Effect of trends on detrended fluctuation analysis

A study on the possible usefulness of detrended fluctuation analysis of the electroencephalogram background activity in Alzheimer's disease

We studied the EEG background activity of Alzheimer's disease (AD) patients with detrended fluctuation analysis (DFA). DFA provides an estimation of the scaling information and long-range correlations in time series. We recorded the EEG in 11 AD patients and 11 age-matched controls. Our results showed two scaling regions in all subjects' channels (for limited time scales from 0.01 to 0.04 s and from 0.08 to 0.43 s, respectively), with a clear bend when their corresponding slopes (alpha(1) and alpha(2)) were different. No significant differences between groups were found with alpha(1). However, alpha(2) values were significantly lower in control subjects at electrodes T5, T6, and O1 (p < 0.01, Student's t-test). These findings suggest that the scaling behavior of the EEG is sensitive to AD. Although alpha(2) values allowed us to separate AD patients and controls, accuracies were lower than with spectral analysis. However, a forward stepwise linear discriminant analysis with a leave-one-out cross-validation procedure showed that the combined use of DFA and spectral analysis could improve the diagnostic accuracy of each individual technique. Thus, although spectral analysis outperforms DFA, the combined use of both techniques may increase the insight into brain dysfunction in AD.

Serial correlation of detrended time series

A preliminary essential procedure in time series analysis is the separation of the deterministic component from the random one. If the signal is the result of superposing a noise over a deterministic trend, then the first one must estimate and remove the trend from the signal to obtain an estimation of the stationary random component. The errors accompanying the estimated trend are conveyed as well to the estimated noise, taking the form of detrending errors. Therefore the statistical errors of the estimators of the noise parameters obtained after detrending are larger than the statistical errors characteristic to the noise considered separately. In this paper we study the detrending errors by means of a Monte Carlo method based on automatic numerical algorithms for nonmonotonic trends generation and for construction of estimated polynomial trends alike to those obtained by subjective methods. For a first order autoregressive noise we show that in average the detrending errors of the noise parameters evaluated by means of the autocovariance and autocorrelation function are almost uncorrelated to the statistical errors intrinsic to the noise and they have comparable magnitude. For a real time series with significant trend we discuss a recursive method for computing the errors of the estimated parameters after detrending and we show that the detrending error is larger than the half of the total error.

The endogenous circadian pacemaker imparts a scale-invariant pattern of heart rate fluctuations across time scales spanning minutes to 24 hours

Heartbeat fluctuations in mammals display a robust temporal structure characterized by scale-invariant/fractal patterns. These scale-invariant patterns likely confer physiological advantage because they change with cardiovascular disease and these changes are associated with reduced survival. Models of physical systems imply that to produce scale-invariant patterns, factors influencing the system at different time scales must be coupled via a network of feedback interactions. A similar cardiac control network is hypothesized to be responsible for the scale-invariant pattern in heartbeat dynamics, although the essential network components have not been determined. Here is shown that scale-invariant cardiac control occurs across time scales from minutes to approximately 24 h, and that lesioning the mammalian circadian pacemaker (suprachiasmatic nucleus; SCN) completely abolishes the scale-invariant pattern at time scales>or approximately 4 h. At time scales<or approximately 4 h, the scale invariance persisted following SCN lesion but with a different pattern. These results indicate previously unrecognized multiscale influences of the SCN on heart rate fluctuations that cannot be explained by a simple pacemaker of 24-h rhythmicity. The conclusion is that the SCN serves as a major node in the cardiac control network and imparts scale-invariant cardiac control across a wide range of time scales with strongest effects between approximately 4 and 24 h. These results demonstrate that experimental manipulations (e.g., SCN lesion) can be used to begin to model and understand the origin of scale-invariant behavior in a neurophysiological system.

Efficient scheme for parametric fitting of data in arbitrary dimensions

We propose an efficient scheme for parametric fitting expressed in terms of the Legendre polynomials. For continuous systems, our scheme is exact and the derived explicit expression is very helpful for further analytical studies. For discrete systems, our scheme is almost as accurate as the method of singular value decomposition. Through a few numerical examples, we show that our algorithm costs much less CPU time and memory space than the method of singular value decomposition. Thus, our algorithm is very suitable for a large amount of data fitting. In addition, the proposed scheme can also be used to extract the global structure of fluctuating systems. We then derive the exact relation between the correlation function and the detrended variance function of fluctuating systems in arbitrary dimensions and give a general scaling analysis.

The circadian pacemaker generates similar circadian rhythms in the fractal structure of heart rate in humans and rats

AIMS

Adverse cardiovascular events in humans occur with a day/night pattern, presumably related to a daily pattern of behaviours or endogenous circadian rhythms in cardiovascular variables. Healthy humans possess a scale-invariant/fractal structure in heartbeat fluctuations that exhibits an endogenous circadian rhythm and changes towards the structure observed in cardiovascular disease at the circadian phase corresponding to the time of the broad peak of adverse cardiovascular events (at about 10 AM). To explore the relationship between the rest/activity cycle, endogenous circadian rhythmicity, and cardiac vulnerability, we tested whether the fractal structure of heart rate exhibits a similar circadian rhythm in a mammalian species that is nocturnally active (Wistar rats) compared with diurnally active humans, and how this fractal structure changes after lesioning the circadian pacemaker (suprachiasmatic nucleus, SCN) in rats.

METHODS AND RESULTS

Analysis of heart rate data collected over 10 days in eight intact and six SCN-lesioned Wistar rats during constant darkness revealed that: (i) as with humans, rats exhibit an endogenous circadian rhythm in the scaling exponent characterizing the hourly fractal structure of heart rate (P = 0.0005) with larger exponents during the biological day (inactive phase for rats; active phase for humans); (ii) SCN lesioning abolished the rhythm in the fractal structure of heart rate and systematically increased the scaling exponent (P = 0.01).

CONCLUSION

Rats possess a circadian rhythm of fractal structure of heart rate with a similar temporal pattern as previously observed in humans despite opposite rest/activity cycles between the two species. The SCN imparts this endogenous rhythm. Moreover, lesioning the SCN in rats results in a larger scaling exponent, as occurs with cardiovascular disease in humans.

Multifractal detrended cross-correlation analysis for two nonstationary signals

We propose a method called multifractal detrended cross-correlation analysis to investigate the multifractal behaviors in the power-law cross-correlations between two time series or higher-dimensional quantities recorded simultaneously, which can be applied to diverse complex systems such as turbulence, finance, ecology, physiology, geophysics, and so on. The method is validated with cross-correlated one- and two-dimensional binomial measures and multifractal random walks. As an example, we illustrate the method by analyzing two financial time series.

Ecophysiological significance of scale-dependent patterns in prokaryotic genomes unveiled by a combination of statistic and genometric analyses

We combined genometric (DNA walks) and statistical (detrended fluctuation analysis) methods on 456 prokaryotic chromosomes from 309 different bacterial and archaeal species to look for specific patterns and long-range correlations along the genome and relate them to ecological lifestyles. The position of each nucleotide along the complete genome sequence was plotted on an orthogonal plane (DNA landscape), and fluctuation analysis applied to the DNA walk series showed a long-range correlation in contrast to the lack of correlation for artificially generated genomes. Different features in the DNA landscapes among genomes from different ecological and metabolic groups of prokaryotes appeared with the combined analysis. Transition from hyperthermophilic to psychrophilic environments could have been related to more complex structural adaptations in microbial genomes, whereas for other environmental factors such as pH and salinity this effect would have been smaller. Prokaryotes with domain-specific metabolisms, such as photoautotrophy in Bacteria and methanogenesis in Archaea, showed consistent differences in genome correlation structure. Overall, we show that, beyond the relative proportion of nucleotides, correlation properties derived from their sequential position within the genome hide relevant phylogenetic and ecological information. This can be studied by combining genometric and statistical physics methods, leading to a reduction of genome complexity to a few useful descriptors.

Altered Phase Interactions between Spontaneous Blood Pressure and Flow Fluctuations in Type 2 Diabetes Mellitus: Nonlinear Assessment of Cerebral Autoregulation

Cerebral autoregulation (CA) is an important mechanism that involves dilation and constriction in arterioles to maintain relatively s cerebral blood flow in response to changes of systemic blood pressure. Traditional assessments of CA focus on the changes of cerebral blood flow velocity in response to large blood pressure fluctuations induced by interventions. This approach is not feasible for patients with impaired autoregulation or cardiovascular regulation. Here we propose a newly developed technique-the multimodal pressure-flow (MMPF) analysis, which assesses CA by quantifying nonlinear phase interactions between spontaneous oscillations in blood pressure and flow velocity during resting conditions. We show that CA in healthy subjects can be characterized by specific phase shifts between spontaneous blood pressure and flow velocity oscillations, and the phase shifts are significantly reduced in diabetic subjects. Smaller phase shifts between oscillations in the two variables indicate more passive dependence of blood flow velocity on blood pressure, thus suggesting impaired cerebral autoregulation. Moreover, the reduction of the phase shifts in diabetes is observed not only in previously-recognized effective region of CA (<0.1Hz), but also over the higher frequency range from ~0.1 to 0.4Hz. These findings indicate that Type 2 diabetes alters cerebral blood flow regulation over a wide frequency range and that this alteration can be reliably assessed from spontaneous oscillations in blood pressure and blood flow velocity during resting conditions. We also show that the MMPF method has better performance than traditional approaches based on Fourier transform, and is more sui for the quantification of nonlinear phase interactions between nonstationary biological signals such as blood pressure and blood flow.

Discriminating brain activity from task-related artifacts in functional MRI: fractal scaling analysis simulation and application

Functional magnetic resonance imaging (fMRI) signal changes can be separated from background noise by various processing algorithms, including the well-known deconvolution method. However, discriminating signal changes due to task-related brain activities from those due to task-related head motion or other artifacts correlated in time to the task has been little addressed. We examine whether three exploratory fractal scaling analyses correctly classify these possibilities by capturing temporal self-similarity; namely, fluctuation analysis, wavelet multi-resolution analysis, and detrended fluctuation analysis (DFA). We specifically evaluate whether these fractal analytic methods can be effective and reliable in discriminating activations from artifacts. DFA is indeed robust for such classification. Brain activation maps derived by DFA are similar, but not identical, to maps derived by deconvolution. Deconvolution explicitly utilizes task timing to extract the signals whereas DFA does not, so these methods reveal somewhat different information from the data. DFA is better than deconvolution for distinguishing fMRI activations from task-related artifacts, although a combination of these approaches is superior to either one taken alone. We also present a method for estimating noise levels in fMRI data, validated with numerical simulations suggesting that Birn's model is effective for simulating fMRI signals. Simulations further corroborate that DFA is excellent at discriminating signal changes due to task-related brain activities from those due to task-related artifacts, under a range of conditions.

Influence of filters in the detrended fluctuation analysis of digital electroencephalographic data

The technique named detrended fluctuation analysis (DFA) has been used to reveal the presence of long-range temporal correlations (LRTC) and scaling behavior (SB) in electroencephalographic (EEG) recordings. The occurrence of these phenomena seems to be a salient characteristic of the healthy human brain and alterations in different pathologies has been described. Here we show how the filtering stages implemented in the systems for digital EEG influence the estimation of the DFA parameters used to characterize the brain signals. In consequence, we conclude that it is important to consider these filtering effects before interpreting the results obtained from digital EEG recordings.

Assessing a signal model and identifying brain activity from fMRI data by a detrending-based fractal analysis

One of the major challenges of functional magnetic resonance imaging (fMRI) data analysis is to develop simple and reliable methods to correlate brain regions with functionality. In this paper, we employ a detrending-based fractal method, called detrended fluctuation analysis (DFA), to identify brain activity from fMRI data. We perform three tasks: (a) Estimating noise level from experimental fMRI data; (b) Assessing a signal model recently introduced by Birn et al.; and (c) Evaluating the effectiveness of DFA for discriminating brain activations from artifacts. By computing the receiver operating characteristic (ROC) curves, we find that the ROC curve for experimental data is similar to the curve for simulated data with similar signal-to-noise ratio (SNR). This suggests that the proposed algorithm for estimating noise level is very effective and that Birn's model fits our experimental data very well. The brain activation maps for experimental data derived by DFA are similar to maps derived by deconvolution using a widely used software, AFNI. Considering that deconvolution explicitly uses the information about the experimental paradigm to extract the activation patterns whereas DFA does not, it remains to be seen whether one can effectively integrate the two methods to improve accuracy for detecting brain areas related to functional activity.

Endogenous circadian rhythm in human motor activity uncoupled from circadian influences on cardiac dynamics

The endogenous circadian pacemaker influences key physiologic functions, such as body temperature and heart rate, and is normally synchronized with the sleep/wake cycle. Epidemiological studies demonstrate a 24-h pattern in adverse cardiovascular events with a peak at approximately 10 a.m. It is unknown whether this pattern in cardiac risk is caused by a day/night pattern of behaviors, including activity level and/or influences from the internal circadian pacemaker. We recently found that a scaling index of cardiac vulnerability has an endogenous circadian peak at the circadian phase corresponding to approximately 10 a.m., which conceivably could contribute to the morning peak in cardiac risk. Here, we test whether this endogenous circadian influence on cardiac dynamics is caused by circadian-mediated changes in motor activity or whether activity and heart rate dynamics are decoupled across the circadian cycle. We analyze high-frequency recordings of motion from young healthy subjects during two complementary protocols that decouple the sleep/wake cycle from the circadian cycle while controlling scheduled behaviors. We find that static activity properties (mean and standard deviation) exhibit significant circadian rhythms with a peak at the circadian phase corresponding to 5-9 p.m. ( approximately 9 h later than the peak in the scale-invariant index of heartbeat fluctuations). In contrast, dynamic characteristics of the temporal scale-invariant organization of activity fluctuations (long-range correlations) do not exhibit a circadian rhythm. These findings suggest that endogenous circadian-mediated activity variations are not responsible for the endogenous circadian rhythm in the scale-invariant structure of heartbeat fluctuations and likely do not contribute to the increase in cardiac risk at approximately 10 a.m.

Algorithm to estimate the Hurst exponent of high-dimensional fractals

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

The suprachiasmatic nucleus functions beyond circadian rhythm generation

We recently discovered that human activity possesses a complex temporal organization characterized by scale-invariant/self-similar fluctuations from seconds to approximately 4 h-(statistical properties of fluctuations remain the same at different time scales). Here, we show that scale-invariant activity patterns are essentially identical in humans and rats, and exist for up to approximately 24 h: six-times longer than previously reported. Theoretically, such scale-invariant patterns can be produced by a neural network of interacting control nodes-system components with feedback loops-operating at different time scales. However such control nodes have not yet been identified in any neurophysiological model of scale invariance/self-similarity in mammals. Here we demonstrate that the endogenous circadian pacemaker (suprachiasmatic nucleus; SCN), known to modulate locomotor activity with a periodicity of approximately 24 h, also acts as a major neural control node responsible for the generation of scale-invariant locomotor patterns over a broad range of time scales from minutes to at least 24 h (rather than solely at approximately 24 h). Remarkably, we found that SCN lesion in rats completely abolished the scale-invariant locomotor patterns between 4 and 24 h and significantly altered the patterns at time scales <4 h. Identification of the control nodes of a neural network responsible for scale invariance is the critical first step in understanding the neurophysiological origin of scale invariance/self-similarity.

Fluctuation and synchronization of gait intervals and gait force profiles distinguish stages of Parkinson's disease

We study the effects of Parkinson's disease (PD) on the long-term fluctuation and phase synchronization properties of gait timing (series of interstride intervals) as well as gait force profiles (series characterizing the morphological changes between the steps). We find that the fluctuations in the gait timing are significantly larger for PD patients and early PD patients, who were not treated yet with medication, compared to age-matched healthy controls. Simultaneously, the long-term correlations and the phase synchronization of right and left leg are significantly reduced in both types of PD patients. Surprisingly, long-term correlations of the gait force profiles are relatively weak for treated PD patients and healthy controls, while they are significantly larger for early PD patients. The results support the idea that timing and morphology of recordings obtained from a complex system can contain complementary information.

Fractal scale-invariant and nonlinear properties of cardiac dynamics remain stable with advanced age: a new mechanistic picture of cardiac control in healthy elderly

Heart beat fluctuations exhibit temporal structure with robust long-range correlations, fractal and nonlinear features, which have been found to break down with pathologic conditions, reflecting changes in the mechanism of neuroautonomic control. It has been hypothesized that these features change and even break down also with advanced age, suggesting fundamental alterations in cardiac control with aging. Here we test this hypothesis. We analyze heart beat interval recordings from the following two independent databases: 1) 19 healthy young (average age 25.7 yr) and 16 healthy elderly subjects (average age 73.8 yr) during 2 h under resting conditions from the Fantasia database; and 2) 29 healthy elderly subjects (average age 75.9 yr) during approximately 8 h of sleep from the sleep heart health study (SHHS) database, and the same subjects recorded 5 yr later. We quantify: 1) the average heart rate (<R-R>); 2) the SD sigma(R-R) and sigma(DeltaR-R) of the heart beat intervals R-R and their increments DeltaR-R; 3) the long-range correlations in R-R as measured by the scaling exponent alpha(R-R) using the Detrended Fluctuation Analysis; 4) fractal linear and nonlinear properties as represented by the scaling exponents alpha(sgn) and alpha(mag) for the time series of the sign and magnitude of DeltaR-R; and 5) the nonlinear fractal dimension D(k) of R-R using the fractal dimension analysis. We find: 1) No significant difference in (P > 0.05); 2) a significant difference in sigma(R-R) and sigma(DeltaR-R) for the Fantasia groups (P < 10(-4)) but no significant change with age between the elderly SHHS groups (P > 0.5); and 3) no significant change in the fractal measures alpha(R-R) (P > 0.15), alpha(sgn) (P > 0.2), alpha(mag) (P > 0.3), and D(k) with age. Our findings do not support the hypothesis that fractal linear and nonlinear characteristics of heart beat dynamics break down with advanced age in healthy subjects. Although our results indeed show a reduced SD of heart beat fluctuations with advanced age, the inherent temporal fractal and nonlinear organization of these fluctuations remains stable. This indicates that the coupled cascade of nonlinear feedback loops, which are believed to underlie cardiac neuroautonomic regulation, remains intact with advanced age.

Temporally resolved fluctuation analysis of sleep ECG

The correlation behavior in the heart beat rate significantly differs with respect to light sleep, deep sleep, and REM sleep. We investigate whether fluctuations of the heart beat rhythm may serve as a surrogate parameter for rapidly changing sleep phenomena, and if these changes are accessible by progressive beat-by-beat analysis of the sleep electrocardiogram (ECG).

Relation between volatility correlations in financial markets and Omori processes occurring on all scales

We analyze the memory in volatility by studying volatility return intervals, defined as the time between two consecutive fluctuations larger than a given threshold, in time periods following stock market crashes. Such an aftercrash period is characterized by the Omori law, which describes the decay in the rate of aftershocks of a given size with time t by a power law with exponent close to 1. A shock followed by such a power law decay in the rate is here called Omori process. We find self-similar features in the volatility. Specifically, within the aftercrash period there are smaller shocks that themselves constitute Omori processes on smaller scales, similar to the Omori process after the large crash. We call these smaller shocks subcrashes, which are followed by their own aftershocks. We also show that the Omori law holds not only after significant market crashes as shown by Lillo and Mantegna [Phys. Rev. E 68, 016119 (2003)], but also after "intermediate shocks." By appropriate detrending we remove the influence of the crashes and subcrashes from the data, and find that this procedure significantly reduces the memory in the records. Moreover, when studying long-term correlated fractional Brownian motion and autoregressive fractionally integrated moving average artificial models for volatilities, we find Omori-type behavior after high volatilities. Thus, our results support the hypothesis that the memory in the volatility is related to the Omori processes present on different time scales.

Exploiting nonlinear recurrence and fractal scaling properties for voice disorder detection

BACKGROUND

Voice disorders affect patients profoundly, and acoustic tools can potentially measure voice function objectively. Disordered sustained vowels exhibit wide-ranging phenomena, from nearly periodic to highly complex, aperiodic vibrations, and increased "breathiness". Modelling and surrogate data studies have shown significant nonlinear and non-Gaussian random properties in these sounds. Nonetheless, existing tools are limited to analysing voices displaying near periodicity, and do not account for this inherent biophysical nonlinearity and non-Gaussian randomness, often using linear signal processing methods insensitive to these properties. They do not directly measure the two main biophysical symptoms of disorder: complex nonlinear aperiodicity, and turbulent, aeroacoustic, non-Gaussian randomness. Often these tools cannot be applied to more severe disordered voices, limiting their clinical usefulness.

METHODS

This paper introduces two new tools to speech analysis: recurrence and fractal scaling, which overcome the range limitations of existing tools by addressing directly these two symptoms of disorder, together reproducing a "hoarseness" diagram. A simple bootstrapped classifier then uses these two features to distinguish normal from disordered voices.

RESULTS

On a large database of subjects with a wide variety of voice disorders, these new techniques can distinguish normal from disordered cases, using quadratic discriminant analysis, to overall correct classification performance of 91.8 +/- 2.0%. The true positive classification performance is 95.4 +/- 3.2%, and the true negative performance is 91.5 +/- 2.3% (95% confidence). This is shown to outperform all combinations of the most popular classical tools.

CONCLUSION

Given the very large number of arbitrary parameters and computational complexity of existing techniques, these new techniques are far simpler and yet achieve clinically useful classification performance using only a basic classification technique. They do so by exploiting the inherent nonlinearity and turbulent randomness in disordered voice signals. They are widely applicable to the whole range of disordered voice phenomena by design. These new measures could therefore be used for a variety of practical clinical purposes.

Cardiovascular and respiratory dynamics during normal and pathological sleep

Sleep is an active and regulated process with restorative functions for physical and mental conditions. Based on recordings of brain waves and the analysis of characteristic patterns and waveforms it is possible to distinguish wakefulness and five sleep stages. Sleep and the sleep stages modulate autonomous nervous system functions such as body temperature, respiration, blood pressure, and heart rate. These functions consist of a sympathetic tone usually related to activation and to parasympathetic (or vagal) tone usually related to inhibition. Methods of statistical physics are used to analyze heart rate and respiration to detect changes of the autonomous nervous system during sleep. Detrended fluctuation analysis and synchronization analysis and their applications to heart rate and respiration during sleep in healthy subjects and patients with sleep disorders are presented. The observed changes can be used to distinguish sleep stages in healthy subjects as well as to differentiate normal and disturbed sleep on the basis of heart rate and respiration recordings without direct recording of brain waves. Of special interest are the cardiovascular consequences of disturbed sleep because they present a risk factor for cardiovascular disorders such as arterial hypertension, cardiac ischemia, sudden cardiac death, and stroke. New derived variables can help to find indicators for these health risks.

Identifying characteristic scales in the human genome

The scale-free, long-range correlations detected in DNA sequences contrast with characteristic lengths of genomic elements, being particularly incompatible with the isochores (long, homogeneous DNA segments). By computing the local behavior of the scaling exponent alpha of detrended fluctuation analysis (DFA), we discriminate between sequences with and without true scaling, and we find that no single scaling exists in the human genome. Instead, human chromosomes show a common compositional structure with two characteristic scales, the large one corresponding to the isochores and the other to small and medium scale genomic elements.

Automatic algorithm for monotone trend removal

The available numerical algorithms for trend removal require a direct subjective intervention in choosing critical parameters. In this paper an algorithm is presented, which needs no initial subjective assumptions. Monotone trends are approximated by piecewise linear curves obtained by dividing into subintervals the signal values interval, not the time interval. The slope of each linear segment of the estimated trend is proportional to the average one-step displacement of the signal values included into the corresponding subinterval. The evaluation of the trend removal is performed on statistical ensembles of artificial time series with the random component given by realizations of autoregressive of order one stochastic processes or by fractional Brownian motions. The accuracy of the algorithm is compared with that of two well-tested methods: polynomial fitting and a nonparametric method based on moving average. For stationary noise the results of the algorithm are slightly better, but for nonstationary noise the preliminary results indicate that the polynomial fitting has the best accuracy. As a verification on a real time series, the time periods with monotone variation of global average temperature over the last 1800years are established. The removal of a nonmonotone trend is also briefly discussed.

Technical Note: Long-term memory effect in the atmospheric CO2concentration at Mauna Loa

Abstract. The monthly mean values of the atmospheric carbon dioxide concentration derived from in-situ air samples collected at Mauna Loa Observatory, Hawaii, USA during 1958–2004 (the longest continuous record available in the world) are analyzed by employing the detrended fluctuation analysis to detect scaling behavior in this time series. The main result is that the fluctuations of carbon dioxide concentrations exhibit long-range power-law correlations (long memory) with lag times ranging from four months to eleven years, which correspond to 1/f noise. This result indicates that random perturbations in the carbon dioxide concentrations give rise to noise, characterized by a frequency spectrum following a power-law with exponent that approaches to one; the latter shows that the correlation times grow strongly. This feature is pointing out that a correctly rescaled subset of the original time series of the carbon dioxide concentrations resembles the original time series. Finally, the power-law relationship derived from the real measurements of the carbon dioxide concentrations could also serve as a tool to improve the confidence of the atmospheric chemistry-transport and global climate models.

Detrended Fluctuation Analysis: A Suitable Long-term Measure of HRV Signals in Children with Sleep Disordered Breathing

On the body surface the electric field generated by the cardiac muscles consists of electric potential maxima and minima that increase and decrease during each cardiac cycle. The recording of these electric potentials as a function of time is called electrocardiography, and the resulting signal is called the electrocardiogram (ECG). The ECG signal is used extensively as a low cost diagnostic tool to provide information concerning the heart's state of health. Reliable and accurate detection of the QRS complex and R wave peak in ECG signals is essential in computer-based ECG analysis. In this paper we evaluate the significance of Detrended Fluctuation Analysis (DFA) for studying heart rate variability in children with sleep disordered breathing. An Enhanced Hilbert Transform (EHT) algorithm was used to derive the Heart Rate Variability (HRV) signal. We compare the DFA values with Approximate Entropy and Poincaré Plots of HRV signals as these are very useful in characterization and visualization of HRV data. Our data demonstrated differences in DFA parameters between periods of normal and abnormal breathing and also between sleep stages. These results suggest that DFA is suitable for the long-term analysis of non-stationary time series such as HRV signals and may also be applied in the detection of sleep disordered breathing.

Statistics of return intervals in long-term correlated records

We consider long-term correlated data with several distribution densities (Gaussian, exponential, power law, and log normal) and various correlation exponents gamma (0<gamma<1) , and study the statistics of the return intervals r_{j} between events above some threshold q . We show that irrespective of the distribution, the return intervals are long-term correlated in the same way as the original record, but with additional uncorrelated noise. Due to this noise, the correlations are difficult to observe by the detrended fluctuation analysis (which exhibits a crossover behavior) but show up very clearly in the autocorrelation function. The distribution P_{q}(r) of the return intervals is characterized at large scales by a stretched exponential with exponent gamma , and at short scales by a power law with exponent gamma-1 . We discuss in detail the occurrence of finite-size effects for large threshold values for all considered distributions. We show that finite-size effects are most pronounced in exponentially distributed data sets where they can even mask the stretched exponential behavior in records of up to 10;{6} data points. Finally, in order to quantify the clustering of extreme events due to the long-term correlations in the return intervals, we study the conditional distribution function and the related moments. We find that they show pronounced memory effects, irrespective of the distribution of the original data.

Analysis of long-range correlation in sequences data of proteins

The results presented here suggest the existence of correlations in the sequence data of proteins. 32 proteins, both globular and fibrous, both monomeric and polymeric, were analyzed. The primary structures of these proteins were treated as time series. Three spatial series of data for each sequence of a protein were generated from numerical correspondences between each amino acid and a physical property associated with it, i.e., its electric charge, its polar character and its dipole moment. For each series, the spectral coefficient, the scaling exponent and the Hurst coefficient were determined. The values obtained for these coefficients revealed non-randomness in the series of data. .

Motion sickness preceded by unstable displacements of the center of pressure

We exposed standing participants to optic flow in a moving room. Motion sickness was induced by motion that simulated the amplitude and frequency of standing sway. We identified instabilities in displacements of the center of pressure among participants who became sick; these instabilities occurred before the onset of subjective motion sickness symptoms. Postural differences between Sick and Well participants were observed before exposure to the nauseogenic stimulus. During exposure to the nauseogenic stimulus, sway increased for participants who became sick but also for those who did not. However, at every point during exposure sway was greater for participants who became motion sick. The results reveal that motion sickness is preceded by instabilities in displacements of the center of pressure.

Detrended fluctuation analysis for fractals and multifractals in higher dimensions

One-dimensional detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MFDFA) are widely used in the scaling analysis of fractal and multifractal time series because they are accurate and easy to implement. In this paper we generalize the one-dimensional DFA and MFDFA to higher-dimensional versions. The generalization works well when tested with synthetic surfaces including fractional Brownian surfaces and multifractal surfaces. The two-dimensional MFDFA is also adopted to analyze two images from nature and experiment, and nice scaling laws are unraveled.

Detrended fluctuation analysis of laser Doppler flowmetry time series: the effect of extrinsic and intrinsic factors on the fractal scaling of microvascular blood flow

The relative contribution of extrinsic (central) and intrinsic (local) oscillatory mechanisms to the fractal scaling of blood flow in forearm cutaneous microcirculation is unclear. The aim of this study was to investigate the contributions of these mechanisms to the fractal properties of the blood flow signal by using their frequency spectrum in the analyses. To evoke local oscillatory components, acetylcholine (ACh) was iontophoresed into the forearm and cutaneous perfusion was measured by a laser Doppler flowmeter (LDF) at rest. Depending on the involved factors in ACh-induced vasodilatation, central, cardiac and respiratory, signals have also increasingly appeared in LDF. The detrended fluctuation analysis (DFA) of filtered LDF time series demonstrated that the LDF was fractal with three distinct scaling regions. Furthermore, the findings of the present study indicated that these regions are related to the frequency bands of well-known control systems of blood flow and were called cardiac, cardio-respiratory and local regions. The mean scaling exponent increased with vasodilatation in the cardiac region but decreased and even changed its sign in the cardio-respiratory region. Inhibition of a local vasodilator mechanism not only decreased the scaling exponent of the local region but also eliminated the effect of respiratory coupling on fractal scaling. These findings suggest that the scaling exponents might have a diagnostic value for detecting pathological dynamics in vascular beds.

Detrended fluctuation analysis in x-ray photon correlation spectroscopy for determining coarsening dynamics in alloys

We study the dynamics of precipitate coarsening in phase-separating alloys at late stages of phase separation by x-ray photon correlation spectroscopy (XPCS). For analyzing time series of fluctuating speckle intensities from small-angle scattering of coherent x rays, the method of detrended fluctuation analysis (DFA), which is ideal for determining power-law correlations, is applied. We discuss the application of DFA with respect to XPCS data by means of simulated time series. In particular, the effects of different signal-to-noise ratios are examined. Results from measurements of the two model systems Al-6 at. % Ag at 140 degrees C and Al-9 at. % Zn at 0 degrees C are presented. Since the DFA effectively removes adulterating trends in the data, quantitative agreement with Monte Carlo simulations is obtained. It is verified that two different coarsening mechanisms are predominant in the two systems--coarsening either by diffusion of single atoms or by movement of whole precipitates.

Long-memory processes in ozone and temperature variations at the region 60° S–60° N

Abstract. Global column ozone and tropospheric temperature observations made by ground-based (1964–2004) and satellite-borne (1978–2004) instrumentation are analyzed. Ozone and temperature fluctuations in small time-intervals are found to be positively correlated to those in larger time-intervals in a power-law fashion. For temperature, the exponent of this dependence is larger in the mid-latitudes than in the tropics at long time scales, while for ozone, the exponent is larger in tropics than in the mid-latitudes. In general, greater persistence could be a result of either stronger positive feedbacks or larger inertia. Therefore, the increased slope of the power distribution of temperature in mid-latitudes at long time scales compared to the slope in the tropics could be connected to the poleward increase in climate sensitivity predicted by the global climate models. The detrended fluctuation analysis of model and observed time series provides a helpful tool for visualizing errors in the treatment of long-range correlations, whose correct modeling would greatly enhance confidence in long-term climate and atmospheric chemistry modeling.

Understanding volatility correlation behavior with a magnitude cross-correlation function

We propose an approach for analyzing the basic relation between correlation properties of the original signal and its magnitude fluctuations by decomposing the original signal into its positive and negative fluctuation components. We use this relation to understand the following phenomenon found in many naturally occurring time series: the magnitude of the signal exhibits long-range correlation, whereas the original signal is short-range correlated. The applications of our approach to heart rate variability signals and high-frequency foreign exchange rates reveal that the difference between the correlation properties of the original signal and its magnitude fluctuations is induced by the time organization structure of the correlation function between the magnitude fluctuations of positive and negative components. We show that this correlation function can be described well by a stretched-exponential function and is related to the nonlinearity and the multifractal structure of the signals.

Breakdown of long-range temporal correlations in theta oscillations in patients with major depressive disorder

Neuroimaging has revealed robust large-scale patterns of high neuronal activity in the human brain in the classical eyes-closed wakeful rest condition, pointing to the presence of a baseline of sustained endogenous processing in the absence of stimulus-driven neuronal activity. This baseline state has been shown to differ in major depressive disorder. More recently, several studies have documented that despite having a complex temporal structure, baseline oscillatory activity is characterized by persistent autocorrelations for tens of seconds that are highly replicable within and across subjects. The functional significance of these long-range temporal correlations has remained unknown. We recorded neuromagnetic activity in patients with a major depressive disorder and in healthy control subjects during eyes-closed wakeful rest and quantified the long-range temporal correlations in the amplitude fluctuations of different frequency bands. We found that temporal correlations in the theta-frequency band (3-7 Hz) were almost absent in the 5-100 s time range in the patients but prominent in the control subjects. The magnitude of temporal correlations over the left temporocentral region predicted the severity of depression in the patients. These data indicate that long-range temporal correlations in theta oscillations are a salient characteristic of the healthy human brain and may have diagnostic potential in psychiatric disorders. We propose a link between the abnormal temporal structure of theta oscillations in the depressive patients and the systems-level impairments of limbic-cortical networks that have been identified in recent anatomical and functional studies of patients with major depressive disorder.

Scaling and memory of intraday volatility return intervals in stock markets

We study the return interval tau between price volatilities that are above a certain threshold q for 31 intraday data sets, including the Standard and Poor's 500 index and the 30 stocks that form the Dow Jones Industrial index. For different threshold q, the probability density function Pq(tau)scales with the mean interval tau as [Formula: see text], similar to that found in daily volatilities. Since the intraday records have significantly more data points compared to the daily records, we could probe for much higher thresholds and still obtain good statistics. We find that the scaling function f(x)is consistent for all 31 intraday data sets in various time resolutions, and the function is well-approximated by the stretched exponential, f(x) similar to e(-ax)(gamma), with gamma=0.38+/-0.05 and a=3.9+/-0.5, which indicates the existence of correlations. We analyze the conditional probability distribution Pq(tau/tau0) for tau following a certain interval tau0, and find Pq(tau/tau0) depends on tau0, which demonstrates memory in intraday return intervals. Also, we find that the mean conditional interval (tau/tau0) increases with tau0, consistent with the memory found for Pq(tau/tau0). Moreover, we find that return interval records, in addition to having short-term correlations as demonstrated by Pq(tau/tau0), have long-term correlations with correlation exponents similar to that of volatility records.

Critical fluctuation of wind reversals in convective turbulence

The irregular reversals of wind direction in convective turbulence are found to have fluctuating intervals that can be related, under certain circumstances, to critical behavior. In particular, by focusing on its temporal evolution, the net magnetization of a two-dimensional Ising lattice of finite size is observed to fluctuate in the same way. Detrended fluctuation analysis of the wind reversal time series results in a scaling behavior that agrees remarkably well with that of the Ising problem. The specific properties found here, as well as the lack of an external tuning parameter, also suggest that the wind reversal phenomenon exhibits signs of self-organized criticality.

GPS analysis of human locomotion: further evidence for long-range correlations in stride-to-stride fluctuations of gait parameters

During free walking, gait is automatically adjusted to provide optimal mechanical output and minimal energy expenditure; gait parameters, such as cadence, fluctuate from one stride to the next around average values. It was described that this fluctuation exhibited long-range correlations and fractal-like patterns. In addition, it was suggested that these long-range correlations disappeared if the participant followed the beep of metronome to regulate his or her pace. Until now, these fractal fluctuations were only observed for stride interval, because no technique existed to adequately analyze an extended time of free walking. The aim of the present study was to measure walking speed (WS), step frequency (SF) and step length (SL) with high accuracy (<1 cm) satellite positioning method (global positioning system or GPS) in order to detect long-range correlations in the stride-to-stride fluctuations. Eight participants walked 30 min under free and constrained (metronome) conditions. Under free walking conditions, DFA (detrended fluctuation analysis) and surrogate data tests showed that the fluctuation of WS, SL and SF exhibited a fractal pattern (i.e., scaling exponent alpha: 0.5 < alpha < 1) in a large majority of participants (7/8). Under constrained conditions (metronome), SF fluctuations became significantly anti-correlated (alpha < 0.5) in all participants. However, the scaling exponent of SL and WS was not modified. We conclude that, when the walking pace is controlled by an auditory signal, the feedback loop between the planned movement (at supraspinal level) and the sensory inputs induces a continual shifting of SF around the mean (persistent anti-correlation), but with no effect on the fluctuation dynamics of the other parameters (SL, WS).

Wavelet analysis and scaling properties of time series

We propose a wavelet based method for the characterization of the scaling behavior of nonstationary time series. It makes use of the built-in ability of the wavelets for capturing the trends in a data set, in variable window sizes. Discrete wavelets from the Daubechies family are used to illustrate the efficacy of this procedure. After studying binomial multifractal time series with the present and earlier approaches of detrending for comparison, we analyze the time series of averaged spin density in the 2D Ising model at the critical temperature, along with several experimental data sets possessing multifractal behavior.

How useful is satellite positioning system (GPS) to track gait parameters? A review

Over the last century, numerous techniques have been developed to analyze the movement of humans while walking and running. The combined use of kinematics and kinetics methods, mainly based on high speed video analysis and forceplate, have permitted a comprehensive description of locomotion process in terms of energetics and biomechanics. While the different phases of a single gait cycle are well understood, there is an increasing interest to know how the neuro-motor system controls gait form stride to stride. Indeed, it was observed that neurodegenerative diseases and aging could impact gait stability and gait parameters steadiness. From both clinical and fundamental research perspectives, there is therefore a need to develop techniques to accurately track gait parameters stride-by-stride over a long period with minimal constraints to patients. In this context, high accuracy satellite positioning can provide an alternative tool to monitor outdoor walking. Indeed, the high-end GPS receivers provide centimeter accuracy positioning with 5–20 Hz sampling rate: this allows the stride-by-stride assessment of a number of basic gait parameters – such as walking speed, step length and step frequency – that can be tracked over several thousand consecutive strides in free-living conditions. Furthermore, long-range correlations and fractal-like pattern was observed in those time series. As compared to other classical methods, GPS seems a promising technology in the field of gait variability analysis. However, relative high complexity and expensiveness – combined with a usability which requires further improvement – remain obstacles to the full development of the GPS technology in human applications.

Fractionally integrated process with power-law correlations in variables and magnitudes

Motivated by the fact that many empirical time series--including changes of heartbeat intervals, physical activity levels, intertrade times in finance, and river flux values--exhibit power-law anticorrelations in the variables and power-law correlations in their magnitudes, we propose a simple stochastic process that can account for both types of correlations. The process depends on only two parameters, where one controls the correlations in the variables and the other controls the correlations in their magnitudes. We apply the process to time series of heartbeat interval changes and air temperature changes and find that the statistical properties of the modeled time series are in agreement with those observed in the data.