursory analysis of the multifrac-tal spectrum indicates heterogeneity in the scaling properties. Particularly,the width of the spectrum, together with its strong right-sided asymmetry( A α = − . igure 10: Analysis of the synthetic time-series, with multifractal spectra (left) and relativefrequency histogram f r of the H¨older exponents (right). (a) L´evy process, (b) pseudo-periodic signal with sudden jumps (mimics w variable in the Saito chaos generator), (c)pseudo-multifractal process based on fractional Gaussian noise. As a last example, we analyzed an artefactually-generated stochastic pro-cess with the singularity spectrum derived analytically. In this respect, weconsidered the square transform of fractional Brownian motion (Fig. 11a),which represents a bi-Hlder process whose spectrum is given by the followingrelation [72]: f ( α ) = α = H − H if α = 2 H −∞ elsewhere . (20)In our study, we generated fBm having a length of 10 points with a Hurstexponent H = 0 .
7. Fluctuations functions F q ( s ) (Fig. 11b) obtained throughMFDFA show non-homogeneous scaling, which suggests a multiscaling be-havior of the data. This is even more clearly visible in the singularity spec-trum, which reproduces a concave hull supported by the interval in the rangefrom H to 2 H (Fig. 11c). Thus, based only on the Legendre-based method-ology, a flawed conclusion on the multifractal structure of the data wouldbe drawn. However, consideration of the histogram of H¨older exponents(Fig. 11d) estimated through wavelet analysis reveals the true bi-fractal na-22ure of the process, with exponents corresponding to the theoretical expec-tations. This leads to the conclusion that the exponents identified throughMFDFA and WL, except the two extreme values, are an artifact caused bya methodological limitation and do not contain any true information aboutthe analysed process. Figure 11: Analysis of the square transform of fractional Brownian motion. (a) Rep-resentative fragment of the times series. (b) Fluctuation functions F q ( s ) from MFDFAalgorithm. (c) Multifractal spectra estimated for the analysed signal. (d) Correspondingrelative frequency histogram of the H¨older exponents.
4. Discussion
The present study was initially conceived as the search for a determin-istic non-linear dynamical system, relatively low-dimensional, which couldgenuinely generate multifractal structures. The Saito chaos generator, rep-resenting a hysteretic oscillator which can be easily simulated numericallyas well as realized experimentally in an electronic circuit, was identified as apromising candidate. Indeed, an initial investigation of its dynamics basedon the commonly-accepted MFDFA technique yielded a spectrum suggestiveof fully-developed multifractality; similar conclusions were drawn from theWL analysis. However, concerningly, these results could only be obtained23or the dynamical variable w , which features discontinuities in the form ofjumps, and not for the others. A clear indication of a flaw came from theobservation of largely overlapping multifractal spectra between the hyper-chaotic and quasiperiodic regimes: since in the latter there is no turbulence,this is unexpected.Closer visual inspection of the time-series was instrumental in resolvingthis issue, because it demonstrated the absence of a cascaded hierarchy ofsingularities, fractally nested over the consecutive scales of magnification; in-stead, it revealed merely a sequence of isolated singularities, associated witha restricted number of distinct H¨older exponents. In the Saito chaos gen-erator, these isolated singularities are produced by twofold dynamics of thesystem, namely a continuous manifold and sudden jumps. The dynamics ofthese components, whose organization is quantified by single scaling expo-nents, are not strongly interrelated: this results in a fluctuation structuredevoid of hierarchical organization. An explicit step-by-step wavelet-basedestimation of the same indicated values limited to two-well separated, narrowintervals (Fig. 3): this is in stark contrast with the results given by both theMFDFA and WL algorithms, which generated a broad distribution of f ( α ),comprising the two H¨older exponents seen locally, but mistakenly suggest-ing coverage of the entire interval between these extremes. Tentatively, acombination of processes without intrinsic convolution can be indicated as anecessary condition for the emergence of artefactual multifractality relatedto isolated singularities. Thus, our methodology can be applied to dynamicalsystems revealing different forms of dynamics (cf. Fig. 7); for instance, theSaito system in the (quasi-)periodic regime is characterized by two types ofbehavior. One, related to the periodic component, and the other, producedby hysteresis. However, these dynamics are not hierarchically nested, whichleads to a set of isolated singularities, which in turn masquerade as a uniformmultifractal structure. Further synthetic signals revealed that the danger ofover-evaluating the multifractal content is not confined to the dynamics ofthis particular oscillator: quite on the contrary, the diversity of these signalspoints to a potentially pervasive problem.At the same time, for the genuine multifractals considered, such as the bi-nomial cascade, the MFDFA and WL methods reproduced remarkably closelythe explicitly-determined distribution of underlying H¨older exponents. Simi-larly, a good correspondence was found for the other cases of fully-developedmultifractality, such as the series of heart inter-beat intervals or sentencelength variability in “Finnegans Wake”. These methods, then, are obviously24ot per-se flawed, but their results need to be interpreted much more cau-tiously than is generally done.However, precise numerical determination of the singularity spectrum forexperimental signals knowingly remains an open problem [73]. Due to theintricate nature of the multifractal formalism, there are no theoretical proofsof the mathematical validity of the algorithms used to estimate the singu-larity spectrum: these algorithms are treated simply as reasonable numer-ical approximations of the underlying ground truth [74]. In particular, thecommonly used formulas for the multifractal spectrum have been proposedheuristically, through analogy with thermodynamic formalism, and verifiednumerically for specific cases based on multifractal measures [75]. Generally,the multifractal formalism yields the upper bound of the singularity spec-trum as was shown in [76], but not the exact spectrum. Therefore, it doesnot lend itself to formal proof. The present demonstration of what casesunder which the MFDFA and WL algorithms produce a flawed indicationof multifractality provides essential inspiration and input for further, moremathematically rigorous inquiry into the validity of these methods.It is important to point out that the observed disagreement between themultifractal spectra and the true distribution of the underlying H¨older expo-nents is not merely down to avoidable algorithmic implementation choices,but stems straight from the foundational assumptions upon which the mul-tifractal analyses under consideration are built. The MFDFA and WL ap-proaches represent two related advances of the Parisi-Frisch procedure [77],aiming for a compromise between reflecting the mathematical definition ofthe H¨older exponent and obtaining an estimate which is practically viablein terms of both computational load and stability. An averaging procedurerelated to the partition function (or its equivalent, cf. Eqs. (5) and (14)) andLegendre transform (cf. Eq. (15)) is central to their framework. Althoughnumerically stable, this methodology has one serious drawback. Namely, theLegendre transform by construction imposes the concavity of the multifrac-tal spectrum [73] a priori even when this is not true: if non-concave H¨oldercharacteristics are considered, it provides only the upper bound of the H¨olderspectrum [78]. Therefore, it is patently impossible to say without additionaltests whether an observed concave spectrum is a valid representation of thedata, or simply reflects the limitations of the methodology. To overcome thisproblem, multifractal methods not based on the Legendre transform haveto be applied [37, 79]. However, in many cases, these methods suffer fromissues of practical implementation and computational stability. Therefore, in25his work, the distribution of the H¨older exponents was instead directly ana-lyzed. In the presence of non-concave multifractal characteristics, it formedtwo clearly-separable clusters: this inexorably illuminates the true organiza-tion of the data and thus demonstrates the risk of faulty interpretation ofMFDFA- and WL-based results.To conclude, while these techniques offer very efficient and practical toolsfor quantifying the principal characteristics of multifractal patterns both intime and in space, extreme care needs to be taken when a suspicion arisesthat the pattern does not stem from true, uniform multifractality. The rea-son is that the methods are, by design, forced to assume a multifractal formfor the singularity spectrum. This aspect points to a definite necessity tobe addressed in future developments, namely, mitigating this strong priorassumption. For now, inspection of the distribution of the local H¨older ex-ponents as given by wavelet-based techniques is crucial in identifying suchinstances and classifying them correctly. Acknowledgement
L. Minati gratefully acknowledges funding by the World Research HubInitiative (WRHI), Institute of Innovative Research (IIR), Tokyo Instituteof Technology, Tokyo, Japan. L. Zunino was supported by the ArgentineanConsejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET).
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