Biases in the Simulation and Analysis of Fractal Processes

A guide to Whittle maximum likelihood estimator in MATLAB

The assessment of physiological complexity via the estimation of monofractal exponents or multifractal spectra of biological signals is a recent field of research that allows detection of relevant and original information for health, learning, or autonomy preservation. This tutorial aims at introducing Whittle's maximum likelihood estimator (MLE) that estimates the monofractal exponent of time series. After introducing Whittle's maximum likelihood estimator and presenting each of the steps leading to the construction of the algorithm, this tutorial discusses the performance of this estimator by comparing it to the widely used detrended fluctuation analysis (DFA). The objective of this tutorial is to propose to the reader an alternative monofractal estimation method, which has the advantage of being simple to implement, and whose high accuracy allows the analysis of shorter time series than those classically used with other monofractal analysis methods.

Considerations for Applying Entropy Methods to Temporally Correlated Stochastic Datasets

The goal of this paper is to highlight considerations and provide recommendations for analytical issues that arise when applying entropy methods, specifically Sample Entropy (SampEn), to temporally correlated stochastic datasets, which are representative of a broad range of biomechanical and physiological variables. To simulate a variety of processes encountered in biomechanical applications, autoregressive fractionally integrated moving averaged (ARFIMA) models were used to produce temporally correlated data spanning the fractional Gaussian noise/fractional Brownian motion model. We then applied ARFIMA modeling and SampEn to the datasets to quantify the temporal correlations and regularity of the simulated datasets. We demonstrate the use of ARFIMA modeling for estimating temporal correlation properties and classifying stochastic datasets as stationary or nonstationary. We then leverage ARFIMA modeling to improve the effectiveness of data cleaning procedures and mitigate the influence of outliers on SampEn estimates. We also emphasize the limitations of SampEn to distinguish among stochastic datasets and suggest the use of complementary measures to better characterize the dynamics of biomechanical variables. Finally, we demonstrate that parameter normalization is not an effective procedure for increasing the interoperability of SampEn estimates, at least not for entirely stochastic datasets.

Better than DFA? A Bayesian Method for Estimating the Hurst Exponent in Behavioral Sciences

Detrended Fluctuation Analysis (DFA) is the most popular fractal analytical technique used to evaluate the strength of long-range correlations in empirical time series in terms of the Hurst exponent, H. Specifically, DFA quantifies the linear regression slope in log-log coordinates representing the relationship between the time series' variability and the number of timescales over which this variability is computed. We compared the performance of two methods of fractal analysis-the current gold standard, DFA, and a Bayesian method that is not currently well-known in behavioral sciences: the Hurst-Kolmogorov (HK) method-in estimating the Hurst exponent of synthetic and empirical time series. Simulations demonstrate that the HK method consistently outperforms DFA in three important ways. The HK method: (i) accurately assesses long-range correlations when the measurement time series is short, (ii) shows minimal dispersion about the central tendency, and (iii) yields a point estimate that does not depend on the length of the measurement time series or its underlying Hurst exponent. Comparing the two methods using empirical time series from multiple settings further supports these findings. We conclude that applying DFA to synthetic time series and empirical time series during brief trials is unreliable and encourage the systematic application of the HK method to assess the Hurst exponent of empirical time series in behavioral sciences.

Temperature and precipitation in the US states: long memory, persistence, and time trend

This paper investigates the time series properties of the temperature and precipitation anomalies in the contiguous USA by using fractional differentiation. This methodology allows to capture time trend components along with properties such as long-range dependence and the degree of persistence. For aggregated data, we find out that long memory is present in both precipitation and temperature since the integration order is significantly positive in the two cases. The time trend is also positive, being higher for the temperature. In addition, observing disaggregated data by states, for the temperature, there are only seven states where the time trend is not significant, with most of them located in Southeast areas, while for the rest of cases, the time trend is significantly positive. All cases exhibit long-range dependence, though the differencing parameter substantially changes from one state to another, ranging from 0.09 in Nebraska and Kansas to 0.18 in Florida and Michigan. For precipitation, the time trend is insignificant in a large number of cases, and the integration order is smaller than for the temperature. In fact, short memory cannot be rejected in fourteen states, and the highest orders of differencing are obtained in Arizona (d = 0.11) and Texas (0.12). In general, we highlight that one cannot draw conclusions about persistence and trends in these two climate-related variables based on aggregate information of the overall USA, given widespread heterogeneity across the states. Tentatively, the degree of dependence across the states seems to be negatively correlated with their level of climate-related risks and the associated preparedness in terms of handling climate change, but this conclusion requires more elaborate research in the future.

Self-potential time series reveal emergent behavior in soil organic matter dynamics

The active cycling of carbon between soil organic matter and the atmosphere is of critical importance to global climate change. An extensive body of research exists documenting the capricious nature of soil organic matter (SOM) dynamics, which is symptomatic of an intricate network of interactions between diverse groups of heterotrophic microorganisms, complex organic substrates, and highly variable local environmental conditions. These attributes are consistent with elements of complex system theory and the temporal evolution of otherwise unpredictable patterns of behavior that emerge from long range dependency on initial conditions. Here we show that vertical depth profile of self-potential (SP) time series measurements responds in a quantitative manner to variations in soil moisture, SOM concentrations, and relative rates of microbial activity. Application of detrended fluctuation analysis (DFA) of self potential time series data is shown additionally to reveal the presence of long-range dependence and emergence of anomalous electrochemical diffusion behavior, both of which diminish with depth as SOM specific energy densities decline.

Entropy fluctuation and correlation transfer in tunable discrete-time quantum walk with fractional Gaussian noise

We study the time correlation in the von Neumann entropy fluctuation of the tunable discrete-time quantum walk in one dimension, induced by the coin disorder arising from the temporal fractional Gaussian noise (fGn). The fGn is characterized by the Hurst exponent H, which provides three different correlation scenarios, namely antipersistent (0<H<0.5), memoryless (H=0.5), and persistent (0.5<H<1). We show the correlation of fGn is transferred to the coin's degree of entanglement and eventually transpires in the time correlation of the von Neumann entropy fluctuation. This study hints at the potential of using noise correlation as a resource to sustain information backflow via the interaction of quantum system with the noisy environment.

Fast Hand Movements Unveil Multifractal Roots of Adaptation in the Visuomotor Cognitive System

Beyond apparent simplicity, visuomotor dexterity actually requires the coordination of multiple interactions across a complex system that links the brain, the body and the environment. Recent research suggests that a better understanding of how perceptive, cognitive and motor activities cohere to form executive control could be gained from multifractal formalisms applied to movement behavior. Rather than a central executive "talking" to encapsuled components, the multifractal intuition suggests that eye-hand coordination arises from multiplicative cascade dynamics across temporal scales of activity within the whole system, which is reflected in movement time series. Here we examined hand movements of sport students performing a visuomotor task in virtual reality (VR). The task involved hitting spatially arranged targets that lit up on a virtual board under critical time pressure. Three conditions were compared where the visual search field changed: whole board (Standard), half-board lower view field (LVF) and upper view field (UVF). Densely sampled (90 Hz) time series of hand motions captured by VR controllers were analyzed by a focus-based multifractal detrended fluctuation analysis (DFA). Multiplicative rather than additive interactions across temporal scales were evidenced by testing comparatively phase-randomized surrogates of experimental series, which confirmed nonlinear processes. As main results, it was demonstrated that: (i) the degree of multifractality in hand motion behavior was minimal in LVF, a familiar visual search field where subjects correlatively reached their best visuomotor response times (RTs); (ii) multifractality increased in the less familiar UVF, but interestingly only for the non-dominant hand; and (iii) multifractality increased further in Standard, for both hands indifferently; in Standard, the maximal expansion of the visual search field imposed the highest demand as evidenced by the worst visuomotor RTs. Our observations advocate for visuomotor dexterity best described by multiplicative cascades dynamics and a system-wide distributed control rather than a central executive. More importantly, multifractal metrics obtained from hand movements behavior, beyond the confines of the brain, offer a window on the fine organization of control architecture, with high sensitivity to hand-related control behavior under specific constraints. Appealing applications may be found in movement learning/rehabilitation, e.g., in hemineglect people, stroke patients, maturing children or athletes.

Multifractal Dynamics in Executive Control When Adapting to Concurrent Motor Tasks

There is some evidence that an improved understanding of executive control in the human movement system could be gained from explorations based on scale-free, fractal analysis of cyclic motor time series. Such analyses capture non-linear fractal dynamics in temporal fluctuations of motor instances that are believed to reflect how executive control enlist a coordination of multiple interactions across temporal scales between the brain, the body and the task environment, an essential architecture for adaptation. Here by recruiting elite rugby players with high motor skills and submitting them to the execution of rhythmic motor tasks involving legs and arms concurrently, the main attempt was to build on the multifractal formalism of movement control to show a marginal need of effective adaptation in concurrent tasks, and a preserved adaptability despite complexified motor execution. The present study applied a multifractal analytical approach to experimental time series and added surrogate data testing based on shuffled, ARFIMA, Davies&Harte and phase-randomized surrogates, for assessing scale-free behavior in repeated motor time series obtained while combining cycling with finger tapping and with circling. Single-tasking was analyzed comparatively. A focus-based multifractal-DFA approach provided Hurst exponents (H) of individual time series over a range of statistical moments H(q), q = [−15 15]. H(2) quantified monofractality and H(-15)-H(15) provided an index of multifractality. Despite concurrent tasking, participants showed great capacity to keep the target rhythm. Surrogate data testing showed reasonable reliability in using multifractal formalism to decipher movement control behavior. The global (i.e., monofractal) behavior in single-tasks did not change when adapting to dual-task. Multifractality dominated in cycling and did not change when cycling was challenged by upper limb movements. Likewise, tapping and circling behaviors were preserved despite concurrent cycling. It is concluded that the coordinated executive control when adapting to dual-motor tasking is not modified in people having developed great motor skills through physical training. Executive control likely emerged from multiplicative interactions across temporal scales which puts emphasis on multifractal approaches of the movement system to get critical cues on adaptation. Extending such analyses to less skilled people is appealing in the context of exploring healthy and diseased movement systems.

Immersive Virtual Reality to Restore Natural Long-Range Autocorrelations in Parkinson's Disease Patients' Gait During Treadmill Walking

Effects of treadmill walking on Parkinson’s disease (PD) patients’ spatiotemporal gait parameters and stride duration variability, in terms of magnitude [coefficient of variation (CV)] and temporal organization [long range autocorrelations (LRA)], are known. Conversely, effects on PD gait of adding an optic flow during treadmill walking using a virtual reality headset, to get closer to an ecological walk, is unknown. This pilot study aimed to compare PD gait during three conditions: Overground Walking (OW), Treadmill Walking (TW), and immersive Virtual Reality on Treadmill Walking (iVRTW). Ten PD patients completed the three conditions at a comfortable speed. iVRTW consisted in walking at the same speed as TW while wearing a virtual reality headset reproducing an optic flow. Gait parameters assessed were: speed, step length, cadence, magnitude (CV) and temporal organization (evenly spaced averaged Detrended Fluctuation Analysis, α exponent) of stride duration variability. Motion sickness was assessed after TW and iVRTW using the Simulator Sickness Questionnaire (SSQ). Step length was greater (p = 0.008) and cadence lower (p = 0.009) during iVRTW compared to TW while CV was similar (p = 0.177). α exponent was similar during OW (0.77 ± 0.07) and iVRTW (0.76 ± 0.09) (p = 0.553). During TW, α exponent (0.85 ± 0.07) was higher than during OW (p = 0.039) and iVRTW (p = 0.016). SSQ was similar between TW and iVRTW (p = 0.809). iVRTW is tolerable, could optimize TW effects on spatiotemporal parameters while not increasing CV in PD. Furthermore, iVRTW could help to capture the natural LRA of PD gait in laboratory settings and could potentially be a challenging second step in PD gait rehabilitation.

Fractal Analysis of Human Gait Variability via Stride Interval Time Series

Fractal analysis of stride interval time series is a useful tool in human gait research which could be used as a marker for gait adaptability, gait disorder, and fall risk among patients with movement disorders. This study is designed to systematically and comprehensively investigate two practical aspects of fractal analysis which significantly affect the outcome: the series length and the parameters used in the algorithm. The Hurst exponent, scaling exponent, and/or fractal dimension are computed from both simulated and experimental data using three fractal methods, namely detrended fluctuation analysis, box-counting dimension, and Higuchi's fractal dimension. The advantages and drawbacks of each method are discussed, in terms of biases and variability. The results demonstrate that a careful selection of fractal analysis methods and their parameters is required, which is dependent on the aim of study (either analyzing differences between experimental groups or estimating an accurate determination of fractal features). A set of guidelines for the selection of the fractal methods and the length of stride interval time series is provided, along with the optimal parameters for a robust implementation for each method.