Effect of trends on detrended fluctuation analysis

Identification of Criticality in Neuronal Avalanches: II. A Theoretical and Empirical Investigation of the Driven Case

The observation of apparent power laws in neuronal systems has led to the suggestion that the brain is at, or close to, a critical state and may be a self-organised critical system. Within the framework of self-organised criticality a separation of timescales is thought to be crucial for the observation of power-law dynamics and computational models are often constructed with this property. However, this is not necessarily a characteristic of physiological neural networks-external input does not only occur when the network is at rest/a steady state. In this paper we study a simple neuronal network model driven by a continuous external input (i.e. the model does not have an explicit separation of timescales from seeding the system only when in the quiescent state) and analytically tuned to operate in the region of a critical state (it reaches the critical regime exactly in the absence of input-the case studied in the companion paper to this article). The system displays avalanche dynamics in the form of cascades of neuronal firing separated by periods of silence. We observe partial scale-free behaviour in the distribution of avalanche size for low levels of external input. We analytically derive the distributions of waiting times and investigate their temporal behaviour in relation to different levels of external input, showing that the system's dynamics can exhibit partial long-range temporal correlations. We further show that as the system approaches the critical state by two alternative 'routes', different markers of criticality (partial scale-free behaviour and long-range temporal correlations) are displayed. This suggests that signatures of criticality exhibited by a particular system in close proximity to a critical state are dependent on the region in parameter space at which the system (currently) resides.

Impact of stock market structure on intertrade time and price dynamics

We analyse times between consecutive transactions for a diverse group of stocks registered on the NYSE and NASDAQ markets, and we relate the dynamical properties of the intertrade times with those of the corresponding price fluctuations. We report that market structure strongly impacts the scale-invariant temporal organisation in the transaction timing of stocks, which we have observed to have long-range power-law correlations. Specifically, we find that, compared to NYSE stocks, stocks registered on the NASDAQ exhibit significantly stronger correlations in their transaction timing on scales within a trading day. Further, we find that companies that transfer from the NASDAQ to the NYSE show a reduction in the correlation strength of transaction timing on scales within a trading day, indicating influences of market structure. We also report a persistent decrease in correlation strength of intertrade times with increasing average intertrade time and with corresponding decrease in companies' market capitalization–a trend which is less pronounced for NASDAQ stocks. Surprisingly, we observe that stronger power-law correlations in intertrade times are coupled with stronger power-law correlations in absolute price returns and higher price volatility, suggesting a strong link between the dynamical properties of intertrade times and the corresponding price fluctuations over a broad range of time scales. Comparing the NYSE and NASDAQ markets, we demonstrate that the stronger correlations we find in intertrade times for NASDAQ stocks are associated with stronger correlations in absolute price returns and with higher volatility, suggesting that market structure may affect price behavior through information contained in transaction timing. These findings do not support the hypothesis of universal scaling behavior in stock dynamics that is independent of company characteristics and stock market structure. Further, our results have implications for utilising transaction timing patterns in price prediction and risk management optimization on different stock markets.

Noninvasive fractal biomarker of clock neurotransmitter disturbance in humans with dementia

Human motor activity has a robust, intrinsic fractal structure with similar patterns from minutes to hours. The fractal activity patterns appear to be physiologically important because the patterns persist under different environmental conditions but are significantly altered/reduced with aging and Alzheimer's disease (AD). Here, we report that dementia patients, known to have disrupted circadian rhythmicity, also have disrupted fractal activity patterns and that the disruption is more pronounced in patients with more amyloid plaques (a marker of AD severity). Moreover, the degree of fractal activity disruption is strongly associated with vasopressinergic and neurotensinergic neurons (two major circadian neurotransmitters) in postmortem suprachiasmatic nucleus (SCN), and can better predict changes of the two neurotransmitters than traditional circadian measures. These findings suggest that the SCN impacts human activity regulation at multiple time scales and that disrupted fractal activity may serve as a non-invasive biomarker of SCN neurodegeneration in dementia.

Econophysics: Comments on a Few Applications, Successes, Methods and Models

For this special issue, the article aims at discussing a few econophysics problems studied so far rather successfully. The following ‘applications’ in micro-econophysics are considered: (i) financial crashes; it is emphasized that one can distinguish between endogenous and exogenous causes; (ii) portofolio control, selection and inherent risk measure; (iii) foreign currency exchanges, also distinguishing endogenous and exogenous money control; (iv) price and asset evolution values. It is shown that some macro-econo-physics problems have been also tackled, like geographic/ political constraints, the globalization of the economy and country clustering. Moreover, it is daring to suggest prospect for studies and researches, whence presenting some selection of a few interesting perspectives.

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.

Assessing complexity of skin blood flow oscillations in response to locally applied heating and pressure in rats: implications for pressure ulcer risk

The objective of this study was to investigate the effects of local heating on complexity of skin blood flow oscillations (BFO) under prolonged surface pressure in rats. Eleven Sprague-Dawley rats were studied: 7 rats underwent surface pressure with local heating (Δt = 10 °C) and 4 rats underwent pressure without heating. A pressure of 700 mmHg was applied to the right trochanter area of rats for 3 h. Skin blood flow was measured using laser Doppler flowmetry. The loading period was divided into nonoverlapping 30 min epochs. For each epoch, multifractal detrended fluctuation analysis (MDFA) was utilized to compute DFA coefficients and complexity of endothelia related metabolic, neurogenic, and myogenic frequencies of BFO. The results showed that under surface pressure, local heating led to a significant decrease in DFA coefficients of myogenic frequency during the initial epoch of loading period, a sustained decrease in complexity of myogenic frequency, and a significantly higher degree of complexity of metabolic frequency during the later phase of loading period. Surrogate tests showed that the reduction in complexity of myogenic frequency was associated with a loss of nonlinearity whereas increased complexity of metabolic frequency was associated with enhanced nonlinearity. Our results indicate that increased metabolic activity and decreased myogenic response due to local heating manifest themselves not only in magnitudes of metabolic and myogenic frequencies but also in their structural complexity. This study demonstrates the feasibility of using complexity analysis of BFO to monitor the ischemic status of weight-bearing skin and risk of pressure ulcers.

Multifractal detrended fluctuation analysis of human gait diseases

In this paper multifractal detrended fluctuation analysis (MFDFA) is used to study the human gait time series for normal and diseased sets. It is observed that long range correlation is primarily responsible for the origin of multifractality. The study reveals that the degree of multifractality is more for normal set compared to diseased set. However, the method fails to distinguish between the two diseased sets.

Metabolic imaging in multiple time scales

We report here a novel combination of time-resolved imaging methods for probing mitochondrial metabolism in multiple time scales at the level of single cells. By exploiting a mitochondrial membrane potential reporter fluorescence we demonstrate the single cell metabolic dynamics in time scales ranging from microseconds to seconds to minutes in response to glucose metabolism and mitochondrial perturbations in real time. Our results show that in comparison with normal human mammary epithelial cells, the breast cancer cells display significant alterations in metabolic responses at all measured time scales by single cell kinetics, fluorescence recovery after photobleaching and by scaling analysis of time-series data obtained from mitochondrial fluorescence fluctuations. Furthermore scaling analysis of time-series data in living cells with distinct mitochondrial dysfunction also revealed significant metabolic differences thereby suggesting the broader applicability (e.g. in mitochondrial myopathies and other metabolic disorders) of the proposed strategies beyond the scope of cancer metabolism. We discuss the scope of these findings in the context of developing portable, real-time metabolic measurement systems that can find applications in preclinical and clinical diagnostics.

The complexity of standing postural control in older adults: a modified detrended fluctuation analysis based upon the empirical mode decomposition algorithm

Human aging into senescence diminishes the capacity of the postural control system to adapt to the stressors of everyday life. Diminished adaptive capacity may be reflected by a loss of the fractal-like, multiscale complexity within the dynamics of standing postural sway (i.e., center-of-pressure, COP). We therefore studied the relationship between COP complexity and adaptive capacity in 22 older and 22 younger healthy adults. COP magnitude dynamics were assessed from raw data during quiet standing with eyes open and closed, and complexity was quantified with a new technique termed empirical mode decomposition embedded detrended fluctuation analysis (EMD-DFA). Adaptive capacity of the postural control system was assessed with the sharpened Romberg test. As compared to traditional DFA, EMD-DFA more accurately identified trends in COP data with intrinsic scales and produced short and long-term scaling exponents (i.e., α(Short), α(Long)) with greater reliability. The fractal-like properties of COP fluctuations were time-scale dependent and highly complex (i.e., α(Short) values were close to one) over relatively short time scales. As compared to younger adults, older adults demonstrated lower short-term COP complexity (i.e., greater α(Short) values) in both visual conditions (p>0.001). Closing the eyes decreased short-term COP complexity, yet this decrease was greater in older compared to younger adults (p<0.001). In older adults, those with higher short-term COP complexity exhibited better adaptive capacity as quantified by Romberg test performance (r(2) = 0.38, p<0.001). These results indicate that an age-related loss of COP complexity of magnitude series may reflect a clinically important reduction in postural control system functionality as a new biomarker.

Skin blood flow dynamics and its role in pressure ulcers

Pressure ulcers are a significant healthcare problem affecting the quality of life in wheelchair bounded or bed-ridden people and are a major cost to the healthcare system. Various assessment tools such as the Braden scale have been developed to quantify the risk level of pressure ulcers. These tools have provided an initial guideline on preventing pressure ulcers while additional assessments are needed to improve the outcomes of pressure ulcer prevention. Skin blood flow function that determines the ability of the skin in response to ischemic stress has been proposed to be a good indicator for identifying people at risk of pressure ulcers. Wavelet spectral and nonlinear complexity analyses have been performed to investigate the influences of the metabolic, neurogenic and myogenic activities on microvascular regulation in people with various pathological conditions. These findings have contributed to the understanding of the role of ischemia and viability on the development of pressure ulcers. The purpose of the present review is to provide an introduction of the basic concepts and approaches for the analysis of skin blood flow oscillations, and present an overview of the research results obtained so far. We hope this information may contribute to the development of better clinical guidelines for the prevention of pressure ulcers.

Microstructure identification via detrended fluctuation analysis of ultrasound signals

We describe an algorithm for simulating ultrasound propagation in random one-dimensional media, mimicking different microstructures by choosing physical properties such as domain sizes and mass densities from probability distributions. By combining a detrended fluctuation analysis (DFA) of the simulated ultrasound signals with tools from the pattern-recognition literature, we build a Gaussian classifier which is able to associate each ultrasound signal with its corresponding microstructure with a very high success rate. Furthermore, we also show that DFA data can be used to train a multilayer perceptron which estimates numerical values of physical properties associated with distinct microstructures.

Effects of autonomic ganglion blockade on fractal and spectral components of blood pressure and heart rate variability in free-moving rats

Fractal analysis is a promising tool for assessing autonomic influences on heart rate (HR) and blood pressure (BP) variability. The temporal spectrum of scale coefficients, α(t), was recently proposed to describe the cardiovascular fractal dynamics. Aim of our work is to evaluate sympathetic influences on cardiovascular variability analyzing α(t) and spectral powers of HR and BP after ganglionic blockade. BP was recorded in 11 rats before and after autonomic blockade by hexamethonium infusion (HEX). Systolic and diastolic BP, pulse pressure and pulse interval were derived beat-by-beat. Segments longer than 5 min were selected at baseline and HEX to estimate power spectra and α(t). Comparisons were made by paired t-test. HEX reduced all spectral components of systolic and diastolic BP, the reduction being particularly significant around the frequency of Mayer waves; it induced a reduction on α(t) coefficients at t<2s and an increase on coefficients at t>8s. HEX reduced only slower components of pulse interval power spectrum, but decreased significantly faster scale coefficients (t<8s). HEX only marginally affected pulse pressure variability. Results indicate that the sympathetic outflow contributes to BP fractal dynamics with fractional Gaussian noise (α<1) at longer scales and fractional Brownian motion (α>1) at shorter scales. Ganglionic blockade also removes a fractional Brownian motion component at shorter scales from HR dynamics. Results may be explained by the characteristic time constants between sympathetic efferent activity and cardiovascular effectors. Therefore fractal analysis may complete spectral analysis with information on the correlation structure of the data.

Size effects on correlation measures

The detection and quantification of long-range correlations in time series is a fundamental tool to characterize the properties of different dynamical systems, and is applied in many different fields, including physics, biology or engineering. Due to the diversity of applications, many techniques for measuring correlations have been designed. Here, we study systematically the influence of the length of a time series on the results obtained from several techniques commonly used to detect and quantify long-range correlations: the autocorrelation analysis, Hurst's analysis, and detrended fluctuation analysis (DFA). Using the Fourier filtering method, we generate artificial time series with known and controlled long-range correlations and with a broad range of lengths, and apply on them the different correlation measures we have studied. Our results indicate that while the DFA method is practically unaffected by the length of the time series, and almost always provides accurate results, the results from Hurst's analysis and the autocorrelation analysis strongly depend on the length of the time series.

Long-range temporal correlations in the subthalamic nucleus of patients with Parkinson's disease

Neuronal activity in the subthalamic nucleus (STN) of patients with Parkinson's disease (PD) is characterised by excessive neuronal synchronization, particularly in the beta frequency range. However, less is known about the temporal dynamics of neuronal oscillations in PD. In this respect long-range temporal correlations (LRTC) are of special interest as they quantify the neuronal dynamics on different timescales and have been shown to be relevant for optimal information processing in the brain. While the presence of LRTC has been demonstrated in cortical data, their existence in deep brain structures remains an open question. We investigated (i) whether LRTC are present in local field potentials (LFP) recorded bilaterally from the STN at wakeful rest in ten patients with PD after overnight withdrawal of levodopa (OFF) and (ii) whether LRTC can be modulated by levodopa treatment (ON). Detrended fluctuation analysis was utilised in order to quantify the temporal dynamics in the amplitude fluctuations of LFP oscillations. We demonstrated for the first time the presence of LRTC (extending up to 50 s) in the STN. Importantly, the ON state was characterised by significantly stronger LRTC than the OFF state, both in beta (13-35 Hz) and high-frequency (> 200 Hz) oscillations. The existence of LRTC in subcortical structures such as STN provides further evidence for their ubiquitous nature in the brain. The weaker LRTC in the OFF state might indicate limited information processing in the dopamine-depleted basal ganglia. The present results implicate LRTC as a potential biomarker of pathological neuronal processes in PD.

Effects of fetal respiratory movements on the short-term fractal properties of heart rate variability

We evaluated the effect of fetal respiratory movements (RM) on the heart rate (HR) fractal dynamics.Abdominal ECG recordings were collected from low-middle-risk pregnant woman at rest. Mean gestational age was 34.8 ± 3.7 weeks. Ultrasound images were simultaneously acquired determining if RM were exhibited by fetuses. 13 pairs of HR series were compared. Each pair included 5 min of data from the same fetus either during the manifestation of RM or when there was no persistent indication of them. Detrended fluctuation analysis was applied to these series for obtaining the scaling exponent α1. HR series were also assessed using the conventional parameters RMSSD and HF power.The main findings of this contribution were the lack of significant changes in the scaling exponent α1 of fetal HR fluctuations as a result of RM. By contrast, HF power and RMSSD did show significant changes associated with the manifestation of fetal RM (p < 0.001 and p < 0.05, respectively). Yet the scaling exponent was the only parameter showing a significant relationship with the particular frequency of fetal RM (r s  = 0.6, p < 0.03). Given the invariability of α1 regarding the manifestation of fetal RM, we consider that the HR short-term fractal properties are convenient for assessing the cardiovascular prenatal regulation.

Detrended fluctuation analysis: a scale-free view on neuronal oscillations

Recent years of research have shown that the complex temporal structure of ongoing oscillations is scale-free and characterized by long-range temporal correlations. Detrended fluctuation analysis (DFA) has proven particularly useful, revealing that genetic variation, normal development, or disease can lead to differences in the scale-free amplitude modulation of oscillations. Furthermore, amplitude dynamics is remarkably independent of the time-averaged oscillation power, indicating that the DFA provides unique insights into the functional organization of neuronal systems. To facilitate understanding and encourage wider use of scaling analysis of neuronal oscillations, we provide a pedagogical explanation of the DFA algorithm and its underlying theory. Practical advice on applying DFA to oscillations is supported by MATLAB scripts from the Neurophysiological Biomarker Toolbox (NBT) and links to the NBT tutorial website http://www.nbtwiki.net/. Finally, we provide a brief overview of insights derived from the application of DFA to ongoing oscillations in health and disease, and discuss the putative relevance of criticality for understanding the mechanism underlying scale-free modulation of oscillations.

Fractal patterns of neural activity exist within the suprachiasmatic nucleus and require extrinsic network interactions

The mammalian central circadian pacemaker (the suprachiasmatic nucleus, SCN) contains thousands of neurons that are coupled through a complex network of interactions. In addition to the established role of the SCN in generating rhythms of ~24 hours in many physiological functions, the SCN was recently shown to be necessary for normal self-similar/fractal organization of motor activity and heart rate over a wide range of time scales--from minutes to 24 hours. To test whether the neural network within the SCN is sufficient to generate such fractal patterns, we studied multi-unit neural activity of in vivo and in vitro SCNs in rodents. In vivo SCN-neural activity exhibited fractal patterns that are virtually identical in mice and rats and are similar to those in motor activity at time scales from minutes up to 10 hours. In addition, these patterns remained unchanged when the main afferent signal to the SCN, namely light, was removed. However, the fractal patterns of SCN-neural activity are not autonomous within the SCN as these patterns completely broke down in the isolated in vitro SCN despite persistence of circadian rhythmicity. Thus, SCN-neural activity is fractal in the intact organism and these fractal patterns require network interactions between the SCN and extra-SCN nodes. Such a fractal control network could underlie the fractal regulation observed in many physiological functions that involve the SCN, including motor control and heart rate regulation.

Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series

Notwithstanding the significant efforts to develop estimators of long-range correlations (LRC) and to compare their performance, no clear consensus exists on what is the best method and under which conditions. In addition, synthetic tests suggest that the performance of LRC estimators varies when using different generators of LRC time series. Here, we compare the performances of four estimators [Fluctuation Analysis (FA), Detrended Fluctuation Analysis (DFA), Backward Detrending Moving Average (BDMA), and Centred Detrending Moving Average (CDMA)]. We use three different generators [Fractional Gaussian Noises, and two ways of generating Fractional Brownian Motions]. We find that CDMA has the best performance and DFA is only slightly worse in some situations, while FA performs the worst. In addition, CDMA and DFA are less sensitive to the scaling range than FA. Hence, CDMA and DFA remain "The Methods of Choice" in determining the Hurst index of time series.

Evaluation of scale invariance in physiological signals by means of balanced estimation of diffusion entropy

By means of the concept of the balanced estimation of diffusion entropy, we evaluate the reliable scale invariance embedded in different sleep stages and stride records. Segments corresponding to waking, light sleep, rapid eye movement (REM) sleep, and deep sleep stages are extracted from long-term electroencephalogram signals. For each stage the scaling exponent value is distributed over a considerably wide range, which tell us that the scaling behavior is subject and sleep cycle dependent. The average of the scaling exponent values for waking segments is almost the same as that for REM segments (∼0.8). The waking and REM stages have a significantly higher value of the average scaling exponent than that for light sleep stages (∼0.7). For the stride series, the original diffusion entropy (DE) and the balanced estimation of diffusion entropy (BEDE) give almost the same results for detrended series. The evolutions of local scaling invariance show that the physiological states change abruptly, although in the experiments great efforts have been made to keep conditions unchanged. The global behavior of a single physiological signal may lose rich information on physiological states. Methodologically, the BEDE can evaluate with considerable precision the scale invariance in very short time series (∼10^{2}), while the original DE method sometimes may underestimate scale-invariance exponents or even fail in detecting scale-invariant behavior. The BEDE method is sensitive to trends in time series. The existence of trends may lead to an unreasonably high value of the scaling exponent and consequent mistaken conclusions.

An ensemble system for automatic sleep stage classification using single channel EEG signal

The present work aims at automatic identification of various sleep stages like, sleep stages 1, 2, slow wave sleep (sleep stages 3 and 4), REM sleep and wakefulness from single channel EEG signal. Automatic scoring of sleep stages was performed with the help of pattern recognition technique which involves feature extraction, selection and finally classification. Total 39 numbers of features from time domain, frequency domain and from non-linear analysis were extracted. After extraction of features, SVM based recursive feature elimination (RFE) technique was used to find the optimum number of feature subset which can provide significant classification performance with reduced number of features for the five different sleep stages. Finally for classification, binary SVMs were combined with one-against-all (OAA) strategy. Careful extraction and selection of optimum feature subset helped to reduce the classification error to 8.9% for training dataset, validated by k-fold cross-validation (CV) technique and 10.61% in the case of independent testing dataset. Agreement of the estimated sleep stages with those obtained by expert scoring for all sleep stages of training dataset was 0.877 and for independent testing dataset it was 0.8572. The proposed ensemble SVM-based method could be used as an efficient and cost-effective method for sleep staging with the advantage of reducing stress and burden imposed on subjects.

Detection of a long-range correlation with an adaptive detrending method

We propose a methodology of estimating the scaling exponent for a long-range correlation in a nonstationary time series from the perspective of the regression analysis. By an adaptive degree determination of a regression polynomial, the proposed methodology is designed to properly remove various types of trends embedded in the nonstationary signal so that the scaling exponent can be estimated without artificial crossovers. To show the validity of the proposed methodology, we applied it to the detrended fluctuation analysis and tested it out against correlated data superimposed by various types of trends. It turned out that, unlike the conventional technique, our approach was capable of eliminating artificial crossovers. We also discuss the statistical characteristics of the proposed method with regard to the estimation of the scaling exponent.

Scaling of seismic memory with earthquake size

It has been observed that discrete earthquake events possess memory, i.e., that events occurring in a particular location are dependent on the history of that location. We conduct an analysis to see whether continuous real-time data also display a similar memory and, if so, whether such autocorrelations depend on the size of earthquakes within close spatiotemporal proximity. We analyze the seismic wave form database recorded by 64 stations in Japan, including the 2011 "Great East Japan Earthquake," one of the five most powerful earthquakes ever recorded, which resulted in a tsunami and devastating nuclear accidents. We explore the question of seismic memory through use of mean conditional intervals and detrended fluctuation analysis (DFA). We find that the wave form sign series show power-law anticorrelations while the interval series show power-law correlations. We find size dependence in earthquake autocorrelations: as the earthquake size increases, both of these correlation behaviors strengthen. We also find that the DFA scaling exponent α has no dependence on the earthquake hypocenter depth or epicentral distance.

Adaptive time-varying detrended fluctuation analysis

Detrended fluctuation analysis (DFA) is a technique commonly used to assess and quantify the presence of long-range temporal correlations (LRTCs) in neurophysiological time series. Convergence of the method is asymptotic only and therefore its application assumes a constant scaling exponent. However, most neurophysiological data are likely to involve either spontaneous or experimentally induced scaling exponent changes. We present a novel extension of the DFA method that permits the characterisation of time-varying scaling exponents. The effectiveness of the methodology in recovering known changes in scaling exponents is demonstrated through its application to synthetic data. The dependence of the method on its free parameters is systematically explored. Finally, application of the methodology to neurophysiological data demonstrates that it provides experimenters with a way to identify previously un-recognised changes in the scaling exponent in the data. We suggest that this methodology will make it possible to go beyond a simple demonstration of the presence of scaling to an appreciation of how it may vary in response to either intrinsic changes or experimental perturbations.

Segmentation of time series with long-range fractal correlations

Segmentation is a standard method of data analysis to identify change-points dividing a nonstationary time series into homogeneous segments. However, for long-range fractal correlated series, most of the segmentation techniques detect spurious change-points which are simply due to the heterogeneities induced by the correlations and not to real nonstationarities. To avoid this oversegmentation, we present a segmentation algorithm which takes as a reference for homogeneity, instead of a random i.i.d. series, a correlated series modeled by a fractional noise with the same degree of correlations as the series to be segmented. We apply our algorithm to artificial series with long-range correlations and show that it systematically detects only the change-points produced by real nonstationarities and not those created by the correlations of the signal. Further, we apply the method to the sequence of the long arm of human chromosome 21, which is known to have long-range fractal correlations. We obtain only three segments that clearly correspond to the three regions of different G + C composition revealed by means of a multi-scale wavelet plot. Similar results have been obtained when segmenting all human chromosome sequences, showing the existence of previously unknown huge compositional superstructures in the human genome.

Statistical laws governing fluctuations in word use from word birth to word death

We analyze the dynamic properties of 10(7) words recorded in English, Spanish and Hebrew over the period 1800-2008 in order to gain insight into the coevolution of language and culture. We report language independent patterns useful as benchmarks for theoretical models of language evolution. A significantly decreasing (increasing) trend in the birth (death) rate of words indicates a recent shift in the selection laws governing word use. For new words, we observe a peak in the growth-rate fluctuations around 40 years after introduction, consistent with the typical entry time into standard dictionaries and the human generational timescale. Pronounced changes in the dynamics of language during periods of war shows that word correlations, occurring across time and between words, are largely influenced by coevolutionary social, technological, and political factors. We quantify cultural memory by analyzing the long-term correlations in the use of individual words using detrended fluctuation analysis.

Revisiting detrended fluctuation analysis

Half a century ago Hurst introduced Rescaled Range (R/S) Analysis to study fluctuations in time series. Thousands of works have investigated or applied the original methodology and similar techniques, with Detrended Fluctuation Analysis becoming preferred due to its purported ability to mitigate nonstationaries. We show Detrended Fluctuation Analysis introduces artifacts for nonlinear trends, in contrast to common expectation, and demonstrate that the empirically observed curvature induced is a serious finite-size effect which will always be present. Explicit detrending followed by measurement of the diffusional spread of a signals' associated random walk is preferable, a surprising conclusion given that Detrended Fluctuation Analysis was crafted specifically to replace this approach. The implications are simple yet sweeping: there is no compelling reason to apply Detrended Fluctuation Analysis as it 1) introduces uncontrolled bias; 2) is computationally more expensive than the unbiased estimator; and 3) cannot provide generic or useful protection against nonstationaries.

Robustness of estimators of long-range dependence and self-similarity under non-Gaussianity

Long-range dependence (LRD) and non-Gaussianity are ubiquitous in many natural systems such as ecosystems, biological systems and climate. However, it is not always appreciated that the two phenomena may occur together in natural systems and that self-similarity in a system can be a superposition of both phenomena. These features, which are common in complex systems, impact the attribution of trends and the occurrence and clustering of extremes. The risk assessment of systems with these properties will lead to different outcomes (e.g. return periods) than the more common assumption of independence of extremes. Two paradigmatic models are discussed that can simultaneously account for LRD and non-Gaussianity: autoregressive fractional integrated moving average (ARFIMA) and linear fractional stable motion (LFSM). Statistical properties of estimators for LRD and self-similarity are critically assessed. It is found that the most popular estimators can be biased in the presence of important features of many natural systems like trends and multiplicative noise. Also the LRD and non-Gaussianity of two typical natural time series are discussed.

Geometrical exponents of contour loops on synthetic multifractal rough surfaces: multiplicative hierarchical cascade p model

In this paper, we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces, which are generated by multiplicative hierarchical cascading processes. To this end, two different classes of multifractal rough surfaces are numerically simulated. As the first group, singular measure multifractal rough surfaces are generated by using the p model. The smoothened multifractal rough surface then is simulated by convolving the first group with a so-called Hurst exponent, H*. The generalized multifractal dimension of isoheight lines (contours), D(q), correlation exponent of contours, x(l), cumulative distributions of areas, ξ, and perimeters, η, are calculated for both synthetic multifractal rough surfaces. Our results show that for both mentioned classes, hyperscaling relations for contour loops are the same as that of monofractal systems. In contrast to singular measure multifractal rough surfaces, H* plays a leading role in smoothened multifractal rough surfaces. All computed geometrical exponents for the first class depend not only on its Hurst exponent but also on the set of p values. But in spite of multifractal nature of smoothened surfaces (second class), the corresponding geometrical exponents are controlled by H*, the same as what happens for monofractal rough surfaces.

Long-range temporal correlations in the EEG bursts of human preterm babies

The electrical activity in the very early human preterm brain, as recorded by scalp EEG, is mostly discontinuous and has bursts of high-frequency oscillatory activity nested within slow-wave depolarisations of high amplitude. The temporal organisation of the occurrence of these EEG bursts has not been previously investigated. We analysed the distribution of the EEG bursts in 11 very preterm (23-30 weeks gestational age) human babies through two estimates of the Hurst exponent. We found long-range temporal correlations (LRTCs) in the occurrence of these EEG bursts demonstrating that even in the very immature human brain, when the cerebral cortical structure is far from fully developed, there is non-trivial temporal structuring of electrical activity.

Culturomics meets random fractal theory: insights into long-range correlations of social and natural phenomena over the past two centuries

Culturomics was recently introduced as the application of high-throughput data collection and analysis to the study of human culture. Here, we make use of these data by investigating fluctuations in yearly usage frequencies of specific words that describe social and natural phenomena, as derived from books that were published over the course of the past two centuries. We show that the determination of the Hurst parameter by means of fractal analysis provides fundamental insights into the nature of long-range correlations contained in the culturomic trajectories, and by doing so offers new interpretations as to what might be the main driving forces behind the examined phenomena. Quite remarkably, we find that social and natural phenomena are governed by fundamentally different processes. While natural phenomena have properties that are typical for processes with persistent long-range correlations, social phenomena are better described as non-stationary, on-off intermittent or Lévy walk processes.

Multiscale multifractal analysis of heart rate variability recordings with a large number of occurrences of arrhythmia

Human heart rate variability, in the form of time series of intervals between heart beats, shows complex, fractal properties. Recently, it was demonstrated many times that the fractal properties vary from point to point along the series, leading to multifractality. In this paper, we concentrate not only on the fact that the human heart rate has multifractal properties but also that these properties depend on the time scale in which the multifractality is measured. This time scale is related to the frequency band of the signal. We find that human heart rate variability appears to be far more complex than hitherto reported in the studies using a fixed time scale. We introduce a method called multiscale multifractal analysis (MMA), which allows us to extend the description of heart rate variability to include the dependence on the magnitude of the variability and time scale (or frequency band). MMA is relatively immune to additive noise and nonstationarity, including the nonstationarity due to inclusions into the time series of events of a different dynamics (e.g., arrhythmic events in sinus rhythm). The MMA method may provide new ways of measuring the nonlinearity of a signal, and it may help to develop new methods of medical diagnostics.

Ivanov P, Carpena P. Phase transitions in the first-passage time of scale-invariant correlated processes

A key quantity describing the dynamics of complex systems is the first-passage time (FPT). The statistical properties of FPT depend on the specifics of the underlying system dynamics. We present a unified approach to account for the diversity of statistical behaviors of FPT observed in real-world systems. We find three distinct regimes, separated by two transition points, with fundamentally different behavior for FPT as a function of increasing strength of the correlations in the system dynamics: stretched exponential, power-law, and saturation regimes. In the saturation regime, the average length of FPT diverges proportionally to the system size, with important implications for understanding electronic delocalization in one-dimensional correlated-disordered systems.

On the limits of the air pollution predictability: the case of the surface ozone at Athens, Greece

PURPOSE

The aim of this study is to investigate the potential effects of increased urbanization in the Athens city, Greece on the intrinsic features of the temporal fluctuations of the surface ozone concentration (SOC).

METHODS

The detrended fluctuation analysis was applied to the mean monthly values of SOC derived from ground-based observations collected at the centre of Athens basin during 1901-1940 and 1987-2007.

RESULTS

Despite the present-day SOC doubling in respect to SOC historic levels, its fluctuations exhibit long-range power-law persistence, with similar features in both time periods. This contributes to an improved understanding of our predictive powers and enables better environmental management and more efficient decision-making processes.

CONCLUSIONS

The extensive photochemistry enhancement observed in the Athens basin from the beginning of the twentieth century until the beginning of the twenty-first century seems not to have affected the long memory of SOC correlations. The strength of this memory stems from its temporal evolution and provides the limits of the air pollution predictability at various time scales.

A FAST ALGORITHM FOR COMPUTING SAMPLE ENTROPY

Sample entropy is a widely used tool for quantifying complexity of a biological system. Computing sample entropy directly using its definition requires large computational costs. We propose a fast algorithm based on a k-d tree data structure for computing sample entropy. We prove that the time complexity of the proposed algorithm is [Formula: see text] and its space complexity is O(N log N), where N is the length of the input time series and m is the length of its pattern templates. We present a numerical experiment that demonstrates significant improvement of the proposed algorithm in computing time.

CONSTRUCTING CROSSOVER-FRACTALS USING INTRINSIC MODE FUNCTIONS

Real nonstationary time sequences are in general not monofractals. That is, they cannot be characterized by a single value of fractal dimension. It has been shown that many real-time sequences are crossover-fractals: sequences with two fractal dimensions — one for the short and the other for long ranges. Here, we use the empirical mode decomposition (EMD) to decompose monofractals into several intrinsic mode functions (IMFs) and then use partial sums of the IMFs decomposed from two monofractals to construct crossover-fractals. The scale-dependent fractal dimensions of these crossover-fractals are checked by the inverse random midpoint displacement method (IRMD).

Effects of coarse-graining on the scaling behavior of long-range correlated and anti-correlated signals

We investigate how various coarse-graining (signal quantization) methods affect the scaling properties of long-range power-law correlated and anti-correlated signals, quantified by the detrended fluctuation analysis. Specifically, for coarse-graining in the magnitude of a signal, we consider (i) the Floor, (ii) the Symmetry and (iii) the Centro-Symmetry coarse-graining methods. We find that for anti-correlated signals coarse-graining in the magnitude leads to a crossover to random behavior at large scales, and that with increasing the width of the coarse-graining partition interval Δ, this crossover moves to intermediate and small scales. In contrast, the scaling of positively correlated signals is less affected by the coarse-graining, with no observable changes when Δ < 1, while for Δ > 1 a crossover appears at small scales and moves to intermediate and large scales with increasing Δ. For very rough coarse-graining (Δ > 3) based on the Floor and Symmetry methods, the position of the crossover stabilizes, in contrast to the Centro-Symmetry method where the crossover continuously moves across scales and leads to a random behavior at all scales; thus indicating a much stronger effect of the Centro-Symmetry compared to the Floor and the Symmetry method. For coarse-graining in time, where data points are averaged in non-overlapping time windows, we find that the scaling for both anti-correlated and positively correlated signals is practically preserved. The results of our simulations are useful for the correct interpretation of the correlation and scaling properties of symbolic sequences.

Self-similarity of higher-order moving averages

In this work, higher-order moving average polynomials are defined by straightforward generalization of the standard moving average. The self-similarity of the polynomials is analyzed for fractional Brownian series and quantified in terms of the Hurst exponent H by using the detrending moving average method. We prove that the exponent H of the fractional Brownian series and of the detrending moving average variance asymptotically agree for the first-order polynomial. Such asymptotic values are compared with the results obtained by the simulations. The higher-order polynomials correspond to trend estimates at shorter time scales as the degree of the polynomial increases. Importantly, the increase of polynomial degree does not require to change the moving average window. Thus trends at different time scales can be obtained on data sets with the same size. These polynomials could be interesting for those applications relying on trend estimates over different time horizons (financial markets) or on filtering at different frequencies (image analysis).

Fractal and nonlinear changes in the long-term baseline fluctuations of fetal heart rate

The interpretation of heart rate patterns obtained by fetal monitoring relies on the definition of a baseline, which is considered as the running average heart rate in the absence of external stimuli during periods of fetal rest. We present a study along gestation of the baseline's fluctuations, in relation to fractal and nonlinear properties, to assess these fluctuations according with time-varying attracting levels introduced by maturing regulatory mechanisms. A low-risk pregnancy was studied weekly from the 17th to 38th week of gestation during long-term recording sessions at night (>6 h). Fetal averaged pulse rate samples and corresponding baseline series were obtained from raw abdominal ECG ambulatory data. The fractal properties of these series were evaluated by applying detrended fluctuation analysis. The baseline series were also explored to evaluate nonlinear properties and time ordering by applying the scaling magnitude and sign analyses. Our main findings are that the baseline shows fractal and even nonlinear anticorrelated fluctuations. This condition was specially the case before mid-gestation, as revealed by α values near to unit, yet becoming significantly more complex after 30 weeks of gestation as indicated by α(mag) values >0.5. The structured (i.e. not random) fluctuations and particular nonlinear changes that we found thus suggest that the baseline provides on itself information concerning the functional integration of cardiac regulatory mechanisms.

Distribution of natural trends in long-term correlated records: a scaling approach

We estimate the exceedance probability W(x,α;L) that, in a long-term correlated Gaussian-distributed (sub) record of length L characterized by a fluctuation exponent α between 0.5 and 1.5, a relative increase Δ/σ(t) of size larger than x occurs, where Δ is the total observed increase measured by linear regression and σ(t) is the standard deviation around the regression line. We consider L between 500 and 2000, which is the typical length scale of monthly local and reconstructed annual global temperature records. We use scaling theory to obtain an analytical expression for W(x,α;L). From this expression, we can determine analytically, for a given confidence probability Q, the boundaries ±x(Q)(α,L) of the confidence interval. In the presence of an external linear trend, the total observed increase is the sum of the natural and the external increase. An observed relative increase Δ/σ(t) is considered unnatural when it is above x(Q)(α,L). In this case, the size of the external relative increase is bounded by Δ/σ(t)±x(Q)(α,L). We apply this approach to various global and local climate data and discuss the different results for the significance of the observed trends.