Effect of trends on detrended fluctuation analysis

1/f(alpha) spectra in elementary cellular automata and fractal signals

We systematically compute the power spectra of the one-dimensional elementary cellular automata introduced by Wolfram. On the one hand our analysis reveals that one automaton displays 1/f spectra though considered as trivial, and on the other hand that various automata classified as chaotic or complex display no 1/f spectra. We model the results generalizing the recently investigated Sierpinski signal to a class of fractal signals that are tailored to produce 1/f(alpha) spectra. From the widespread occurrence of (elementary) cellular automata patterns in chemistry, physics, and computer sciences, there are various candidates to show spectra similar to our results.

Empirical mode decomposition and correlation properties of long daily ozone records

Correlations for daily data of total ozone column are investigated by detrended fluctuation analysis (DFA). The removal of annual periodicity does not result in a background-free signal for the tropical station Mauna Loa. In order to identify the remaining quasiperiodic constituent, the relatively new method of empirical mode decomposition (EMD) is tested. We found that the so-called intrinsic mode functions do not represent real signal components of the ozone time series, their amplitude modulation is very sensitive to local changes such as random data removal or smoothing. Tests on synthetic data further corroborate the limitations of decomposing quasiperiodic signals from noise with EMD. Nevertheless the EMD algorithm helps to identify dominating frequencies in the time series, which allows to separate fluctuations from the remaining background. We demonstrate that DFA analysis for the cleaned Mauna Loa record yields scaling comparable to a mid-latitude station.

Quantifying signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques

Detrended fluctuation analysis (DFA) and detrended moving average (DMA) are two scaling analysis methods designed to quantify correlations in noisy nonstationary signals. We systematically study the performance of different variants of the DMA method when applied to artificially generated long-range power-law correlated signals with an a priori known scaling exponent alpha(0) and compare them with the DFA method. We find that the scaling results obtained from different variants of the DMA method strongly depend on the type of the moving average filter. Further, we investigate the optimal scaling regime where the DFA and DMA methods accurately quantify the scaling exponent alpha(0) , and how this regime depends on the correlations in the signal. Finally, we develop a three-dimensional representation to determine how the stability of the scaling curves obtained from the DFA and DMA methods depends on the scale of analysis, the order of detrending, and the order of the moving average we use, as well as on the type of correlations in the signal.

Long-range temporal correlations in electroencephalographic oscillations: Relation to topography, frequency band, age and gender

Presence of long-range temporal correlations in neuronal oscillations is thought to be beneficial for a reliable transfer of information in neuronal networks. In the present study long-range temporal correlations in electroencephalographic (EEG) neuronal oscillations were characterized with respect to their topography, frequency-band specificity (alpha and beta oscillations), gender and age. EEG was recorded in 91 normal subjects (age 20-65 years) in a resting condition. The amplitude of ongoing alpha and beta oscillations was extracted with band-pass filtering and Hilbert transform, and long-range temporal correlations were analyzed with detrended fluctuation analysis. The topography of long-range temporal correlations was comparable for alpha and beta oscillations, showing largest scaling exponents in the occipital and parietal areas. This topography was partially similar to that of the power distribution and a weak positive correlation was observed between long-range temporal correlations and power of neuronal oscillations. Long-range temporal correlations were stronger in alpha than beta oscillations, but only in a few electrode locations in the left hemisphere. In both frequency bands long-range temporal correlations were stronger in males than in females and were largely unaffected by the age of the subjects. It is hypothesized that the idling state of the occipital areas in the closed-eyes condition may explain both large power values and pronounced long-range temporal correlations in this region.

Effect of nonlinear filters on detrended fluctuation analysis

When investigating the dynamical properties of complex multiple-component physical and physiological systems, it is often the case that the measurable system's output does not directly represent the quantity we want to probe in order to understand the underlying mechanisms. Instead, the output signal is often a linear or nonlinear function of the quantity of interest. Here, we investigate how various linear and nonlinear transformations affect the correlation and scaling properties of a signal, using the detrended fluctuation analysis (DFA) which has been shown to accurately quantify power-law correlations in nonstationary signals. Specifically, we study the effect of three types of transforms: (i) linear ( y(i) =a x(i) +b) , (ii) nonlinear polynomial ( y(i) =a x(k)(i) ) , and (iii) nonlinear logarithmic [ y(i) =log ( x(i) +Delta) ] filters. We compare the correlation and scaling properties of signals before and after the transform. We find that linear filters do not change the correlation properties, while the effect of nonlinear polynomial and logarithmic filters strongly depends on (a) the strength of correlations in the original signal, (b) the power k of the polynomial filter, and (c) the offset Delta in the logarithmic filter. We further apply the DFA method to investigate the "apparent" scaling of three analytic functions: (i) exponential [exp (+/-x+a) ] , (ii) logarithmic [log (x+a) ] , and (iii) power law [ (x+a)(lambda) ] , which are often encountered as trends in physical and biological processes. While these three functions have different characteristics, we find that there is a broad range of values for parameter a common for all three functions, where the slope of the DFA curves is identical. We further note that the DFA results obtained for a class of other analytic functions can be reduced to these three typical cases. We systematically test the performance of the DFA method when estimating long-range power-law correlations in the output signals for different parameter values in the three types of filters and the three analytic functions we consider.

Endogenous circadian rhythm in an index of cardiac vulnerability independent of changes in behavior

There exists a robust day/night pattern in the incidence of adverse cardiac events with a peak at approximately 10 a.m. This peak traditionally has been attributed to day/night patterns in behaviors affecting cardiac function in vulnerable individuals. However, influences from the endogenous circadian pacemaker independent from behaviors may also affect cardiac control. Heartbeat dynamics under healthy conditions exhibit robust complex fluctuations characterized by self-similar temporal structures, which break down under pathologic conditions. We hypothesize that these dynamical features of the healthy human heartbeat have an endogenous circadian rhythm that brings the features closer to those observed under pathologic conditions at the endogenous circadian phase corresponding to approximately 10 a.m. We investigate heartbeat dynamics in healthy subjects recorded throughout a 10-day protocol wherein the sleep/wake and behavior cycles are desynchronized from the endogenous circadian cycle, enabling assessment of circadian factors while controlling for behavior-related factors. We demonstrate that the scaling exponent characterizing temporal correlations in heartbeat dynamics does exhibit a significant circadian rhythm (with a sharp peak at the circadian phase corresponding to approximately 10 a.m.), which is independent from scheduled behaviors and mean heart rate. Cardiac dynamics under pathologic conditions such as congestive heart failure also are associated with a larger value of the scaling exponent of the interbeat interval. Thus, the sharp peak in the scaling exponent at the circadian phase coinciding with the period of highest cardiac vulnerability observed in epidemiological studies suggests that endogenous circadian-mediated influences on cardiac control may be involved in the day/night pattern of adverse cardiac events in vulnerable individuals.

Sierpinski signal generates 1/f alpha spectra

We investigate the row sum of the binary pattern generated by the Sierpinski automaton: Interpreted as a time series we calculate the power spectrum of this Sierpinski signal analytically and obtain a unique rugged fine structure with underlying power law decay with an exponent of approximately 1.15. Despite the simplicity of the model, it can serve as a model for 1/f(alpha) spectra in a certain class of experimental and natural systems such as catalytic reactions and mollusc patterns.

Anomalous scaling of superrough growing surfaces: from correlation functions to residual local interfacial widths and scaling exponents

A study on the (1+1) -dimensional superrough growth processes is undertaken. We first work out the exact relations among the local interfacial width w , the correlation function G , and the pth degree residual local interfacial width w(p) with p=1,2,3,... . The relations obtained are exact and thus can be applied to any (1+1) -dimensional growth processes in the continuum limit, no matter whether the interface is superrough or not. Then we investigate the influence of the macroscopic structure formation on the scaling behavior of the superrough growth processes. Moreover, we show analytically that the residual local interfacial width w(p) excludes only the influence of the macroscopic structure on the scaling behavior of the system and retains the true scaling behavior originating from the stochastic nature of the system. Finally, we analyze and simulate some superrough growth models for demonstration.

Long-range temporal correlations in alpha and beta oscillations: effect of arousal level and test–retest reliability

OBJECTIVE

The aim of the present study was to evaluate test-retest reliability and condition sensitivity of long-range temporal correlations in the amplitude dynamics of electroencephalographic alpha and beta oscillations.

METHODS

Twelve normal subjects were measured two times with a test-retest interval of several days. Open- and closed-eyes conditions were used, representing different levels of arousal. The amplitude of the alpha and beta oscillations was extracted with bandpass filtering and the Hilbert transform. The long-range temporal correlations were quantified with detrended fluctuation analysis.

RESULTS

The amplitude dynamics of the alpha and beta oscillations demonstrated power-law long-range temporal correlations lasting for tens of seconds. These correlations were degraded in the open- compared to the closed-eyes condition. Test-retest statistics demonstrated that the long-range temporal correlations had significant reliability, which was greatest in the closed-eyes condition.

CONCLUSIONS

The presence of long-range temporal correlations indicates that the amplitude of neuronal oscillations at a given time is dependent on the amplitude at times as remote in the past as tens of seconds. The reliability of long-range temporal correlations suggests that the mechanisms generating the amplitude fluctuations are not perturbed over several days. The systematic changes in the scaling exponents at different levels of arousal indicate that these changes occur on many time scales (5-80 s) as a result of modifications in the intrinsic dynamics of the neuronal oscillations.

SIGNIFICANCE

This study demonstrates that the dynamics of spontaneous neuronal oscillations possess long-range temporal correlations with properties suitable for functional and clinical studies.

Changes in the Hurst exponent of heartbeat intervals during physical activity

The fractal scaling properties of the heartbeat time series are studied in different controlled ergometric regimes using both the improved Hurst rescaled range (R/S) analysis and the detrended fluctuation analysis (DFA). The long-time "memory effect" quantified by the value of the Hurst exponent H>0.5 is found to increase during progressive physical activity in healthy subjects, in contrast to those having stable angina pectoris, where it decreases. The results are also supported by the detrended fluctuation analysis. We argue that this finding may be used as a useful new diagnostic parameter for short heartbeat time series.

Detrended fluctuation analysis: a suitable method for studying fetal heart rate variability?

We evaluate the suitability of an enhanced detrended fluctuation analysis for studying fetal heart rate series involving imperfect quality of information. Our results indicate that to explore persistent long-range correlations, or fractality, the collection requirements of the data can be relaxed by allowing the possibility of using averaged fetal heart rate series. In addition, it also appears feasible to employ, without producing major alterations in the long-range scaling behaviour, fragmented fetal heart rate series involving up to 50% of random missing values, or up to 50 min of consecutive missing samples in recordings of approximately equal to 8 h length. These are crucial advantages to overcome the often variable quality of fetal data. Consequently, these findings may open the possibility of obtaining information concerning the development of neural processes from fetal heart rate series, despite their non-stationary and fragmented nature.

Temporal series analysis approach to spectra of complex networks

The spacing of nearest levels of the spectrum of a complex network can be regarded as a time series. Joint use of the multifractal detrended fluctuation approach (MF-DFA) and diffusion entropy (DE) is employed to extract characteristics from this time series. For the Watts-Strogatz small-world model, there exists a critical point at rewiring probability P(r) =0.32. For a network generated in the range 0< P(r) <0.32, the correlation exponent is in the range of 1.0-1.64. Above this critical point, all the networks behave similar to that at p(r) =1. For the Erdos-Renyi model, the time series behaves like fractional Brownian motion noise at p(ER) =1/N. For the growing random network (GRN) model, the values of the long-range correlation exponent are in the range of 0.74-0.83. For most of the GRN networks the probability distribution function of a constructed time series obeys a Gaussian form. In the joint use of MF-DFA and DE, the shuffling procedure in DE is essential to obtain a reliable result.

Non-random fluctuations and multi-scale dynamics regulation of human activity

We investigate if known extrinsic and intrinsic factors fully account for the complex features observed in recordings of human activity as measured from forearm motion in subjects undergoing their regular daily routine. We demonstrate that the apparently random forearm motion possesses dynamic patterns characterized by robust scale-invariant and nonlinear features. These patterns remain stable from one subject to another and are unaffected by changes in the average activity level that occur within individual subjects throughout the day and on different days of the week, since they persist during daily routine and when the same subjects undergo time-isolation laboratory experiments designed to account for the circadian phase and to control the known extrinsic factors. Further, by modeling the scheduled events imposed throughout the laboratory protocols, we demonstrate that they cannot account for the observed scaling patterns in activity fluctuations. We attribute these patterns to a previously unrecognized intrinsic nonlinear multi-scale control mechanism of human activity that is independent of known extrinsic factors such as random and scheduled events, as well as the known intrinsic factors which possess a single characteristic time scale such as circadian and ultradian rhythms.

Common scaling patterns in intertrade times of U. S. stocks

We analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the U.S. economy over a period of 4 yrs. For all companies we find that: (i) the probability density function of intertrade times may be fit by a Weibull distribution, (ii) when appropriately rescaled the probability densities of all companies collapse onto a single curve implying a universal functional form, (iii) the intertrade times exhibit power-law correlated behavior within a trading day and a consistently greater degree of correlation over larger time scales, in agreement with the correlation behavior of the absolute price returns for the corresponding company, and (iv) the magnitude series of intertrade time increments is characterized by long-range power-law correlations suggesting the presence of nonlinear features in the trading dynamics, while the sign series is anticorrelated at small scales. Our results suggest that independent of industry sector, market capitalization and average level of trading activity, the series of intertrade times exhibit possibly universal scaling patterns, which may relate to a common mechanism underlying the trading dynamics of diverse companies. Further, our observation of long-range power-law correlations and a parallel with the crossover in the scaling of absolute price returns for each individual stock, support the hypothesis that the dynamics of transaction times may play a role in the process of price formation.

Extraction of backgrounds in fluctuating systems

We undertake an extensive analytical study on the "generalized detrended fluctuation analysis" method, designed to detect the scaling behaviors of fluctuating systems but exclude out the influences of the backgrounds (or the trends). Through our extensive studies, we systematically extract out the exact backgrounds (or the trends) of the fluctuating systems to any order, expressed in terms of the Legendre polynomial. Our results are exact and can be applied to any (1+1)-dimensional continuous fluctuating systems.

Nonuniversal atmospheric persistence: different scaling of daily minimum and maximum temperatures

An extensive investigation of 61 daily temperature records by means of detrended fluctuation analysis has revealed that the value of correlation exponent is not universal, contrary to earlier claims. Furthermore, statistically significant differences are found for daily minimum and maximum temperatures measured at the same station, suggesting different degrees of long-range correlations for the two extremes. Numerical tests on synthetic time series demonstrate that a correlated signal interrupted by uncorrelated segments exhibits an apparently lower exponent value over the usual time window of empirical data analysis. In order to find statistical differences between the two daily extreme temperatures, high frequency (10 min) records were evaluated for two distant locations. The results show that daily maxima characterize better the dynamic equilibrium state of the atmosphere than daily minima, for both stations. This provides a conceptual explanation why scaling analysis can yield different exponent values for minima and maxima.

Analysis of clusters formed by the moving average of a long-range correlated time series

We analyze the stochastic function C(n)(i) identical with y(i)-y(n)(i), where y(i) is a long-range correlated time series of length N(max) and y(n)(i) identical with (1/n) Sigma(n-1)(k=0)y(i-k) is the moving average with window n. We argue that C(n)(i) generates a stationary sequence of self-affine clusters C with length l, lifetime tau, and area s. The length and the area are related to the lifetime by the relationships l approximately tau(psi(l)) and s approximately tau(psi(s)), where psi(l)=1 and psi(s)=1+H. We also find that l, tau, and s are power law distributed with exponents depending on H: P(l) approximately l(-alpha), P(tau) approximately tau(-beta), and P(s) approximately s(-gamma), with alpha=beta=2-H and gamma=2/(1+H). These predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned Hurst exponent H (ranging from 0.05 to 0.95) of length up to N(max)=2(21) and n up to 2(13).

Comparison of detrended fluctuation analysis and spectral analysis for heart rate variability in sleep and sleep apnea

Sleep has been regarded as a testing situation for the autonomic nervous system, because its activity is modulated by sleep stages. Sleep-related breathing disorders also influence the autonomic nervous system and can cause heart rate changes known as cyclical variation. We investigated the effect of sleep stages and sleep apnea on autonomic activity by analyzing heart rate variability (HRV). Since spectral analysis is suited for the identification of cyclical variations and detrended fluctuation analysis can analyze the scaling behavior and detect long-range correlations, we compared the results of both complementary techniques in 14 healthy subjects, 33 patients with moderate, and 31 patients with severe sleep apnea. The spectral parameters VLF, LF, HF, and LF/HF confirmed increasing parasympathetic activity from wakefulness and REM over light sleep to deep sleep, which is reduced in patients with sleep apnea. Discriminance analysis was used on a person and sleep stage basis to determine the best method for the separation of sleep stages and sleep apnea severity. Using spectral parameters 69.7% of the apnea severity assignments and 54.6% of the sleep stage assignments were correct, while using scaling analysis these numbers increased to 74.4% and 85.0%, respectively. We conclude that changes in HRV are better quantified by scaling analysis than by spectral analysis.

Power-law persistence and trends in the atmosphere: a detailed study of long temperature records

We use several variants of the detrended fluctuation analysis to study the appearance of long-term persistence in temperature records, obtained at 95 stations all over the globe. Our results basically confirm earlier studies. We find that the persistence, characterized by the correlation C(s) of temperature variations separated by s days, decays for large s as a power law, C(s) approximately s(-gamma). For continental stations, including stations along the coastlines, we find that gamma is always close to 0.7. For stations on islands, we find that gamma ranges between 0.3 and 0.7, with a maximum at gamma=0.4. This is consistent with earlier studies of the persistence in sea surface temperature records where gamma is close to 0.4. In all cases, the exponent gamma does not depend on the distance of the stations to the continental coastlines. By varying the degree of detrending in the fluctuation analysis we obtain also information about trends in the temperature records.

Dynamical model and nonextensive statistical mechanics of a market index on large time windows

The shape and tails of partial distribution functions (PDF) for a financial signal, i.e., the S&P500 and the turbulent nature of the markets are linked through a model encompassing Tsallis nonextensive statistics and leading to evolution equations of the Langevin and Fokker-Planck type. A model originally proposed to describe the intermittent behavior of turbulent flows describes the behavior of normalized log returns for such a financial market index, for small and large time windows, and both for small and large log returns. These turbulent market volatility (of normalized log returns) distributions can be sufficiently well fitted with a chi(2) distribution. The transition between the small time scale model of nonextensive, intermittent process, and the large scale Gaussian extensive homogeneous fluctuation picture is found to be at ca. a 200 day time lag. The intermittency exponent kappa in the framework of the Kolmogorov log-normal model is found to be related to the scaling exponent of the PDF moments, thereby giving weight to the model. The large value of kappa points to a large number of cascades in the turbulent process. The first Kramers-Moyal coefficient in the Fokker-Planck equation is almost equal to zero, indicating "no restoring force." A comparison is made between normalized log returns and mere price increments.

Long-memory analysis of time series with missing values

The estimation of long memory is often restricted by missing data. We examine the effects on the estimation of long memory of three simple gap-filling techniques: interpolation, random, and mean filling. Numerical simulations show that the gap-filling techniques introduce significant deviations from the expected scaling behavior for both persistent and antipersistent time series. For persistent time series the interpolation method provides a reliable estimation of long memory for scales longer than the largest likely gap.

Interpretation of heart rate variability via detrended fluctuation analysis and alphabeta filter

Detrended fluctuation analysis (DFA), suitable for the analysis of nonstationary time series, has confirmed the existence of persistent long-range correlations in healthy heart rate variability data. In this paper, we present the incorporation of the alphabeta filter to DFA to determine patterns in the power-law behavior that can be found in these correlations. Well-known simulated scenarios and real data involving normal and pathological circumstances were used to evaluate this process. The results presented here suggest the existence of evolving patterns, not always following a uniform power-law behavior, that cannot be described by scaling exponents estimated using a linear procedure over two predefined ranges. Instead, the power law is observed to have a continuous variation with segment length. We also show that the study of these patterns, avoiding initial assumptions about the nature of the data, may confer advantages to DFA by revealing more clearly abnormal physiological conditions detected in congestive heart failure patients related to the existence of dominant characteristic scales.

Reply to "Comment on 'Power-law correlations in the southern-oscillation-index fluctuations characterizing El Niño'"

Earlier [Phys. Rev. E 63, 047201 (2001)] we studied the southern oscillation index (SOI). Our findings tended to favor specific physical models for the El Niño description. The Comment by Metzler [Phys. Rev. E 67, 018201 (2003)] on this publication does not give any argument in favor of another El Niño physical model. In contrast, the Comment points out that statistical properties of the SOI data can be explained with a model based on a linear autoregressive process, but such a modeling does not help in identifying the relevant physical mechanisms.

THE GLOBAL EFFECTS OF STROKE ON THE HUMAN ELECTROENCEPHALOGRAM

The scaling properties of the fluctuations of EEG time series are used in an investigation of acute stroke in humans. We use detrended fluctuation analysis to characterize the fluctuations in 10-second time series in terms of two dimensionless scaling exponents. The statistics of these scaling exponents across 129 scalp sites define measures which may be used to distinguish normal subjects from those with acute cerebral ischemia. By their nature, these statistics emphasize the global properties of EEG dynamics. Simulation of a focal anomaly which accurately reproduces the mean scaling exponents for stroke subjects contradicts the data for the variances, which we take as evidence that the effect of stroke on EEG is global.

Scaling of atmosphere and ocean temperature correlations in observations and climate models

Power-law scaling of near surface air temperature fluctuations and its geographical distribution is analyzed in 100-yr observations and in a 1000-yr simulation of the present-day climate with a complex atmosphere-ocean model. In observations and simulation detrended fluctuation analysis leads to the scaling exponent alpha approximately 1 over the oceans, alpha approximately 0.5 over the inner continents, and alpha approximately 0.65 in transition regions [spectrum S(f) approximately f(-beta),beta=2alpha-1]. Scaling up to decades is demonstrated in observations and coupled atmosphere-ocean models with complex and mixed-layer oceans. Only with the complex ocean model the simulated power laws extend up to centuries.

Interpretation of scaling properties of electroencephalographic fluctuations via spectral analysis and underlying physiology

Detrended fluctuation analysis has recently demonstrated the existence of two approximate temporal scaling regimes in locally detrended human electroencephalographic (EEG) fluctuations, and has suggested a connection between the location of the breakpoint between regimes and the alpha resonance near 10 Hz. It is shown here that these scalings can be explained in terms of the filtering of the underlying power spectrum implied by the detrending process. Using a recent physiologically based model of EEG generation, the main features of the scalings, and deviations from them, are related to the underlying physiology of dendritic propagation and muscle electrical activity, and it is concluded that the effects of such physiological features are usually clearer in spectra.

Long-range correlations in the electric signals that precede rupture: further investigations

The correlations within the time series of the seismic electric signal (SES) activities have been studied in a previous paper [P. Varotsos, N. Sarlis, and E. Skordas, Phys. Rev. E 66, 011902 (2002)]. Here, we analyze the time series of successive high- and low-level states' durations. The existence of correlation between the states is investigated by means of Hurst and detrended fluctuation analysis (DFA). The multifractal DFA (MF-DFA) is also employed. The results point to a stronger correlation, and hence longer memory, in the series of the high-level states. Furthermore, an analysis in the "natural" time domain reveals that certain power spectrum characteristics seem to distinguish SES activities from "artificial" (man-made) electric noises. More precisely, for natural frequencies 0<phi<0.5, the curves of the SES activities and artificial noises lie above and below, respectively, that of the "uniform" distribution (UD). A classification of these two types of electric signals (SES activities, artificial noises), cannot be achieved on the basis of the values of the power-law exponents alone, if the Hurst analysis, DFA, and MF-DFA are applied to the original time series. The latter two methods, however, seem to allow a distinction between the SES activities and artificial noises when treating them (not in conventional the time frame, but) in the natural time domain. To further test the techniques, a time series produced by another system was examined. We chose a signal of ion current fluctuations in membrane channels (ICFMCs). The following conclusions, among others, have been obtained: First, the power spectrum analysis in the natural time domain shows that the ICFMC curve almost coincides (in the range 0<phi<0.5) with that of the UD, and hence ICFMC lies just in the boundary between the SES activities and artificial noises. Second, MF-DFA indicates monofractality for the ICFMCs with a generalized Hurst exponent h=0.84+/-0.03 in the range 7-70 ms.

Study of statistical correlations in DNA sequences

Here we present a study of statistical correlations among different positions in DNA sequences and their implications by directly using the autocorrelation function. Such an analysis is possible now because of the availability of large sequences or even complete genomes of many organisms. After describing the way in which the autocorrelation function can be applied to DNA-sequence analysis, we show that long-range correlations, implying scale independence, appear in several bacterial genomes as well as in long human chromosome contigs. The source for such correlations in bacteria, which may extend up to 60 kb in Bacillus subtilis, may be related to massive lateral transfer of compositionally biased genes from other genomes. In the human genome, correlations extend for more than five decades and may be related to the evolution of the 'neogenome', a modern evolutionary acquisition composed by GC-rich isochores displaying long-range correlations and scale invariance.

Scaling detection in time series: diffusion entropy analysis

The methods currently used to determine the scaling exponent of a complex dynamic process described by a time series are based on the numerical evaluation of variance. This means that all of them can be safely applied only to the case where ordinary statistical properties hold true even if strange kinetics are involved. We illustrate a method of statistical analysis based on the Shannon entropy of the diffusion process generated by the time series, called diffusion entropy analysis (DEA). We adopt artificial Gauss and Lévy time series, as prototypes of ordinary and anomalous statistics, respectively, and we analyze them with the DEA and four ordinary methods of analysis, some of which are very popular. We show that the DEA determines the correct scaling exponent even when the statistical properties, as well as the dynamic properties, are anomalous. The other four methods produce correct results in the Gauss case but fail to detect the correct scaling in the case of Lévy statistics.

Metal-insulator transition in chains with correlated disorder

According to Bloch's theorem, electronic wavefunctions in perfectly ordered crystals are extended, which implies that the probability of finding an electron is the same over the entire crystal. Such extended states can lead to metallic behaviour. But when disorder is introduced in the crystal, electron states can become localized, and the system can undergo a metal-insulator transition (also known as an Anderson transition). Here we theoretically investigate the effect on the physical properties of the electron wavefunctions of introducing long-range correlations in the disorder in one-dimensional binary solids, and find a correlation-induced metal-insulator transition. We perform numerical simulations using a one-dimensional tight-binding model, and find a threshold value for the exponent characterizing the long-range correlations of the system. Above this threshold, and in the thermodynamic limit, the system behaves as a conductor within a broad energy band; below threshold, the system behaves as an insulator. We discuss the possible relevance of this result for electronic transport in DNA, which displays long-range correlations and has recently been reported to be a one-dimensional disordered conductor.

Scaling breakdown: a signature of aging

We prove that the Lévy walk is characterized by bilinear scaling. This effect mirrors the existence of a form of aging that does not require the adoption of nonstationary conditions.

Stochastic modeling of daily temperature fluctuations

Classical spectral, Hurst, and detrended fluctuation analysis have been revealed asymptotic power-law correlations for daily average temperature data. For short-time intervals, however, strong correlations characterize the dynamics that permits a satisfactory description of temperature changes as a low order linear autoregressive process (dominating the texts on climate research). Here we propose a unifying stochastic model reproducing correlations for all time scales. The concept is an extension of a first-order autoregressive model with power-law correlated noise. The inclusion of a nonlinear "atmospheric response function" conveys the observed skew for the amplitude distribution of temperature fluctuations. While stochastic models cannot help to understand the physics behind atmospheric processes, they are capable to extract useful features promoting to benchmark physical models, an example is shown. Possible applications for other systems of strong short-range and asymptotic power-law correlations are discussed.

Characterization of sleep stages by correlations in the magnitude and sign of heartbeat increments

We study correlation properties of the magnitude and the sign of the increments in the time intervals between successive heartbeats during light sleep, deep sleep, and rapid eye movement (REM) sleep using the detrended fluctuation analysis method. We find short-range anticorrelations in the sign time series, which are strong during deep sleep, weaker during light sleep, and even weaker during REM sleep. In contrast, we find long-range positive correlations in the magnitude time series, which are strong during REM sleep and weaker during light sleep. We observe uncorrelated behavior for the magnitude during deep sleep. Since the magnitude series relates to the nonlinear properties of the original time series, while the sign series relates to the linear properties, our findings suggest that the nonlinear properties of the heartbeat dynamics are more pronounced during REM sleep. Thus, the sign and the magnitude series provide information which is useful in distinguishing between the sleep stages.

Effect of nonstationarities on detrended fluctuation analysis

Detrended fluctuation analysis (DFA) is a scaling analysis method used to quantify long-range power-law correlations in signals. Many physical and biological signals are "noisy," heterogeneous, and exhibit different types of nonstationarities, which can affect the correlation properties of these signals. We systematically study the effects of three types of nonstationarities often encountered in real data. Specifically, we consider nonstationary sequences formed in three ways: (i) stitching together segments of data obtained from discontinuous experimental recordings, or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts-a "cutting" procedure commonly used in preparing data prior to signal analysis; (ii) adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes; and (iii) generating a signal comprised of segments with different properties-e.g., different standard deviations or different correlation exponents. We compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities. We find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on (a) the fraction and size of the parts cut out from the signal, (b) the concentration of spikes and their amplitudes (c) the proportion between segments with different standard deviations or different correlations and (d) the correlation properties of the stationary signal. We show how to develop strategies for preprocessing "raw" data prior to analysis, which will minimize the effects of nonstationarities on the scaling properties of the data, and how to interpret the results of DFA for complex signals with different local characteristics.

Diffusion entropy and waiting time statistics of hard-x-ray solar flares

We show at work a technique of scaling detection based on evaluating the Shannon entropy of the diffusion process obtained by converting the time series under study into trajectories. This method, called diffusion entropy, affords information that cannot be derived from the direct evaluation of waiting times. We apply this method to the analysis of the distribution of time distance tau between two nearest-neighbor solar flares. This traditional part of the analysis is based on the direct evaluation of the distribution function psi(tau), or of the probability Psi(tau), that no time distance smaller than a given tau is found. We adopt the paradigm of the inverse power-law behavior, and we focus on the determination of the inverse power index mu, without ruling out different asymptotic properties that might be revealed, at larger scales, with the help of richer statistics. We then use the DE method, with three different walking rules, and we focus on the regime of transition to scaling. This regime of transition and the value of the scaling parameter itself, delta, depends on the walking rule adopted, a property of interest to shed light on the slow process of transition from dynamics to thermodynamics often occurring under anomalous statistical conditions. With the first two rules the transition regime occurs throughout a large time interval, and the information contained in the time series is transmitted, to a great extent, to it, as well as to the scaling regime. By using the third rule, on the contrary, the same information is essentially conveyed to the scaling regime, which, in fact, emerges very quickly after a fast transition process. We show that the DE method not only causes to emerge the long-range correlation with a given mu < 3, and so a basin of attraction different from the ordinary Gaussian one, but it also reveals the presence of memory effects induced by the time dependence of the solar flare rate. When this memory is annihilated by shuffling, the scaling parameter delta is shown to fit the theoretically expected function of mu. All this leads us to the compelling conclusion that mu = 2.138+/-0.01.

Correlation studies of open and closed state fluctuations in an ion channel: Analysis of ion current through a large-conductance locust potassium channel

Ion current fluctuations occurring within open and closed states of a large-conductance locust potassium channel (BK channel) were investigated for the existence of correlation. Both the time series, extracted from the ion current signal, were studied by the autocorrelation function and the detrended fluctuation analysis (DFA) methods. The persistent character of the short- and middle-range correlations of time series is shown by the slow decay of the autocorrelation function. The DFA exponent alpha is significantly larger than 0.5. The existence of strongly persistent long-range correlations was detected only for closed state fluctuations, with alpha=0.98+/-0.02. The long-range correlation of the BK channel action is therefore determined by the character of closed states. The main outcome of this study reveals that the memory effect is present not only between successive conducting states of the channel but also independently within the open and closed states themselves. As the ion current fluctuations give information about the dynamics of the channel protein, our results point to the correlated character of the protein movement regardless of whether the channel is in its open or closed state.

Fractal dynamics in physiology: alterations with disease and aging

According to classical concepts of physiologic control, healthy systems are self-regulated to reduce variability and maintain physiologic constancy. Contrary to the predictions of homeostasis, however, the output of a wide variety of systems, such as the normal human heartbeat, fluctuates in a complex manner, even under resting conditions. Scaling techniques adapted from statistical physics reveal the presence of long-range, power-law correlations, as part of multifractal cascades operating over a wide range of time scales. These scaling properties suggest that the nonlinear regulatory systems are operating far from equilibrium, and that maintaining constancy is not the goal of physiologic control. In contrast, for subjects at high risk of sudden death (including those with heart failure), fractal organization, along with certain nonlinear interactions, breaks down. Application of fractal analysis may provide new approaches to assessing cardiac risk and forecasting sudden cardiac death, as well as to monitoring the aging process. Similar approaches show promise in assessing other regulatory systems, such as human gait control in health and disease. Elucidating the fractal and nonlinear mechanisms involved in physiologic control and complex signaling networks is emerging as a major challenge in the postgenomic era.

From 1/f noise to multifractal cascades in heartbeat dynamics

We explore the degree to which concepts developed in statistical physics can be usefully applied to physiological signals. We illustrate the problems related to physiologic signal analysis with representative examples of human heartbeat dynamics under healthy and pathologic conditions. We first review recent progress based on two analysis methods, power spectrum and detrended fluctuation analysis, used to quantify long-range power-law correlations in noisy heartbeat fluctuations. The finding of power-law correlations indicates presence of scale-invariant, fractal structures in the human heartbeat. These fractal structures are represented by self-affine cascades of beat-to-beat fluctuations revealed by wavelet decomposition at different time scales. We then describe very recent work that quantifies multifractal features in these cascades, and the discovery that the multifractal structure of healthy dynamics is lost with congestive heart failure. The analytic tools we discuss may be used on a wide range of physiologic signals. (c) 2001 American Institute of Physics.