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Fractal and Multifractal Time Series

Jan W. Kantelhardt

arXiv:0804.0747v1physics.data-an

TL;DR

Fractal and multifractal scaling in complex-system time series requires methods that distinguish genuine dynamics from artifacts such as non-stationarity. This review presents analysis and modelling approaches, showing that fractal measures can characterize systems, detect transitions, and inform extreme-event risk estimation.

  • Problem

    Fractal scaling is not established a priori, while limited data lengths and non-stationarities complicate reliable identification of genuine fractal dynamics.

  • Method

    The review synthesizes Statistical Physics and Applied Mathematics methods for analyzing stationary and non-stationary fractal and multifractal time series, including DFA, WTMM, MF-DFA, and cascade models.

  • Results

    Fractal analysis can characterize complex systems, detect transitions and system states, support model improvement, and estimate risk from clustered extreme events.

  • Takeaways & Limitations

    Fractal and multifractal analysis provides tools for characterizing dynamics, comparing systems and models, and studying extreme-event timing.

  • Takeaways & Limitations

    Practically usable models displaying fractal or transient fractal scaling still need to be developed for many applications.

Abstract

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Data series generated by complex systems exhibit fluctuations on many time scales and/or broad distributions of the values. In both equilibrium and non-equilibrium situations, the natural fluctuations are often found to follow a scaling relation over several orders of magnitude, allowing for a characterisation of the data and the generating complex system by fractal (or multifractal) scaling exponents. In addition, fractal and multifractal approaches can be used for modelling time series and deriving predictions regarding extreme events. This review article describes and exemplifies several methods originating from Statistical Physics and Applied Mathematics, which have been used for fractal and multifractal time series analysis.

Glossary

The glossary defines time series and scaling laws as foundations for describing fractal and multifractal systems. It distinguishes self-affinity, multifractality, crossovers, persistence, correlation ranges, and non-stationarity.

  • Core definitions: A time series is a one-dimensional array of values (x_i), usually measured at equidistant or nearly equidistant times.The series contains N observations indexed by i = 1, . . . , N.
  • Scaling and fractality: A scaling law describes a quantity F as a power of scale s, F(s) ∼s^α, over a large range of scales.The range should cover at least one order of magnitude of s values.
  • Scaling and fractality: Fractal systems have non-integer scaling exponents and statistical self-similarity, whereas self-affine systems use direction-dependent magnifications and the Hurst exponent H.Self-affine time series are commonly called fractal under a less strict terminology.
  • Scaling and fractality: Multifractal systems exhibit infinitely many different fractal exponents whose scaling laws hold over the same scale range.This definition extends the single-exponent characterization of fractal systems.
  • Temporal structure: A crossover changes the applicable scaling exponent between small and large scales, while persistence and correlation terms distinguish temporal dependence by scale and decay rate.Short-term correlations have a characteristic correlation time, whereas long-term correlations decay slowly and show asymptotic power-law scaling; non-stationarity refers to time-varying distributional properties or system dynamics.

1 Definition of the Subject and Its Importance

Complex-system time series often show multiscale fluctuations and broad value distributions governed by scaling laws, which fractal or multifractal exponents characterize. The review presents methods for detecting genuine scaling while distinguishing non-stationarity artifacts and short- from long-term correlations.

  • Definition of the Subject and Its Importance: Complex-system data exhibit fluctuations across wide time scales and/or broad value distributions, often following scaling relations over several orders of magnitude.These scaling laws characterize both the data and the generating system through fractal or multifractal scaling exponents.
  • Definition of the Subject and Its Importance: Several tools have been developed to observe fractal and multifractal scaling in time series beyond older methods that assume stationarity.Recent methods distinguish genuine fractal dynamics from apparent scaling caused by non-stationarities.
  • Definition of the Subject and Its Importance: Unambiguous fractal-scaling analysis requires distinguishing short-term from long-term correlations.The review describes methods originating in Statistical Physics and Applied Mathematics for this analysis.

2 Introduction

Complex systems produce time series with fluctuations across many time scales and broad value distributions, motivating fractal and multifractal analysis. The section introduces applications, predictive uses, methodological challenges, and the review’s organization around stationary, non-stationary, and multifractal data.

  • Long time series of selected observables provide dimensionally reduced representations for studying complex systems whose dynamics cannot be decomposed without altering their properties.
  • Fractal and multifractal scaling has been reported across geophysical, medical, physiological, DNA, astrophysical, technical, social, and physics data.
  • Identifying characteristic scaling exponents can support predictions of future behaviour and responses to perturbations, while changes in fractal dynamics can indicate phase transitions in regulation.
  • Fractal scaling must be established rather than assumed, requiring refined methods that distinguish genuine dynamics from apparent scaling caused by non-stationarity.
  • The review covers properties and primary quantities, methods for stationary and non-stationary self-affine data, and techniques for multifractal time series.

3 Fractal and Multifractal Time Series

The section introduces self-affine and multifractal descriptions of time series, emphasizing scaling exponents, persistence, and the need to distinguish crossovers from genuine multifractality. It explains how Hurst scaling characterizes correlations across time scales.

  • Self-affinity and Hurst scaling: Self-affinity accounts for unequal scaling of time and measured values, with the Hurst exponent H characterizing the scaling relation.The framework differs from fractal-dimension analysis because the time and value axes are not equivalent.
  • Self-affinity and Hurst scaling: H = 0.5 characterizes a random-walk trace, so rescaling time by 4 requires rescaling position by 2.The Hurst exponent can also be retrieved from mean-square fluctuations following ⟨x2(t)⟩∼t2H.
  • Persistence and correlations: Self-affine series exhibit persistence, while their increments may be persistent, independent, or anti-persistent.Increment correlations can be assessed using auto-covariance or auto-correlation functions; an AR example has t× = 48.
  • Crossovers and non-stationarity: H = 0.5 describes the asymptotic scaling of integrated series with short-range correlated increments after correlations decay, despite H > 0.5 at small scales.Finite correlation decay times produce crossovers, so such series lack a unique Hurst exponent across all scales.
  • Multifractality: Crossovers must not be confused with multifractality: multifractality requires different moment scaling over the full range of time scales.Crossovers use different exponents in different regimes, whereas multifractality requires many exponents within the same scale range.

4 Methods for Stationary Fractal Time Series Analysis

This section presents four traditional methods for stationary fractal time series, focusing on estimating scaling exponents H or γ and their relation in long-term persistent data. It also outlines the assumptions, procedures, and limitations of autocorrelation, spectral, rescaled-range, and fluctuation analyses.

  • Overview: Four traditional approaches estimate the scaling exponents H or γ for stationary time series, with H = 1−γ/2 in long-term persistent data.Methods addressing non-stationarities are deferred to the next chapter.
  • Autocorrelation analysis: Autocorrelation analysis classifies short-term correlations by exponential decay and long-term correlations by power-law decay C(s) ∝s−γ with 0 < γ < 1.Direct covariance calculation is often unsuitable because noise and unknown non-stationarities obscure the data.
  • Spectral analysis: Spectral analysis fits a power law to the power spectrum S(f) on a double logarithmic plot to obtain the spectral exponent β and correlation exponent γ.Logarithmic binning is required for reliable results, and the data must be stationary.
  • Rescaled range analysis: Hurst’s rescaled range analysis divides the series into segments, removes local averages, and averages segment ranges rescaled by standard deviations to estimate H.It produces smoother curves without binning and also works with piecewise constant trends; H relates to β and γ through 2H ≈1 + β = 2 −γ under stated conditions.
  • Fluctuation analysis: Standard fluctuation analysis integrates the series into a profile and measures segment fluctuations, yielding F2(s) ∼s1/2 for uncorrelated values and power-law growth for long-term correlations.For multifractal data, the fluctuation exponent α need not equal H; reliable analysis requires s < N/10, with 0 < α < 1.

5 Methods for Non-Stationary Fractal Time-Series Analysis

Wavelet-based and detrended fluctuation methods characterize non-stationary fractal time series by scale-dependent fluctuations while removing polynomial trends. Their accuracy and crossover detection depend on detrending order, record length, and method-specific finite-scale effects.

  • Wavelet analysis: Wavelet coefficients resolve local frequency content by depending jointly on time position τ and scale s, with frequency f = 1/s.Daughter wavelets are generated by shifting and stretching a zero-mean mother wavelet; wavelets may be chosen orthogonal to polynomial trends.
  • Wavelet analysis: Wavelet detrending WTn estimates fluctuations from the nth derivative, eliminating trends described by (n −1)st-order polynomials; WT0 corresponds to FA and WT1 resembles rescaled-range analysis.WT1 eliminates linear profile trends and constant data trends, with α ≈ H up to 2.
  • Detrended fluctuation analysis: DFA exhibits intrinsic small-scale deviations from asymptotic scaling, limiting reliable correlation estimates in short records and at small s, especially for large detrending order m.DFA6 is only defined for s ≥8, and DFA uses smax = N/4 compared with smax = N/10 for FA and WT.
  • Detrending and crossovers: 1.6, 2.6, 3.6, 4.5, and 5.4 are the simulated ratios s×(m)/s× for DFA1 through DFA5, respectively, with an error bar of approximately 0.1.The precise ratio depends on how crossover times are fitted, and higher detrending orders produce more significant systematic deviations.
  • Detrending and crossovers: Testing several polynomial detrending orders distinguishes trend-induced crossovers from real crossovers: increasing m moves the former to larger scales or removes it, while its maximum slope is m + 1.If m is too low, F2(s) develops a pronounced large-scale crossover with increased slope; real crossovers retain identical slopes α.
  • Alternative detrending methods: CMA scaling is reported as more stable than DFA1 and MDFA1, potentially enabling reliable α estimates for s < 10 without correction and up to smax = N/2.MDFA detects crossovers more exactly than DFA but retains boundary jumps and a maximum usable scale s < N/4; overlapping polynomial fits could avoid jumps but are time consuming.

6 Methods for Multifractal Time Series Analysis

Multifractal time-series analysis extends partition-function methods through scaling exponents, generalized Hurst exponents, and singularity spectra. WTMM handles non-stationarities, while MF-DFA offers a simpler, slightly more reliable alternative with support and scale limitations.

  • Standard partition-function multifractal formalism characterizes normalized stationary measures but gives incorrect results for non-stationary or non-normalizable series.
  • The multifractal framework uses τ(q), D(q), h(q), and f(α) to describe scaling across moments and singularity strengths.A series is monofractal when τ(q) is linear in q; otherwise it is multifractal, with D(q) = τ(q)/q − 1.
  • Wavelet Transform Modulus Maxima (WTMM): WTMM analyzes wavelet-transform maxima to investigate multifractal scaling under non-stationarities, and its exponents ˆτ(q) theoretically equal τ(q).The maxima procedure prevents near-zero wavelet coefficients from spoiling negative-moment calculations.
  • Multifractal Detrended Fluctuation Analysis (MF-DFA): MF-DFA integrates the series, segments its profile, detrends each segment, and calculates variances before estimating h(q).For large scales s > N/4, too few segments make Fq(s) unreliable; systematic deviations can also occur near s ≈ 10.
  • Method comparison and accuracy: MF-DFA is slightly more reliable than WTMM, especially for negative q and short series, but it is simpler and requires full one-dimensional support.For q = ±10, expected standard deviations reach approximately ±0.1 for N = 10 000 and ±0.05 for N = 100 000.

7 Statistics of Extreme Events in Fractal Time Series

Return-interval statistics characterize temporal scaling and support risk estimation for hazardous extremes. Long-term correlations cluster rare events, alter return-interval distributions and dependencies, and affect extreme-value convergence and prediction.

  • 7 Statistics of Extreme Events in Fractal Time Series: Return intervals between threshold exceedances characterize temporal scaling and help estimate risks of floods, very high temperatures, and earthquakes.Long-term correlations provide a natural mechanism for clustering hazardous events.
  • 7 Statistics of Extreme Events in Fractal Time Series: The return-interval distribution differs from the uncorrelated case, with long-term correlations making intervals far below and above the mean more frequent.The stretched-exponential law has deviations at very small intervals from discretization and additional power-law effects, and at very large intervals from finite-size effects.
  • 7 Statistics of Extreme Events in Fractal Time Series: Return intervals are long-term power-law correlated with the original record’s exponent γ, so large and small intervals form clusters.The probability of a return interval depends on the preceding interval, which matters for prediction and risk estimation.
  • 7 Statistics of Extreme Events in Fractal Time Series: Conditional return-interval distributions yield expected waiting times that depend on the preceding interval and elapsed time, unlike uncorrelated records.For uncorrelated records, τq(x|r0)/Rq = 1 apart from discreteness effects; correlated data instead show scaling with r0/Rq and x/Rq.
  • 7 Statistics of Extreme Events in Fractal Time Series: The original data distribution affects convergence toward the Gumbel distribution much more strongly than long-term correlations, which only slightly delay convergence.For i.i.d. Gaussian or exponential data, maxima converge to the Fisher-Tippett-Gumbel distribution as R →∞.
  • 7 Statistics of Extreme Events in Fractal Time Series: Maxima retain long-term correlations and depend on history, especially the previous maximum, motivating conditional maxima distributions for improved predictions.Both the maxima series and their distribution reflect dependence beyond the independent-value assumptions of classical extreme-value statistics.

8 Simple Models for Fractal and Multifractal Time Series

The section presents simple models for generating fractal, multifractal, and broad-distribution time series. Fourier filtering controls long-term correlations and scaling exponents, while cascade and bi-fractal constructions model multifractal behavior with explicit limitations and parameterizations.

  • Fourier-filtering technique: Fourier filtering generates long-term-correlated time series with power-law autocorrelation C(s) ∼ x^−γ for 0 < γ < 1.It can realize scaling exponents α = h(2) ≈ H or β = 2α − 1, including α < 1.
  • Fourier-filtering technique: Fourier filtering always produces Gaussian-distributed values and cannot generate nonlinear or multifractal properties.Modified Fourier filtering using Bessel functions avoids divergence of C(s) at s = 0.
  • Preserving broad distributions: The Schreiber–Schmitz iterative algorithm combines Fourier-filtered correlations with a prescribed broad distribution by repeatedly adjusting spectra and rank-ordering values.Fourier filtering first creates a Gaussian correlated reference, while rank replacement enforces the desired distribution and slightly alters correlations, requiring repetition.
  • Multifractal cascade model: The multiplicative cascade recursively constructs records of length N = 2^nmax by multiplying successive subdivisions by factors a and b.Its scaling function is τ(q) = [−ln(a^q + b^q) + q ln(a + b)]/ln 2.
  • Multifractal cascade model: Because h(1) = 1 for all a and b, the basic cascade is limited to cases where the Hurst exponent equals one.Subtracting ∆h = ln(a+b)/ln(2) from h(q), followed by power-spectrum rescaling, generalizes the process to arbitrary h(1).
  • Bi-fractal model: A bi-fractal model can represent apparently multifractal data using two slopes α1 and α2, with h(q) exhibiting a plateau before hyperbolic decay.Its spectrum is characterized by three parameters and has width ∆α = α1 − α2.

9 Future Directions

Future research should determine the causes and practical consequences of fractal and multifractal scaling while extending analysis to broader data types and inter-series relationships.

  • 9 Future Directions: Future studies should analyse additional complex-system time series, including data indexed by other parameters, higher-dimensional data, and spatially fractal structures.This would test for fractal scaling, particularly long-term correlations, across broader data types.
  • 9 Future Directions: The causes of observed fractal or multifractal scaling remain unclear, motivating comparisons between real and modelled time series to improve models.Fractal or multifractal characterisation may help assess and improve models.
  • 9 Future Directions: Improved methods are needed to characterise linear and nonlinear cross-correlations among several non-stationary time series, since most available methods target stationary data.Natural recordings are described as rarely stationary.
  • 9 Future Directions: Practical research should examine predictions of future values and behaviour in time series and complex systems, especially for hydrology, climate research, and online medical-event prediction.Distinguishing trends from natural fluctuations is crucial in hydrology and climate research, while continuous recordings can support online prediction of dangerous medical events.

F(s) s 1/2

The section illustrates wavelet, DFA, WTMM, and MF-DFA methods for analyzing fractal and multifractal time series, alongside definitions and distributions of return intervals and block maxima. The examples include correlated and uncorrelated data, a binomial multifractal model, and extreme-event statistics.

  • Wavelet and DFA: Wavelet analysis compares uncorrelated, long-term correlated, and short-term correlated data after dividing averaged F2(s) by s1/2.The analysis averages over 20 series with N = 216 points; a horizontal line indicates uncorrelated behaviour.
  • Wavelet and DFA: DFA is applied to the same data examined with the discrete wavelet transform.The figure presents a comparative application of Detrended Fluctuation Analysis to the datasets from Fig. 5.
  • Wavelet and DFA: WTMM is illustrated through the original data, its continuous wavelet transform, and extracted maxima lines.The example displays wavelet-coefficient amplitudes and the maxima lines used by the method.
  • Multifractal and extreme-event analysis: MF-DFA of a binomial multifractal model with a = 0.75 relates the slopes of Fq(s) versus s to h(q).The figure averages 100 configurations and includes dashed theoretical slopes h(±∞) from Eq. (42).
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