Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods

Entropy and fractal analysis of brain-related neurophysiological signals in Alzheimer's and Parkinson's disease

Brain-related neuronal recordings, such as local field potential, electroencephalogram and magnetoencephalogram, offer the opportunity to study the complexity of the human brain at different spatial and temporal scales. The complex properties of neuronal signals are intrinsically related to the concept of 'scale-free' behavior and irregular dynamic, which cannot be fully described through standard linear methods, but can be measured by nonlinear indexes. A remarkable application of these analysis methods on electrophysiological recordings is the deep comprehension of the pathophysiology of neurodegenerative diseases, that has been shown to be associated to changes in brain activity complexity. In particular, a decrease of global complexity has been associated to Alzheimer's disease, while a local increase of brain signals complexity characterizes Parkinson's disease. Despite the recent proliferation of studies using fractal and entropy-based analysis, the application of these techniques is still far from clinical practice, due to the lack of an agreement about their correct estimation and a conclusive and shared interpretation. Along with the aim of helping towards the realization of a multidisciplinary audience to approach nonlinear methods based on the concepts of fractality and irregularity, this survey describes the implementation and proper employment of the mostly known and applied indexes in the context of Alzheimer's and Parkinson's diseases.

Stochastic Interpolation of Sparsely Sampled Time Series via Multipoint Fractional Brownian Bridges

We propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractional Brownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of intermediate and nonequidistant points. Determining the optimal value of the Hurst exponent H_{opt}, appropriate to interpolate the sparse signal, is a very important step of our method. We demonstrate the validity of our method on a signal from fluid turbulence in a high Reynolds number flow and discuss the implications of the non-self-similar character of the signal. The method introduced here could be instrumental in several physical problems, including astrophysics, particle tracking, and specific tailoring of surrogate data, as well as in domains of natural and social sciences.

Long range correlations of the ion current in SV channels. Met3PbCl influence study

The long-range correlations within the current signal time series of the Beta vulgaris vacuolar membrane under the influence of organolead compound (Met3PbCl) are investigated. The current time series is transformed into a dwell time series. Then the rescaled range and detrended fluctuations analyses are used. It is shown that the presence of Met3PbCl in the solution decreases the mean value of the Hurst exponent and therefore influences the long-range correlations in ionic channel current. This observation is statistically significant. An ion channel model is built and the experimental results reconstructed and analysed.

Biases in the Simulation and Analysis of Fractal Processes

Fractal processes have recently received a growing interest, especially in the domain of rehabilitation. More precisely, the evolution of fractality with aging and disease, suggesting a loss of complexity, has inspired a number of studies that tried, for example, to entrain patients with fractal rhythms. This kind of study requires relevant methods for generating fractal signals and for assessing the fractality of the series produced by participants. In the present work, we engaged a cross validation of three methods of generation and three methods of analysis. We generated exact fractal series with the Davies–Harte (DH) algorithm, the spectral synthesis method (SSM), and the ARFIMA simulation method. The series were analyzed by detrended fluctuation analysis (DFA), power spectral density (PSD) method, and ARFIMA modeling. Results show that some methods of generation present systematic biases: DH presented a strong bias toward white noise in fBm series close to the 1/f boundary and SSM produced series with a larger variability around the expected exponent, as compared with other methods. In contrast, ARFIMA simulations provided quite accurate series, without major bias. Concerning the methods of analysis, DFA tended to systematically underestimate fBm series. In contrast, PSD yielded overestimates for fBm series. With DFA, the variability of estimates tended to increase for fGn series as they approached the 1/f boundary and reached unacceptable levels for fBm series. The highest levels of variability were produced by PSD. Finally, ARFIMA methods generated the best series and provided the most accurate and less variable estimates.

The nucleus does not significantly affect the migratory trajectories of amoeba in two-dimensional environments

For a wide range of cells, from bacteria to mammals, locomotion movements are a crucial systemic behavior for cellular life. Despite its importance in a plethora of fundamental physiological processes and human pathologies, how unicellular organisms efficiently regulate their locomotion system is an unresolved question. Here, to understand the dynamic characteristics of the locomotion movements and to quantitatively study the role of the nucleus in the migration of Amoeba proteus we have analyzed the movement trajectories of enucleated and non-enucleated amoebas on flat two-dimensional (2D) surfaces using advanced non-linear physical-mathematical tools and computational methods. Our analysis shows that both non-enucleated and enucleated amoebas display the same kind of dynamic migration structure characterized by highly organized data sequences, super-diffusion, non-trivial long-range positive correlations, persistent dynamics with trend-reinforcing behavior, and move-step fluctuations with scale invariant properties. Our results suggest that the presence of the nucleus does not significantly affect the locomotion of amoeba in 2D environments.

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, h_max] 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, r_σ(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, h_max, and Ĥ(2) were lower in the young group explained by a significantly increased r_σ(s) among elderly at high temporal scales. Regarding the fast component of CBSI-pretreated HbT and that of HbO-HbR covariance, h_max, 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.

Anomalous scaling of stochastic processes and the Moses effect

The state of a stochastic process evolving over a time t is typically assumed to lie on a normal distribution whose width scales like t^{1/2}. However, processes in which the probability distribution is not normal and the scaling exponent differs from 1/2 are known. The search for possible origins of such "anomalous" scaling and approaches to quantify them are the motivations for the work reported here. In processes with stationary increments, where the stochastic process is time-independent, autocorrelations between increments and infinite variance of increments can cause anomalous scaling. These sources have been referred to as the Joseph effect and the Noah effect, respectively. If the increments are nonstationary, then scaling of increments with t can also lead to anomalous scaling, a mechanism we refer to as the Moses effect. Scaling exponents quantifying the three effects are defined and related to the Hurst exponent that characterizes the overall scaling of the stochastic process. Methods of time series analysis that enable accurate independent measurement of each exponent are presented. Simple stochastic processes are used to illustrate each effect. Intraday financial time series data are analyzed, revealing that their anomalous scaling is due only to the Moses effect. In the context of financial market data, we reiterate that the Joseph exponent, not the Hurst exponent, is the appropriate measure to test the efficient market hypothesis.

Dynamic properties of calcium-activated chloride currents in Xenopus laevis oocytes

Chloride is the most abundant permeable anion in the cell, and numerous studies in the last two decades highlight the great importance and broad physiological role of chloride currents mediated anion transport. They participate in a multiplicity of key processes, as for instance, the regulation of electrical excitability, apoptosis, cell cycle, epithelial secretion and neuronal excitability. In addition, dysfunction of Cl⁻ channels is involved in a variety of human diseases such as epilepsy, osteoporosis and different cancer types. Historically, chloride channels have been of less interest than the cation channels. In fact, there seems to be practically no quantitative studies of the dynamics of chloride currents. Here, for the first time, we have quantitatively studied experimental calcium-activated chloride fluxes belonging to Xenopus laevis oocytes, and the main results show that the experimental Cl⁻ currents present an informational structure characterized by highly organized data sequences, long-term memory properties and inherent “crossover” dynamics in which persistent correlations arise at short time intervals, while anti-persistent behaviors become dominant in long time intervals. Our work sheds some light on the understanding of the informational properties of ion currents, a key element to elucidate the physiological functional coupling with the integrative dynamics of metabolic processes.

Silent Reading Fluency and Comprehension in Bilingual Children

This paper focuses on reading fluency by bilingual primary school students, and the relation of text fluency to their reading comprehension. Group differences were examined in a cross-sectional design across the age range when fluency is posed to shift from word-level to text-level. One hundred five bilingual children from primary grades 3, 4, and 5 were assessed for English word reading and decoding fluency, phonological awareness, rapid symbol naming, and oral language proficiency with standardized measures. These skills were correlated with their silent reading fluency on a self-paced story reading task. Text fluency was quantified using non-linear analytic methods: recurrence quantification and fractal analyses. Findings indicate that more fluent text reading appeared by grade 4, similar to monolingual findings, and that different aspects of fluency characterized passage reading performance at different grade levels. Text fluency and oral language proficiency emerged as significant predictors of reading comprehension.

Power-law correlations, related models for long-range dependence and their simulation

Martin and Walker ((1997) J. Appl. Prob.34, 657–670) proposed the power-law ρ(v) = c|v|-β, |v| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(v) = (1 + |v/c|α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

Power-law correlations, related models for long-range dependence and their simulation

Martin and Walker ((1997) J. Appl. Prob. 34, 657–670) proposed the power-law ρ(v) = c|v|-β, |v| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(v) = (1 + |v/c|α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

Fractal regional myocardial blood flows pattern according to metabolism, not vascular anatomy

Regional myocardial blood flows are markedly heterogeneous. Fractal analysis shows strong near-neighbor correlation. In experiments to distinguish control by vascular anatomy vs. local vasomotion, coronary flows were increased in open-chest dogs by stimulating myocardial metabolism (catecholamines + atropine) with and without adenosine. During control states mean left ventricular (LV) myocardial blood flows (microspheres) were 0.5-1 ml·g(-1)·min(-1) and increased to 2-3 ml·g(-1)·min(-1) with catecholamine infusion and to ∼4 ml·g(-1)·min(-1) with adenosine (Ado). Flow heterogeneity was similar in all states: relative dispersion (RD = SD/mean) was ∼25%, using LV pieces 0.1-0.2% of total. During catecholamine infusion local flows increased in proportion to the mean flows in 45% of the LV, "tracking" closely (increased proportionately to mean flow), while ∼40% trended toward the mean. Near-neighbor regional flows remained strongly spatially correlated, with fractal dimension D near 1.2 (Hurst coefficient 0.8). The spatial patterns remain similar at varied levels of metabolic stimulation inferring metabolic dominance. In contrast, adenosine vasodilation increased flows eightfold times control while destroying correlation with the control state. The Ado-induced spatial patterns differed from control but were self-consistent, inferring that with full vasodilation the relaxed arterial anatomy dominates the distribution. We conclude that vascular anatomy governs flow distributions during adenosine vasodilation but that metabolic vasoregulation dominates in normal physiological states.

IQM: an extensible and portable open source application for image and signal analysis in Java

Image and signal analysis applications are substantial in scientific research. Both open source and commercial packages provide a wide range of functions for image and signal analysis, which are sometimes supported very well by the communities in the corresponding fields. Commercial software packages have the major drawback of being expensive and having undisclosed source code, which hampers extending the functionality if there is no plugin interface or similar option available. However, both variants cannot cover all possible use cases and sometimes custom developments are unavoidable, requiring open source applications. In this paper we describe IQM, a completely free, portable and open source (GNU GPLv3) image and signal analysis application written in pure Java. IQM does not depend on any natively installed libraries and is therefore runnable out-of-the-box. Currently, a continuously growing repertoire of 50 image and 16 signal analysis algorithms is provided. The modular functional architecture based on the three-tier model is described along the most important functionality. Extensibility is achieved using operator plugins, and the development of more complex workflows is provided by a Groovy script interface to the JVM. We demonstrate IQM's image and signal processing capabilities in a proof-of-principle analysis and provide example implementations to illustrate the plugin framework and the scripting interface. IQM integrates with the popular ImageJ image processing software and is aiming at complementing functionality rather than competing with existing open source software. Machine learning can be integrated into more complex algorithms via the WEKA software package as well, enabling the development of transparent and robust methods for image and signal analysis.

A comparative analysis of spectral exponent estimation techniques for 1/f(β) processes with applications to the analysis of stride interval time series

BACKGROUND

The time evolution and complex interactions of many nonlinear systems, such as in the human body, result in fractal types of parameter outcomes that exhibit self similarity over long time scales by a power law in the frequency spectrum S(f)=1/f(β). The scaling exponent β is thus often interpreted as a "biomarker" of relative health and decline.

NEW METHOD

This paper presents a thorough comparative numerical analysis of fractal characterization techniques with specific consideration given to experimentally measured gait stride interval time series. The ideal fractal signals generated in the numerical analysis are constrained under varying lengths and biases indicative of a range of physiologically conceivable fractal signals. This analysis is to complement previous investigations of fractal characteristics in healthy and pathological gait stride interval time series, with which this study is compared.

RESULTS

The results of our analysis showed that the averaged wavelet coefficient method consistently yielded the most accurate results.

COMPARISON WITH EXISTING METHODS

Class dependent methods proved to be unsuitable for physiological time series. Detrended fluctuation analysis as most prevailing method in the literature exhibited large estimation variances.

CONCLUSIONS

The comparative numerical analysis and experimental applications provide a thorough basis for determining an appropriate and robust method for measuring and comparing a physiologically meaningful biomarker, the spectral index β. In consideration of the constraints of application, we note the significant drawbacks of detrended fluctuation analysis and conclude that the averaged wavelet coefficient method can provide reasonable consistency and accuracy for characterizing these fractal time series.

Analysis of Intracranial Pressure

The monitoring of intracranial pressure (ICP) is an important tool in medicine for its ability to portray the brain’s compliance status. The bedside monitor displays the ICP waveform and intermittent mean values to guide physicians in the management of patients, particularly those having sustained a traumatic brain injury. Researchers in the fields of engineering and physics have investigated various mathematical analysis techniques applicable to the waveform in order to extract additional diagnostic and prognostic information, although they largely remain limited to research applications. The purpose of this review is to present the current techniques used to monitor and interpret ICP and explore the potential of using advanced mathematical techniques to provide information about system perturbations from states of homeostasis. We discuss the limits of each proposed technique and we propose that nonlinear analysis could be a reliable approach to describe ICP signals over time, with the fractal dimension as a potential predictive clinically meaningful biomarker. Our goal is to stimulate translational research that can move modern analysis of ICP using these techniques into widespread practical use, and to investigate to the clinical utility of a tool capable of simplifying multiple variables obtained from various sensors.

Identifying multiplicative interactions between temporal scales of human movement variability

Conventional scaling analyses of human movement variability such as detrended fluctuation analyses assume that the movement variable can be decomposed into scale-dependent variation. However, these conventional scaling analyses are insensitive to multiplicative interactions within the movement variable. Multiplicative interactions refer to couplings between the scale-dependent variations across multiple scales that generate intermittent changes in the human movement variable. The mathematical concept for intermittent variability generated by multiplicative interactions is called multifractal variability. Multifractal variability is numerically defined by a spectrum of scaling exponents (i.e., a multifractal spectrum) that can be an important feature of coordinated movements and, consequently, relevant when aiming to identify movement disorders. In the current study, a new method is introduced based on detrended fluctuation analysis that can identify the multifractal spectrum from the temporal variation of local scaling exponents. The influence of multiplicative interactions on the local scaling exponents is tested by a Monte Carlo surrogate test. The methods are validated on multiplicative cascading processes with known multiplicative interactions. The application of the new methods is subsequently illustrated on an example of centre of pressure variations during quiet and relaxed standing. The results show that multiplicative interactions are present during periods with large movements of the center of gravity, where the movements of the centre of gravity and centre of pressure couple into coordinative structures. Further application and interpretation of the developed method for the study of human movement variability are discussed.

Pitfalls in Fractal Time Series Analysis: fMRI BOLD as an Exemplary Case

This article will be positioned on our previous work demonstrating the importance of adhering to a carefully selected set of criteria when choosing the suitable method from those available ensuring its adequate performance when applied to real temporal signals, such as fMRI BOLD, to evaluate one important facet of their behavior, fractality. Earlier, we have reviewed on a range of monofractal tools and evaluated their performance. Given the advance in the fractal field, in this article we will discuss the most widely used implementations of multifractal analyses, too. Our recommended flowchart for the fractal characterization of spontaneous, low frequency fluctuations in fMRI BOLD will be used as the framework for this article to make certain that it will provide a hands-on experience for the reader in handling the perplexed issues of fractal analysis. The reason why this particular signal modality and its fractal analysis has been chosen was due to its high impact on today’s neuroscience given it had powerfully emerged as a new way of interpreting the complex functioning of the brain (see “intrinsic activity”). The reader will first be presented with the basic concepts of mono and multifractal time series analyses, followed by some of the most relevant implementations, characterization by numerical approaches. The notion of the dichotomy of fractional Gaussian noise and fractional Brownian motion signal classes and their impact on fractal time series analyses will be thoroughly discussed as the central theme of our application strategy. Sources of pitfalls and way how to avoid them will be identified followed by a demonstration on fractal studies of fMRI BOLD taken from the literature and that of our own in an attempt to consolidate the best practice in fractal analysis of empirical fMRI BOLD signals mapped throughout the brain as an exemplary case of potentially wide interest.

Motor imagery may incorporate trial-to-trial error

The authors tested for 1/f noise in motor imagery (MI). Participants pointed and imagined pointing to a single target (Experiment 1), to targets of varied size (Experiment 2), and switched between pointing and grasping (Experiment 3). Experiment 1 showed comparable patterns of serial correlation in actual and imagined movement. Experiment 2 suggested increased correlation for MI and performance with increased task difficulty, perhaps reflecting adaptation to a more complex environment. Experiment 3 suggested a parallel decrease in correlation with task switching, perhaps reflecting discontinuity of mental set. Although present results do not conclusively reveal 1/f fluctuation, the emergent patterns suggest that MI could incorporate trial-to-trial error across a range of constraints.

Quantitative analysis of cellular metabolic dissipative, self-organized structures

One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life.

The correlation structure of relative phase variability influences the occurrence of phase transition in coordination

The coordination dynamics framework has presumed that fluctuations in relative phase series produced in bimanual coordination are random. However, results from recent studies have shown that relative phase series contain 1/f(beta) noise (persistent long-range correlations) instead. Using an incremental protocol in line with the paradigmatic bimanual coordination framework, the author shows that the movement frequencies at which individuals spontaneously switch from anti-phase to in-phase coordination are significantly correlated with the intensity of long-range correlations but not with the amplitude of baseline fluctuations in relative phase. This finding illustrates the tangible relationship between present theoretical perspectives and accumulating evidence for 1/f(beta) noise. The author underscores the heuristic potential of systematic efforts to bridge the gap between present theories and the pervasive findings of 1/f(beta) noise.

Estimating long-range dependence in time series: an evaluation of estimators implemented in R

Recent studies have shown that many physiological and behavioral processes can be characterized by long-range correlations. The Hurst exponent H of fractal analysis and the fractional-differencing parameter d of the ARFIMA methodology are useful for capturing serial correlations. In this study, we report on different estimators of H and d implemented in R, a popular and freely available software package. By means of Monte Carlo simulations, we analyzed the performance of (1) the Geweke-Porter-Hudak estimator, (2) the approximate maximum likelihood algorithm, (3) the smoothed periodogram approach, (4) the Whittle estimator, (5) rescaled range analysis, (6) a modified periodogram, (7) Higuchi's method, and (8) detrended fluctuation analysis. The findings-confined to ARFIMA (0, d, 0) models and fractional Gaussian noise-identify the best estimators for persistent and antipersistent series. Two examples combining these results with the step-by-step procedure proposed by Delignières et al. (2006) demonstrate how this evaluation can be used as a guideline in a typical research situation.

Nonlinear time series analysis of ground-level ozone dynamics in Southern Taiwan

The ozone pollution at ground level in rural and urban areas has been a long-standing problem in the world. This paper focuses on estimating self-affined nature of nonlinearity of ground-level peak ozone time series, which is analyzed by two nonlinear fractal statistical methods, including R/S analysis and BDS test. To explore the underlying structure of ozone observations at ground level and improve the forecasting capacity in urban region, practical implementation was assessed by a case study via collecting and analyzing the monitoring data at Chaojhou and Zenwu in the Kaohsiung metropolitan region, Taiwan. Based on R/S analysis, the time series can be identified as persistent and long-memory processes with Hurst exponents of both about 0.75. In addition, the V statistics specifies possible fluctuation cycle lengths of 32, 170, and 420 day simultaneously. Such results are consistent with the regional meteorological conditions leading to help characterize the regional scale ozone behavioral trend. Furthermore, the BDS test results confirm a strong nonlinearity of both time series associated with these two cities. Yet in both cases, nonlinearity implies chaos. The R/S analysis and BDS test provide strong evidence for nonlinearity and fractality of ozone time series due to noisy chaos, and we could not rule out the possibility of deterministic chaos in tropospheric ozone system.

Using 1/f noise to examine planning and control in a discrete aiming task

The present study used 1/f noise to examine how spatial, physical, and timing constraints affect planning and control processes in aiming. Participants moved objects of different masses to different distances at preferred speed (Experiment 1) and as quickly as possible (Experiment 2). Power spectral density, standardized dispersion, rescaled range, and an autoregressive fractionally integrated moving average (ARFIMA) model selection procedure were used to quantify 1/f noise. Measures from all four analyses were in reasonable agreement, with more ARFIMA (long-range) models selected at peak velocity in Experiment 1 and fewer selected at peak velocity in Experiment 2. There also was a nonsignificant trend where, at preferred speed, of those participants who showed 1/f noise, more tended to show 1/f noise at peak velocity, when planning and control would overlap most. This trend disappeared for fast movements, where planning and control would have less time to overlap. Summing short-range processes at different timescales can produce 1/f-like noise. As planning is a slower-moving process and control faster, present results suggest that, with enough time for both planning and control, 1/f noise in aiming may arise from a similar summation of processes. Potential limitations of time series length in the present task are discussed.

Algorithm to estimate the Hurst exponent of high-dimensional fractals

We propose an algorithm to estimate the Hurst exponent of high-dimensional fractals, based on a generalized high-dimensional variance around a moving average low-pass filter. As working examples, we consider rough surfaces generated by the random midpoint displacement and by the Cholesky-Levinson factorization algorithms. The surrogate surfaces have Hurst exponents ranging from 0.1 to 0.9 with step 0.1, and different sizes. The computational efficiency and the accuracy of the algorithm are also discussed.

1/f (beta) fluctuations in bimanual coordination: an additional challenge for modeling

We analyzed the correlation structure of discrete relative phase (DRP) series in bimanual in-phase and anti-phase coordination by associating a number of fractal methods and using discrete rather than continuous relative phase measurement. ARFIMA/ARMA modeling provided statistical evidence for the presence of long-range correlation, and the series were unambiguously characterized as 1/f (beta) noise. Diverging accounts of bimanual coordination are defended in the literature. Since the evidence for 1/f (beta) noise provides new insight into the properties of stability in coordination, it should be considered as an empirical criterion for determining which mechanisms are likely to be engaged in bimanual coordination models. We discussed some implications for studying the neural basis of coordination, and we tested the performance of three current models in accounting for 1/f (beta) noise in DRP. None of these models was proven to generate the expected correlation structure.

Detection of long-range dependence and estimation of fractal exponents through ARFIMA modelling

We evaluate the performance of autoregressive, fractionally integrated, moving average (ARFIMA) modelling for detecting long-range dependence and estimating fractal exponents. More specifically, we test the procedure proposed by Wagenmakers, Farrell, and Ratcliff, and compare the results obtained with the Akaike information criterion (AIC) and the Bayes information criterion (BIC). The present studies show that ARFIMA modelling is able to adequately detect long-range dependence in simulated fractal series. Conversely, this method tends to produce a non-negligible rate of false detections in pure autoregressive and moving average (ARMA) series. Generally, ARFIMA modelling has a bias favouring the detection of long-range dependence. AIC and BIC gave dissimilar results, due to the different weights attributed by the two criteria to accuracy and parsimony. Finally, ARFIMA modelling provides good estimates of fractal exponents, and could adequately complement classical methods, such as spectral analysis, detrended fluctuation analysis or rescaled range analysis.

Utility of traditional and alternative EMG-based measures of fatigue during low-moderate level isometric efforts

Traditional electromyographic (EMG) measures (e.g., amplitude, mean and median frequencies of the power spectra) have demonstrated inconsistent abilities in monitoring localized muscle fatigue at relatively low effort levels. In the present study, several alternative EMG-based fatigue indices were evaluated, derived using a logarithmic representation of the power spectrum, the fractal dimension of the raw signal, and a Poisson distribution fit to the power spectrum. These methods, along with traditional approaches, were applied to EMG data obtained from three separate experiments. In the first two experiments, 24 participants performed sustained isometric shoulder abductions and torso extensions at 30% of maximum voluntary strength (MVC). In the third experiment, another group of 12 participants performed similar shoulder exercises at 15% and 30% MVC, with repeatability assessed at 15% MVC. Both traditional and alternative EMG measures were analyzed for their 'utility', in terms of sensitivity to fatigue, variability, repeatability, and predictive ability. Results demonstrated that parameters derived from fractal analysis and the Poisson distribution demonstrated high utility. These alternative approaches appear promising as fatigue indices for low level isometric tasks.

Long-range correlations in rabbit brain neural activity

We have analyzed the presence of persistence properties in rabbit brain electrical signals by means of non-equilibrium statistical physics tools. To measure long-memory properties of these experimental signals, we have first determined whether the data are fractional Gaussian noise (fGn) or fractional Brownian motion (fBm) by calculating the slope of the power spectral density plot of the series. The results show that the series correspond to fBm. Then, the data were studied by means of the bridge detrended scaled windowed variance analysis, detecting long-term correlation. Three different types of experimental signals have been studied: neural basal activity without stimulation, the response induced by a single flash light stimulus and the average of the activity evoked by 200 flash light stimulations. Analysis of the series revealed the existence of persistent behavior in all cases. Moreover, the results also exhibited an increasing correlation in the level of long-term memory from recordings without stimulation, to one sweep recording or 200 sweeps averaged recordings. Thus, brain neural electrical activity is affected not only by its most recent states, but also by previous states much more distant in the past.

Provenance of correlations in psychological data

Two distinct families of statistical processes are considered in the production of psychophysical time series data (Gilden, 1997, 2001; Gilden, Thornton, & Mallon, 1995). We inquire whether the spectral signatures of the underlying dynamics are better described in terms of short-range autoregressive moving-average (ARMA) processes or long-range fractal processes. A thorough presentation of both families is given so as to clarify the scope and generalizability of the models as descriptions of choice reaction time data. Analyses of data are supplemented by the construction of a spectral likelihood classifier that discriminates between the two families of processes. The classifier has sufficient sensitivity to ensure that fractals are correctly identified and that ARMA processes will rarely be misconstrued as belonging to the fractal family. Spectral likelihood classification illustrates an extremely general framework for testing competing spectral hypotheses and is offered for use in measuring the specific character of fluctuations in designed experiments.

Human Cognition and 1/f Scaling

Ubiquitous 1/f scaling in human cognition and physiology suggests a mind-body interaction that contradicts commonly held assumptions. The intrinsic dynamics of psychological phenomena are interaction dominant (rather than component dominant), and the origin of purposive behavior lies with a general principle of self-organization (rather than a special neurocognitive mechanism). E.-J. Wagenmakers, S. Farrell, and R. Ratcliff (2005) raised concerns about the kinds of data and analyses that support generic 1/f scaling. This reply is a defense that furthermore questions the model that Wagenmakers and colleagues endorse and their strategy for addressing complexity.

Time intervals production in tapping and oscillatory motion

We applied spectral analysis on series of time intervals produced in a synchronization-continuation experiment. In the first condition intervals were produced by finger tapping, and in the second by an oscillatory motion of the hand. Results obtained in tapping were consistent with a discrete, event-based timing model. In the oscillatory condition, the spectra suggested a continuous, dynamic timing mechanism, based on the regulation of effector stiffness. It is concluded that the oscillatory character of movement can offer an important resource for timing control. The use of an event-based timing control such as postulated in the Wing-Kristoffersson model could be restricted to a quite limited class of rhythmic tasks, characterized by the concatenation of discrete events.

A methodological note on nonlinear time series analysis: is the open- and closed-loop model of Collins and De Luca (1993) a statistical artifact?

The authors reexamined, theoretically and empirically, the method proposed by J. J. Collins and C. D. De Luca (1993) for the analysis of center-of-pressure trajectories. The main argument in this article is that Collins and De Luca's approach is not adapted to the analysis of bounded time series and leads to statistical artifacts such as underestimation of the diffusion process for long-term intervals. The open- and closed-loop model developed by Collins and De Luca is a direct consequence of those statistical problems. Applying more classical methods, such as rescaled range analysis or detrended fluctuation analysis, the authors show that center-of-pressure trajectories can be modeled as continuous, antipersistent fractional Brownian motion. More specifically, those trajectories behave like 1/f noise, a ubiquitous feature in adaptive biological systems.

Self-organization of cognitive performance

Background noise is the irregular variation across repeated measurements of human performance. Background noise remains after task and treatment effects are minimized. Background noise refers to intrinsic sources of variability, the intrinsic dynamics of mind and body, and the internal workings of a living being. Two experiments demonstrate 1/f scaling (pink noise) in simple reaction times and speeded word naming times, which round out a catalog of laboratory task demonstrations that background noise is pink noise. Ubiquitous pink noise suggests processes of mind and body that change each other's dynamics. Such interaction-dominant dynamics are found in systems that self-organize their behavior. Self-organization provides an unconventional perspective on cognition, but this perspective closely parallels a contemporary interdisciplinary view of living systems.

Fractal characterization of complexity in temporal physiological signals

This review first gives an overview on the concept of fractal geometry with definitions and explanations of the most fundamental properties of fractal structures and processes like self-similarity, power law scaling relationship, scale invariance, scaling range and fractal dimensions. Having laid down the grounds of the basics in terminology and mathematical formalism, the authors systematically introduce the concept and methods of monofractal time series analysis. They argue that fractal time series analysis cannot be done in a conscious, reliable manner without having a model capable of capturing the essential features of physiological signals with regard to their fractal analysis. They advocate the use of a simple, yet adequate, dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). They demonstrate the importance of incorporating a step of signal classification according to the fGn/fBm model prior to fractal analysis by showing that missing out on signal class can result in completely meaningless fractal estimates. Limitation and precision of various fractal tools are thoroughly described and discussed using results of numerical experiments on ideal monofractal signals. Steps of a reliable fractal analysis are explained. Finally, the main applications of fractal time series analysis in biomedical research are reviewed and critically evaluated.

Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study

Anomalous subdiffusion is hindered diffusion in which the mean-square displacement of a diffusing particle is proportional to some power of time less than one. Anomalous subdiffusion has been observed for a variety of lipids and proteins in the plasma membranes of a variety of cells. Fluorescence photobleaching recovery experiments with anomalous subdiffusion are simulated to see how to analyze the data. It is useful to fit the recovery curve with both the usual recovery equation and the anomalous one, and to judge the goodness of fit on log-log plots. The simulations show that the simplest approximate treatment of anomalous subdiffusion usually gives good results. Three models of anomalous subdiffusion are considered: obstruction, fractional Brownian motion, and the continuous-time random walk. The models differ significantly in their behavior at short times and in their noise level. For obstructed diffusion the approach to the percolation threshold is marked by a large increase in noise, a broadening of the distribution of diffusion coefficients and anomalous subdiffusion exponents, and the expected abrupt decrease in the mobile fraction. The extreme fluctuations in the recovery curves at and near the percolation threshold result from extreme fluctuations in the geometry of the percolation cluster.

Evaluating maximum likelihood estimation methods to determine the Hurst coeficient

A maximum likelihood estimation method implemented in S-PLUS (S-MLE) to estimate the Hurst coefficient (H) is evaluated. The Hurst coefficient, with 0.5 < H <1, characterizes long memory time series by quantifying the rate of decay of the autocorrelation function. S-MLE was developed to estimate H for fractionally differenced (fd) processes. However, in practice it is difficult to distinguish between fd processes and fractional Gaussian noise (fGn) processes. Thus, the method is evaluated for estimating H for both fd and fGn processes. S-MLE gave biased results of H for fGn processes of any length and for fd processes of lengths less than 2(10). A modified method is proposed to correct for this bias. It gives reliable estimates of H for both fd and fGn processes of length greater than or equal to 2(11).

Deriving dispersional and scaled windowed variance analyses using the correlation function of discrete fractional Gaussian noise

Methods for estimating the fractal dimension, D, or the related Hurst coefficient, H, for a one-dimensional fractal series include Hurst's method of rescaled range analysis, spectral analysis, dispersional analysis, and scaled windowed variance analysis (which is related to detrended fluctuation analysis). Dispersional analysis estimates H by using the variance of the grouped means of discrete fractional Gaussian noise series (DfGn). Scaled windowed variance analysis estimates H using the mean of grouped variances of discrete fractional Brownian motion (DfBm) series. Both dispersional analysis and scaled windowed variance analysis have small bias and variance in their estimates of the Hurst coefficient. This study demonstrates that both methods derive their accuracy from their strict mathematical relationship to the expected value of the correlation function of DfGn. The expected values of the variance of the grouped means for dispersional analysis on DfGn and the mean of the grouped variance for scaled windowed variance analysis on DfBm are calculated. An improved formulation for scaled windowed variance analysis is given. The expected values using these analyses on the wrong kind of series (dispersional analysis on DfBm and scaled windowed variance analysis on DfGn) are also calculated.