Multifractal Noise Research Articles

Multifractal perturbations to multiplicative cascades promote multifractal nonlinearity with asymmetric spectra.

Biological and psychological processes have been conceptualized as emerging from intricate multiplicative interactions among component processes across various spatial and temporal scales. Among the statistical models employed to approximate these intricate nonlinear interactions across scales, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, several fundamental questions persist concerning the proper interpretations of multifractal evidence of nonlinear cross-scale interactivity. Does multifractal spectrum width depend on multiplicative interactions, constituent noise processes participating in those interactions, or both? We conducted numerical simulations of cascade time series featuring component noise processes characterizing a range of nonlinear temporal correlations: nonlinearly multifractal, linearly multifractal (obtained via the iterative amplitude adjusted wavelet transform of nonlinearly multifractal), phase-randomized linearity (obtained via the iterative amplitude adjustment Fourier transform of nonlinearly multifractal), and phase and amplitude randomized (obtained via shuffling of nonlinearly multifractal). Our findings show that the multiplicative interactions coordinate with the nonlinear temporal correlations of noise components to dictate emergent multifractal properties. Multiplicative cascades with stronger nonlinear temporal correlations make multifractal spectra more asymmetric with wider left sides. However, when considering multifractal spectral differences between the original and surrogate time series, even multiplicative cascades produce multifractality greater than in surrogate time series, even with linearized multifractal noise components. In contrast, additivity among component processes leads to a linear outcome. These findings provide a robust framework for generating multifractal expectations for biological and psychological models in which cascade dynamics flow from one part of an organism to another.

Impact of Healthy Aging on Multifractal Hemodynamic Fluctuations in the Human Prefrontal Cortex.

Fluctuations in resting-state cerebral hemodynamics show scale-free behavior over two distinct scaling ranges. Changes in such bimodal (multi) fractal pattern give insight to altered cerebrovascular or neural function. Our main goal was to assess the distribution of local scale-free properties characterizing cerebral hemodynamics and to disentangle the influence of aging on these multifractal parameters. To this end, we obtained extended resting-state records (N = 214) of oxyhemoglobin (HbO), deoxyhemoglobin (HbR) and total hemoglobin (HbT) concentration time series with continuous-wave near-infrared spectroscopy technology from the brain cortex. 52 healthy volunteers were enrolled in this study: 24 young (30.6 ± 8.2 years), and 28 elderly (60.5 ± 12.0 years) subjects. Using screening tests on power-law, multifractal noise, and shuffled data sets we evaluated the presence of true multifractal hemodynamics reflecting long-range correlation (LRC). Subsequently, scaling-range adaptive bimodal signal summation conversion (SSC) was performed based on standard deviation (σ) of signal windows across a range of temporal scales (s). Building on moments of different order (q) of the measure, σ(s), multifractal SSC yielded generalized Hurst exponent function, H(q), and singularity spectrum, D(h) separately for a fast and slow component (the latter dominating the highest temporal scales). Parameters were calculated reflecting the estimated measure at s = N (focus), degree of LRC [Hurst exponent, H(2) and maximal Hölder exponent, hmax] and measuring strength of multifractality [full-width-half-maximum of D(h) and ΔH15 = H(−15)−H(15)]. Correlation-based signal improvement (CBSI) enhanced our signal in terms of interpreting changes due to neural activity or local/systemic hemodynamic influences. We characterized the HbO-HbR relationship with the aid of fractal scale-wise correlation coefficient, rσ(s) and SSC-based multifractal covariance analysis. In the majority of subjects, cerebral hemodynamic fluctuations proved bimodal multifractal. In case of slow component of raw HbT, hmax, and Ĥ(2) were lower in the young group explained by a significantly increased rσ(s) among elderly at high temporal scales. Regarding the fast component of CBSI-pretreated HbT and that of HbO-HbR covariance, hmax, and focus were decreased in the elderly group. These observations suggest an attenuation of neurovascular coupling reflected by a decreased autocorrelation of the neuronal component concomitant with an accompanying increased autocorrelation of the non-neuronal component in the elderly group.

Discerning a linkage between solar wind turbulence and ionospheric dissipation by a method of confined multifractal motions

[1] We develop a theory of confined (mean reverting) multifractal processes in terms of stochastic equations with multifractal noise and varying drift terms. Within this framework we are able to describe the auroral electrojet index and the z component of the interplanetary magnetic field by stochastic equations with identical multifractal noise terms. This suggests the existence of a physical mechanism that links the intermittency of solar wind turbulence and the corresponding intermittency of fluctuations in the ionospheric dissipation rate on time scales 1–100 min.

Short‐term forecasting of surface layer wind speed using a continuous random cascade model

AbstractThis paper describes a statistical method for short‐term forecasting (1–12 h ahead) of surface layer wind speed using only recent observations, relying on the notion of continuous cascades. Inspired by recent empirical findings that suggest the existence of some cascading process in the mesoscale range, we consider that wind speed can be described by a seasonal component and a fluctuating part represented by a ‘multifractal noise’ associated with a random cascade. Performances of our model are tested on hourly wind speed series gathered at various locations in Corsica (France) and the Netherlands. The obtained results show that a better modeling of the noise term based on cascade process enhances the forecast; furthermore, there is a systematic improvement in the prediction as compared with reference models. Copyright © 2011 John Wiley & Sons, Ltd.

Multifractal analyses of hydraulic conductivity distributions

The concept of a universal multifractal, a generalization of a monofractal, is a recently developed scaling model for natural phenomena characterized by irregularity. Presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity K, which have a significant effect on contaminant transport in the subsurface. Structure function analyses of four K data sets show that some vertical variations of K display multifractal structures, while others are consistent with monofractal behavior. In order to make multifractal concepts more useful, multifractal noise is introduced and defined as the increments of a multifractal. It is concluded that the multifractal formalism of Schertzer and Lovejoy [1987] has provided a rather general approach for modeling In K variations in the vertical. With the exception of horizontal variations of the Borden data, all results fell within the domain of universal multifractal behavior, which includes the monofractal case. Parameters were well‐defined in an empirical sense and easy to calculate, indicating a robust formalism. Results were consistent with the recent finding of Liu and Molz [1997, also Non‐Gaussian and scale‐variant behavior in hydraulic conductivity distributions, submitted to Water Resources Research, 1997, hereinafter referred to as submitted paper] that K variations display increasing heterogeneity at decreasing scales.