Examples of truly multifractal time-series
For comparison with the particular cases considered below, representative instances of real multifractals having diverse properties are firstly presented, based on analyses conducted over the range \(q\in [-4,4]\) [61]. To this end, two mathematical multifractals are considered, namely the binomial cascade and the chaotic metronome derived from the Ikeda map [62], together with several real-world time-series: the inter-beat intervals extracted from electrocardiographic signals (103,885 data points), the sentence length variability of the “Finnegans Wake” book by James Joyce, the logarithmic returns of the American stock market index S&P500 (7440 data points), and the sunspot number variability (43,495 data points) [11, 21, 26, 35, 49]. In all these cases, the multifractal spectrum \(f(\alpha _\mathrm{G})\) assumes the shape of a wide inverted parabola, spanning \(\Delta \alpha _\mathrm{G}>0.2\), indicating a multifractal organization of the data (Fig. 1, left). Yet, the spectra develop different degrees of asymmetry. For the binomial cascade, the inter-beat intervals, and the sentence length variability, the spectra appear almost symmetrical (\(A_\alpha \approx 0\)), which suggests a homogeneous distribution of the correlations over small and large fluctuations. On the other hand, for stock market data and sunspot number variability, the asymmetry is, respectively, positively- and negatively-skewed. Thus, multifractality of the S&P500 price variation is mainly the effect of a complex organization of the large fluctuations, whereas the arrangement of small fluctuations is primarily responsible for multifractality in the time-series of sunspot numbers.
Importantly, the presence of true multifractality is confirmed, for all these cases, via analysis of the local scaling properties (Fig. 1, right). Therein, a continuous distribution of the estimated Hölder exponents spans a range of \(\alpha _\mathrm{L}\) even broader than in the multifractal spectrum, incidentally revealing the higher sensitivity of the wavelet transform on the local scaling properties compared to the global methodology, which mainly reflects the prevalent singularities in the time-series.
Artefactual multifractality in the Saito chaos generator
To illustrate the potential pitfalls inherent in drawing hasty conclusions solely from global measures, let us now consider the case of the Saito chaos generator, which is a four-dimensional non-linear oscillator consisting of the following dimensionless state equations [63]:
$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{x}}=-z-w\\ {\dot{y}}=\gamma (2\delta y+z)\\ {\dot{z}}=\rho (x-y),\\ {\dot{w}}=(x-h(w))/\varepsilon \end{array}\right. } \end{aligned}$$
(16)
wherein
$$\begin{aligned} h(w)= {\left\{ \begin{array}{ll} w-(1+\eta ) &{} \text {if } w \ge \eta \\ -\eta ^{-1}w &{} \text {if } |w| < \eta \quad \\ w+(1+\eta ) &{} \text {if } w \le -\eta . \end{array}\right. } \end{aligned}$$
(17)
Despite its simple form and low dimensionality, this system readily generates rich dynamics spanning periodicity, quasi-periodicity, chaos, and eventually hyperchaos as a function of the parameters \(\gamma \), \(\delta \), \(\varepsilon \), \(\eta \), and \(\rho \). Here, it was initially deemed of interest from the perspective of its hypothetical ability to generate signals having a truly multifractal structure; however, in the course of numerical investigation, another feature was realized to be fundamentally important for the purposes of the present work, namely, the presence of the hysteresis function h(w), which only enters the equation of the state variable w. As a consequence of it, even though all state variables conjointly participate in the temporal dynamics, \(x,\ y,\ z\) have rather smooth an activity, whereas, in the limit of \(\varepsilon \rightarrow 0\), the temporal evolution w is characterized by sudden jumps. As shall become clear, it is these discontinuities, namely the combination of slow and fast motions corresponding to the continuous manifold and sudden jumps, which may lead to a mistaken inference of multifractality. Unless indicated otherwise, the parameters were set for operation in the hyperchaotic regime, that is, \(\gamma =1\), \(\delta =0.94\), \(\varepsilon =0.01\), \(\eta =1\), and \(\rho =14\) [63, 64].
Preliminary examination revealed differences between short- and long-range temporal correlations in the simulated time-series, giving rise to a cross-over in the fluctuation functions. In particular, the multiscale characteristics revealed a strong autocorrelation only over short time scales (i.e., \(s<1000\)), occurring alongside a monofractal organization with weak linear correlations on the larger scales. Thus, in the analyses below, the focus is on the short-range correlations, which are relevant to the search for possible multifractality.
Numerical simulations
Time-series having a length of \(10^6\) points were simulated given Eqs. (16), (17), applying the adaptive step-size Runge–Kutta (4,5) method and returnig the results at a fixed step size of \(\Delta t=0.1\) [65]. All simulations were repeated 10 times with randomized initial conditions. Representative segments for each variable in the hyperchaotic regime are depicted in Fig. 2a. Evidently, the dynamics of \(x,\ y,\ z\) are characterized by the markedly irregular behavior characteristic of chaotic systems. However, the dynamics of w are even more complex, featuring sharp upward and downward jumps. Even though the underlying system is the same, the multifractal properties of the signals, being influenced by the presence of singularities, could then be partially dependent on the variable under consideration. This observation 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 apparent. Moreover, for the latter the majority of fluctuation functions have a slope close to those found close for the extreme values of q, i.e., \(q=\pm 4\); only a minority assume intermediate levels, a fact that already points to a more bifractal-like organization of the data rather than to a well-developed multifractal structure.
Strikingly, rather similar characteristics of \(F_q(s)\) are observed in the quasiperiodic regime, with \(\delta =0.65\) (Fig. 2c; for brevity, results are only shown for w). Though the dynamics are profoundly different compared to the hyperchaotic regime, the heterogeneity of the fluctuations functions remains most pronounced for w, with the distribution of slopes nearly unchanged and characteristic of a bifractal structure (Fig. 2d).
The multifractal analyses for the time-series of w generated as a function of the control parameter \(\delta \) are depicted in Fig. 3. The parameter was swept in \(\delta \in [0.6,1]\), thus allowing the system to develop a wide range of dynamical behaviors comprising both chaotic motions and closed orbits [63]. The corresponding averaged Hurst exponent H and multifractal spectrum width \(\Delta \alpha _\mathrm{G}\) as estimated through the MFDFA and WL algorithms are depicted in Fig. 3a. It is evident that the multifractal characteristics are insensitive to the qualitative features of the system dynamics. The time-series remain strongly persistent, with \(H \approx 1.5\), and feature a wide spectrum with \(\Delta \alpha _\mathrm{G} \approx 2.25\): this could, at the surface, suggest a multifractal organization. In Fig. 3b, the multifractal spectra for the hyperchaotic and quasiperiodic regimes are compared. Their shape is almost identical, with a strong left-sided asymmetry \(A_\alpha \approx 0.5\): importantly, this coexists with an uneven distribution of the points along the spectrum, which concentrate mainly towards its ends. Here, analysis of the local scaling revealed fundamental subtleties of the data organization. The relative frequency histogram \(f_r\) of the Hölder exponents \(\alpha _\mathrm{L}\) forms two separable peaks, whose locations coincide with high-concentration points close to the minimal and maximal values of \(\alpha _\mathrm{G}\) identified on the multifractal spectrum.
The locations of the singularities and their “strength” were recovered, as given by the Hölder exponents, through analysis of the local wavelet transform 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), the maxima form separate lines on the space-scale half-plane (cf. Fig. 4b), which delineate isolated singularities (Fig. 4c). In the presence of a truly multifractal geometry, the maxima would follow a tree-like structure, stemming from the self-similar organization of the fluctuations. By contrast, consideration of the locations and strength of the singularities reveals that two discrete types are present in these time-series, and related to volatile portions of the signal: one reflects instants wherein the hysteretic behavior is apparent (\(\alpha _\mathrm{L} \approx 1.2\)), the other reflects the local extrema of the oscillatory component (\(\alpha _\mathrm{L} \approx 3\)). Thus, a faithful reconstruction of the multifractal spectrum would clustered around two separate points. However, as discussed below, the q-filtering method inherently yields a concave spectrum, and isolated peaks are impossible to obtain. The artifactual result, then, is purely the product of the averaging procedures inherent in the MFDFA methodology: together with the dense sampling of the q parameter, these generate a broad spectrum of Hölder exponents, even when only isolated singularities are present in the signal. Although it is markedly stretched towards the left-hand side, with a high concentration of points towards the two limit values of Hölder exponent, the estimated spectrum resembles an inverted parabola. Instead, the scaling properties of the time-series generated by this system should be represented by a single exponent for the \(x,\ y,\ z\) variables, and by a bifractal organization for the w variable.
Experimental confirmation
To independently confirm that the results presented above stem faithfully from the dynamics of this system, an experimental version of the same was conveniently constructed using two operational amplifiers (type TL082) and a non-linearity based on two anti-parallel series Zener diodes (type BZT52-C5V1). The corresponding circuit diagram is given in Fig. 5a, where \(r_1=r_2=R_1=R_2=R=10\, \text {k}\Omega \), \(r_\mathrm{o}=820\, \Omega \), \(C_1=C_2=3.9\,\text {nF}\), \(L_0=3.3\,\text {mH}\), \(L=32\,\text {mH}\) (two inductors in series), and \(U_\mathrm{Z}=5.1\text { V}\). These component values yield \(\gamma =C_1/C_2=1\), \(\varepsilon =L_0/(r_1^2C_1)=0.0085\), \(\eta =r_1/r_2=1\), and \(\rho =r_1^2C_1/L=12.2\). The signal corresponding to the variable w was digitized from the physical circuit board (Fig. 5b) using a recording oscilloscope at a rate of 1 MSa/s, tuning \(g^{-1}\) to obtain \(\delta = \{0.63, 0.67, 0.71, 0.77, 0.83, 0.91, 1\}\). The corresponding time-series have been made publicly available [66].
In agreement with the simulations, apparent multifractality only arises for the variable w. In Fig. 6, the time-series in the quasiperiodic and hyperchaos regimes are shown alongside the corresponding fluctuation functions. The heterogeneity of the latter is equally apparent in both cases, suggesting that the multifractal properties of the signals are only weakly dependent on the dynamics. The multifractal spectra are shown in Fig. 7: they are wide (\(\Delta \alpha _\mathrm{G} > 2\)) but, in contrast to the simulations, more symmetric (\(A_{\alpha } \approx 0.3\)) (cf. Fig. 7b). On the other hand, analysis of the local scaling properties reveals a singularity organization comparable to the simulations (cf. Fig. 7c). Thus, the variability of the Hölder exponents for the experimental signals is higher; however, the histogram still forms two separable clusters (albeit more dispersed than in the simulations), wherein high values reflect local signal maxima and small ones correspond to the sudden jumps. A possible explanation for the more diverse distribution of the singularities could be sought in the tolerances and non-ideal behaviors of the electronic components (e.g., finite quality factor and self-resonance of the inductors, smooth response of the diodes, etc.), which knowingly give rise to richer dynamics. Altogether, these results confirm the above conclusions, reassuring that they are not an artifact of the numerical integration.
Artefactual multifractality in the Rössler system
The next example of dynamical system that we consider is the Rössler system. Its dynamics are governed by the following system of three differential equations [67]
$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{x}}=-y-z\\ {\dot{y}}=x+ay\\ {\dot{z}}=b+z(x-c).\\ \end{array}\right. } \end{aligned}$$
(18)
Similarly to the Saito generator, the Rössler system reveals rich dynamics spanning periodic and chaotic behaviors. Notably, its dynamical properties depend on state equations without a hysteresis element. They are controlled by parameters that were set to a=0.3, b=0.2, c=5.7, knowingly realizing chaotic behavior with an intermediate level of folding. To ensure statistical reliability, we generated time-series having a length of \(10^6\) points, a representative fragment of which is visible in Fig. 8a. The distribution of the fluctuation functions already suggests a bi-fractal organization (Fig. 8b) [68]. However, the multifractal spectra estimated through the MFDFA and WL algorithms resemble typical multifractal characteristics, with a strong left-sided asymmetry (Fig. 8c). This is in contrast with the local scaling properties (Fig. 8d). The histogram of local Hölder exponents forms, similarly to the Saito generator case, two separate peaks that concentrate in the vicinity of the extreme \(\alpha _\mathrm{G}\) values. Thus, the underlying structure reflects isolated singularities rather than a unified multifractal organization.
Further examples of artefactual multifractality in synthetic signals
The results presented above suggest that singular behavior in the Saito chaos generator can be quantified through just two scaling exponents. Moreover, the subsets corresponding to different singularities index separate components of the time-series, rather than constituting hierarchically-interwoven structures, which are the hallmark of true multifractality. Thus, a naive interpretation of the spectrum width \(\Delta \alpha _\mathrm{G}\) as a signature of multifractality can be faulty. To highlight this issue even more clearly, we finally consider processes that are, by construction, not multifractal. Yet, the methods based on q-filtering, namely the MFDFA and WL, yield misleadingly wide multifractal spectra in these cases.
As an instructive example, results from the multifractal analysis of the Lévy process, which possesses a well-recognized bifractal structure, are firstly presented. The multifractal spectrum of the Lévy time-series consists of two points, whose locations are directly related to the asymptotic behavior of the distribution tail \(P(x)\sim x^{-(\alpha _{\text {Levy}}+1)}\) and are given by [35, 69]:
$$\begin{aligned} \alpha = \left\{ \begin{array}{cc} 1/\alpha _\mathrm{Levy} &{} (q \le \alpha _\mathrm{Levy}) \\ 0 &{} (q> \alpha _\mathrm{Levy}) \end{array} \right. \ \ f(\alpha ) = \left\{ \begin{array}{cc} 1 &{} (q \le \alpha _\mathrm{Levy}) \\ 0 &{} (q > \alpha _\mathrm{Levy}) \end{array} \right. \end{aligned}$$
(19)
where \(\alpha _\mathrm{Levy}\) is the Lévy index and q is q-th moment of the fluctuation function \(F_q(s)\). The multifractal analysis of the Lévy time-series having a length of 50,000 points (Fig. 9a) with \(\alpha _\mathrm{Levy}=1.5\) is reported in Fig. 10a, b. Therein, the bifractal nature of the data is clearly visible in the histogram of Hölder exponents: two peaks, corresponding to the theoretical values, can be readily identified. The dispersion of these peaks is artifact purely due to the finite time-series length. Yet, the multifractal spectrum estimated utilizing the MFDFA and WL methods is wide (i.e., \(\Delta \alpha _\mathrm{G} = 0.7\)) and strongly left-sided asymmetrical (\(A_\alpha \approx 0.38\)): this could lead to the faulty conclusion that the process is multifractal when, in reality, it is not.
Next, a minimal-complexity arrangement which can reproduce qualitative characteristics similar to those observed in the Saito chaos generator is considered: it simply consists of the linear superposition w(t) of two related signals. One is a pseudo-periodic signal given by, e.g., \(u(t)=\sum _i\sin 2\omega t/(p_i/\max p)\) where \(p_i=\{2,3,5,7,11\}\). The other is a sequence of binary fluctuations \(v(t)={\mathscr {W}}[u(t),\xi ]\) generated by a hysteresis operator \({\mathscr {W}}\) acting on that signal, with \(v\in [-1,1]\) and hysteresis parameter \(\xi =0.2\max u\). Their linear combination, e.g., \(w(t)=v(t)+u(t)/\max u\) is, by definition, not hierarchically interwoven and does not obey different scaling exponents [70, 71], hence, the multifractal spectrum should consist of two separable points. To test this hypothesis, 10 time-series segments each having a lengths of \(10^6\) points were generated, and MFDFA was performed. The average of the multifractal spectra and the histogram of the Hölder exponents are depicted in Fig. 10c, d. In this case too, the multifractal spectrum appears well-developed (\(\Delta \alpha _\mathrm{G} = 1.5\)), with a strong left-sided asymmetry (\(A_\alpha = 0.37\)), hence cursory interpretation of these results might suggest a complex multifractal structure. Again, the true nature of the process is revealed by the histogram of the Hölder exponents estimated through the wavelets, which shows that only two discrete types of singularities are present in the time-series. Thus, the analyzed structure is closer to a fractal structure than to multifractality. It is worth noting that the distribution of the Hölder exponents as well as the shape of the multifractal spectrum resemble closely the results for the variable w in the Saito chaos generator; here, however, there was no underlying non-linear dynamical system.
Finally, a pseudo-multifractal process, consisting of the superposition of a fractal time-series with periodic components, is considered. At a first glance, this process resembles the multifractal time-series of sunspot number variability (cf. Fig. 1). However, as demonstrated below, careful inspection of the fractal characteristics illuminates its pseudo-multifractality. To this end, fractional Gaussian noise (fGn) [4] was generated: it represents a well-known example of a stochastic monofractal structure with possible long-range correlations quantified by the Hurst exponent, which has been applied to model phenomena across various fields of science. Namely, simulations produced a fractional Gaussian noise with an arbitrarily-chosen Hurst exponent of \(H=0.8\) (strongly persistent behavior), for which the amplitude of the process was modulated by a cosinusoidal function \(F(i)=A+A\cos (2\pi i/T_0)\) in \(i=1\ldots N\), where A and \(T_0\) are the model parameters (cf. Fig. 9c). Then, the periodic function F(i) was added to this amplitude modulated noise. In our simulations, \(N=10^6\), \(A=0.5\) and \(T_0=4000\) were set. The results of the local scaling analysis, as well as the multifractal spectrum, are shown in Fig. 10e, f. Analysis of the distribution of Hölder exponents confirms that the time-series is a composition of the two independent processes having different singular behaviors. The smaller values of \(\alpha _\mathrm{L}\) concentrate around \(\alpha _\mathrm{L} \approx H = 0.8\), corresponding to the fGn component, whereas the larger ones are related to the periodic trend. A cursory analysis of the multifractal spectrum indicates heterogeneity in the scaling properties. Particularly, the width of the spectrum, together with its strong right-sided asymmetry (\(A_\alpha =-0.85\)), could be taken as the hallmark of a multifractal time-series with a well-developed hierarchy of small fluctuations; however, this is purely an artifact. To avoid mistaking pseudo-multifractality as a valid multifractal structure, the analysis of the global fractal properties should always be corroborated by consideration of the local scaling properties as given by the wavelet transform.
As a last example, we analyzed an artefactually-generated stochastic process with the singularity spectrum derived analytically. In this respect, we considered the square transform of fractional Brownian motion (Fig. 11a), which represents a bi-Hölder process whose spectrum is given by the following relation [72]:
$$\begin{aligned} f(\alpha ) = \left\{ \begin{array}{lcl} 1 &{} \text {if} &{} \alpha = H \\ 1-H &{} \text {if} &{} \alpha =2H \\ -\infty &{} &{} \text {elsewhere}. \end{array} \right. \end{aligned}$$
(20)
In our study, we generated fBm having a length of \(10^6\) points with a Hurst exponent \(H=0.7\). Fluctuations functions \(F_q(s)\) (Fig. 11b) obtained through MFDFA show non-homogeneous scaling, which suggests a multiscaling behavior of the data. This is even more clearly visible in the singularity spectrum, which reproduces a concave hull supported by the interval in the range from H to 2H (Fig. 11c). Thus, based only on the Legendre-based methodology, a flawed conclusion on the multifractal structure of the data would be drawn. However, consideration of the histogram of Hölder exponents (Fig. 11d) estimated through wavelet analysis reveals the true bi-fractal nature of the process, with exponents corresponding to the theoretical expectations. This leads to the conclusion that the exponents identified through MFDFA and WL, except the two extreme values, are an artifact caused by a methodological limitation and do not contain any true information about the analysed process.