Effect of trends on detrended fluctuation analysis

Long-range correlation in cosmic microwave background radiation

We investigate the statistical anisotropy and gaussianity of temperature fluctuations of Cosmic Microwave Background (CMB) radiation data from the Wilkinson Microwave Anisotropy Probe survey, using the Multifractal Detrended Fluctuation Analysis, Rescaled Range, and Scaled Windowed Variance methods. Multifractal Detrended Fluctuation Analysis shows that CMB fluctuations has a long-range correlation function with a multifractal behavior. By comparing the shuffled and surrogate series of CMB data, we conclude that the multifractality nature of the temperature fluctuation of CMB radiation is mainly due to the long-range correlations, and the map is consistent with a gaussian distribution.

Multifractal detrending moving-average cross-correlation analysis

There are a number of situations in which several signals are simultaneously recorded in complex systems, which exhibit long-term power-law cross correlations. The multifractal detrended cross-correlation analysis (MFDCCA) approaches can be used to quantify such cross correlations, such as the MFDCCA based on the detrended fluctuation analysis (MFXDFA) method. We develop in this work a class of MFDCCA algorithms based on the detrending moving-average analysis, called MFXDMA. The performances of the proposed MFXDMA algorithms are compared with the MFXDFA method by extensive numerical experiments on pairs of time series generated from bivariate fractional Brownian motions, two-component autoregressive fractionally integrated moving-average processes, and binomial measures, which have theoretical expressions of the multifractal nature. In all cases, the scaling exponents h(xy) extracted from the MFXDMA and MFXDFA algorithms are very close to the theoretical values. For bivariate fractional Brownian motions, the scaling exponent of the cross correlation is independent of the cross-correlation coefficient between two time series, and the MFXDFA and centered MFXDMA algorithms have comparative performances, which outperform the forward and backward MFXDMA algorithms. For two-component autoregressive fractionally integrated moving-average processes, we also find that the MFXDFA and centered MFXDMA algorithms have comparative performances, while the forward and backward MFXDMA algorithms perform slightly worse. For binomial measures, the forward MFXDMA algorithm exhibits the best performance, the centered MFXDMA algorithms performs worst, and the backward MFXDMA algorithm outperforms the MFXDFA algorithm when the moment order q<0 and underperforms when q>0. We apply these algorithms to the return time series of two stock market indexes and to their volatilities. For the returns, the centered MFXDMA algorithm gives the best estimates of h(xy)(q) since its h(xy)(2) is closest to 0.5, as expected, and the MFXDFA algorithm has the second best performance. For the volatilities, the forward and backward MFXDMA algorithms give similar results, while the centered MFXDMA and the MFXDFA algorithms fail to extract rational multifractal nature.

Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: comparison study with detrended fluctuation analysis and wavelet leaders

In this paper we present an extended version of Hilbert-Huang transform, namely arbitrary-order Hilbert spectral analysis, to characterize the scale-invariant properties of a time series directly in an amplitude-frequency space. We first show numerically that due to a nonlinear distortion, traditional methods require high-order harmonic components to represent nonlinear processes, except for the Hilbert-based method. This will lead to an artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus the power law, if it exists, is contaminated. We then compare the Hilbert method with structure functions (SF), detrended fluctuation analysis (DFA), and wavelet leader (WL) by analyzing fractional Brownian motion and synthesized multifractal time series. For the former simulation, we find that all methods provide comparable results. For the latter simulation, we perform simulations with an intermittent parameter μ=0.15. We find that the SF underestimates scaling exponent when q>3. The Hilbert method provides a slight underestimation when q>5. However, both DFA and WL overestimate the scaling exponents when q>5. It seems that Hilbert and DFA methods provide better singularity spectra than SF and WL. We finally apply all methods to a passive scalar (temperature) data obtained from a jet experiment with a Taylor's microscale Reynolds number Re(λ)≃250. Due to the presence of strong ramp-cliff structures, the SF fails to detect the power law behavior. For the traditional method, the ramp-cliff structure causes a serious artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus DFA and WL underestimate the scaling exponents. However, the Hilbert method provides scaling exponents ξ(θ)(q) quite close to the one for longitudinal velocity, indicating a less intermittent passive scalar field than what was believed before.

New method for analysis of nonstationary signals

BACKGROUND

Analysis of signals by means of symbolic methods consists in calculating a measure of signal complexity, for example informational entropy or Lempel-Ziv algorithmic complexity. For construction of these entropic measures one uses distributions of symbols representing the analyzed signal.

RESULTS

We introduce a new signal characteristic named sequential spectrum that is suitable for analysis of the wide group of signals, including biosignals.The paper contains a brief review of analyses of artificial signals showing features similar to those of biosignals. An example of using sequential spectrum for analyzing EEG signals registered during different stages of sleep is also presented.

CONCLUSIONS

Sequential spectrum is an effective tool for general description of nonstationary signals and it its advantage over Fourier spectrum. Sequential spectrum enables assessment of pathological changes in EEG-signals recorded in persons with epilepsy.

Fluctuation phenotyping based on daily fraction of exhaled nitric oxide values in asthmatic children

BACKGROUND

Fraction of exhaled nitric oxide (Feno), a marker of airway inflammation, has been proposed to be useful for asthma management, but conclusions are inconsistent. This might be due to the failure of mean statistics to characterize individual variability in Feno values, which is possibly a better indicator of asthma control than single measurements.

OBJECTIVE

We characterized fractal fluctuations in daily Feno values over time and the relationship between Feno values and symptom scores. We investigated whether these are associated with asthma severity, control, and exacerbation risk.

METHODS

Daily Feno values and symptom scores over 192 days in 41 atopic asthmatic children from the Childhood Asthma Respiratory Inflammatory Status Monitoring study were analyzed. Two methods of time-series analysis were used: detrended fluctuation analysis to quantify fractal patterns in fluctuations in daily Feno values (α value) and cross-correlation to quantify the strength of the relationship between daily Feno values and symptom scores. The associations of α values and cross-correlation with markers of asthma severity and control were assessed by means of regression analysis.

RESULTS

Daily fluctuations in Feno values exhibited fractal-type long-range correlations. Those subjects receiving higher doses of inhaled corticosteroids at study entry had a significantly lower α value, corresponding to more random fluctuations in Feno values in those with greater inhaled corticosteroid need. The cross-correlation between Feno values and symptom scores was significantly higher in those subjects who had exacerbations.

CONCLUSIONS

Fluctuation in Feno values and their cross-correlation to symptom scores contains information on asthma severity and control. Methods that quantify the complexity of asthma over time might assist in identifying asthmatic subjects with concordance between eosinophilic inflammation and symptoms and thus increased exacerbation risk.

Oscillations in the evaluation of fractal dimension of RR intervals time series

Previously, we have reported the presence of oscillations in the graphs we have used to evaluate the Higuchi's fractal dimension in RR intervals time series of congestive heart failure (CHF) patients in the sleep phase but these oscillations hardly appear in all the six hours of the awake phase. In this paper we report the same analysis for heart rate time series for different groups of healthy subjects; we are looking for the presence of this kind of oscillations in other situations. We analyzed all the time series in the Exaggerated Heart Rate Oscillations database of Physionet during two meditation techniques: volunteers with spontaneous breathing, subjects in meditation, volunteers in a metronomic breathing group and elite athletes. We have found oscillations in the graphs of the Higuchi's fractal dimension in the heart rate time series of subjects in meditation and metronomic breathing and this fact coincides with previous reported results.

High-level organization of isochores into gigantic superstructures in the human genome

Human DNA shows a complex structure with compositional features at many scales; the isochores--long DNA segments (~10⁵ bp) of relatively homogeneous guanine-cytosine (G + C) content--are the largest well-documented and well-analyzed compositional structures. However, we report here on the existence of a high-level compositional organization of isochores in the human genome. By using a segmentation algorithm incorporating the long-range correlations existing in human DNA, we find that every chromosome is composed of a few huge segments (~ 10⁷ bp) of relatively homogeneous G + C content, which become the largest compositional organization of the genome. Finally, we show evidence of the biological relevance of these superstructures, pointing to a large-scale functional organization of the human genome.

Stride dynamics, gait variability and prospective falls risk in active community dwelling older women

BACKGROUND

Measures of walking instability such as stride dynamics and gait variability have been shown to identify future fallers in older adult populations with gait limitations or mobility disorders. This study investigated whether measures of walking instability can predict future fallers (over a prospective 12 month period) in a group of healthy and active older women.

METHODS

Ninety-seven healthy active women aged between 55 and 90 years walked for 7 min around a continuous walking circuit. Gait data recorded by a GAITRite(®) walkway and foot-mounted accelerometers were used to calculate measures of stride dynamics and gait variability. The participant's physical function and balance were assessed. Fall incidence was monitored over the following 12 months.

RESULTS

Inter-limb differences (p≤0.04) in stride dynamics were found for fallers (one or more falls) aged over 70 years, and multiple fallers (two or more falls) aged over 55 years, but not in non-fallers or a combined group of single and non-fallers. No group differences were found in the measures of physical function, balance or gait, including variability. Additionally, no gait variable predicted falls.

CONCLUSIONS

Reduced coordination of inter-limb dynamics was found in active healthy older fallers and multiple fallers despite no difference in other measures of intrinsic falls risk. Evaluating inter-limb dynamics may be a clinically sensitive technique to detect early gait instability and falls risk in high functioning older adults, prior to change in other measures of physical function, balance and gait.

Data series embedding and scale invariant statistics

Data sequences acquired from bio-systems such as human gait data, heart rate interbeat data, or DNA sequences exhibit complex dynamics that is frequently described by a long-memory or power-law decay of autocorrelation function. One way of characterizing that dynamics is through scale invariant statistics or "fractal-like" behavior. For quantifying scale invariant parameters of physiological signals several methods have been proposed. Among them the most common are detrended fluctuation analysis, sample mean variance analyses, power spectral density analysis, R/S analysis, and recently in the realm of the multifractal approach, wavelet analysis. In this paper it is demonstrated that embedding the time series data in the high-dimensional pseudo-phase space reveals scale invariant statistics in the simple fashion. The procedure is applied on different stride interval data sets from human gait measurements time series (Physio-Bank data library). Results show that introduced mapping adequately separates long-memory from random behavior. Smaller gait data sets were analyzed and scale-free trends for limited scale intervals were successfully detected. The method was verified on artificially produced time series with known scaling behavior and with the varying content of noise. The possibility for the method to falsely detect long-range dependence in the artificially generated short range dependence series was investigated.

Aging effects on cardiac and respiratory dynamics in healthy subjects across sleep stages

STUDY OBJECTIVES

Respiratory and heart rate variability exhibit fractal scaling behavior on certain time scales. We studied the short-term and long-term correlation properties of heartbeat and breathing-interval data from disease-free subjects focusing on the age-dependent fractal organization. We also studied differences across sleep stages and night-time wake and investigated quasi-periodic variations associated with cardiac risk.

DESIGN

Full-night polysomnograms were recorded during 2 nights, including electrocardiogram and oronasal airflow.

SETTING

Data were collected in 7 laboratories in 5 European countries.

PARTICIPANTS

180 subjects without health complaints (85 males, 95 females) aged from 20 to 89 years.

INTERVENTIONS

None.

MEASUREMENTS AND RESULTS

Short-term correlations in heartbeat intervals measured by the detrended fluctuation analysis (DFA) exponent alpha1 show characteristic age dependence with a maximum around 50-60 years disregarding the dependence on sleep and wake states. Long-term correlations measured by alpha2 differ in NREM sleep when compared with REM sleep and wake, besides weak age dependence. Results for respiratory intervals are similar to those for alpha2 of heartbeat intervals. Deceleration capacity (DC) decreases with age; it is lower during REM and deep sleep (compared with light sleep and wake).

CONCLUSION

The age dependence of alpha1 should be considered when using this value for diagnostic purposes in post-infarction patients. Pronounced long-term correlations (larger alpha2) for heartbeat and respiration during REM sleep and wake indicate an enhanced control of higher brain regions, which is absent during NREM sleep. Reduced DC possibly indicates an increased cardiovascular risk with aging and during REM and deep sleep.

Studying the time scale dependence of environmental variables predictability using fractal analysis

Prediction of meteorological and air quality variables motivates a lot of research in the atmospheric sciences and exposure assessment communities. An interesting related issue regards the relative predictive power that can be expected at different time scales, and whether it vanishes altogether at certain ranges. An improved understanding of our predictive powers enables better environmental management and more efficient decision making processes. Fractal analysis is commonly used to characterize the self-affinity of time series. This work introduces the Continuous Wavelet Transform (CWT) fractal analysis method as a tool for assessing environmental time series predictability. The high temporal scale resolution of the CWT enables detailed information about the Hurst parameter, a common temporal fractality measure, and thus about time scale variations in predictability. We analyzed a few years records of half-hourly air pollution and meteorological time series from which the trivial seasonal and daily cycles were removed. We encountered a general trend of decreasing Hurst values from about 1.4 (good autocorrelation and predictability), in the sub-daily time scale to 0.5 (which implies complete randomness) in the monthly to seasonal scales. The air pollutants predictability follows that of the meteorological variables in the short time scales but is better at longer scales.

Recurrence interval analysis of trading volumes

We study the statistical properties of the recurrence intervals τ between successive trading volumes exceeding a certain threshold q. The recurrence interval analysis is carried out for the 20 liquid Chinese stocks covering a period from January 2000 to May 2009, and two Chinese indices from January 2003 to April 2009. Similar to the recurrence interval distribution of the price returns, the tail of the recurrence interval distribution of the trading volumes follows a power-law scaling, and the results are verified by the goodness-of-fit tests using the Kolmogorov-Smirnov (KS) statistic, the weighted KS statistic and the Cramér-von Mises criterion. The measurements of the conditional probability distribution and the detrended fluctuation function show that both short-term and long-term memory effects exist in the recurrence intervals between trading volumes. We further study the relationship between trading volumes and price returns based on the recurrence interval analysis method. It is found that large trading volumes are more likely to occur following large price returns, and the comovement between trading volumes and price returns is more pronounced for large trading volumes.

Long-range correlations of different EEG derivations in rats: sleep stage-dependent generators may play a key role

For the electroencephalogram (EEG), topographic differences in the long-range temporal correlations would imply that these signals might be affected by specific mechanisms related to the generation of a given neuronal process. So the properties of the generators of various EEG oscillations might be investigated by their spatial differences of the long-range temporal correlations. In the present study, these correlations were characterized with respect to their topography during different vigilance states by detrended fluctuation analysis (DFA). The results indicated that (1) most of the scaling exponents acquired from different EEG derivations for various oscillations were significantly different in each vigilance state; these differences might be resulted from the different quantities and different locations of sleep stage-dependent generators of various neuronal processes; (2) there might be multiple generators of delta and theta over the brain and many of them were sleep stage-dependent; (3) the best site of the frontal electrode in a fronto-parietal bipolar electrode for sleep staging might be above the anterior midline cortex. We suggest that DFA analysis can be used to explore the properties of the generators of a given neuronal oscillation, and the localizations of these generators if more electrodes are involved.

Exercise modulation of cardiorespiratory variability in humans

Cardiorespiratory variability is the product of the integration of centrally generated rhythms with feedback from central and peripheral sensors. To quantify the effect of increased central drive on scaling patterns of cardiorespiratory activity, breath-to-breath interval (BBI) and heartbeat-to-heartbeat interval (RRI) were recorded from 17 female and 17 male adult subjects at rest and at two levels of mild exercise. Temporal scaling of BBI and RRI was quantified with detrended fluctuation analysis. Relative to a resting state, exercise induced a decrease in the short-term scaling of BBI (p=0.022), an increase in the long-term scaling of RRI (p=0.006), and abolished a significant positive linear relationship in females subjects (p=0.024) and a significant negative relationship in male subjects (p=0.025) in the short-term scaling of BBI and RRI. In conclusion, exercise has opposing effects on the control of breathing and heart rate, and modulates a divergent gender-based coupling of the temporal scaling of cardiorespiratory function.

Analysis of RR intervals time series of congestive heart failure patients with Higuchi's fractal dimension

We study time series of RR intervals of subjects with normal sinus rhythm (NSR) during sleep and wake phases and we also analyze time series of subjects with congestive heart failure (CHF) in both phases with the method of Higuchi's fractal dimension. We have found the presence of oscillations in the plots we have used to evaluate the Higuchi's fractal dimension; these oscillations seem to be associated with the appearance of periodicities in the time series of CHF patients in the sleep phase. These periodicities do not appear in all the six hours of the sleep phase, so this regular behavior is not observed in all sleep stages. It could be possible that patients that always show these periodicities are in worst condition, and it is possible that the disease worsens when periodicities appear in the Higuchi's graph.

Study of the MEG background activity in Alzheimer's disease patients with scaling analysis methods

Alzheimer's disease (AD) is one of the most prominent neurodegenerative disorders. The aim of this research work is to study the magnetoencephalogram (MEG) background activity in AD patients using two scaling analysis methods: detrended fluctuation analysis (DFA) and backward detrended moving average (BDMA). Both measures have been designed to quantify correlations in noisy and non-stationary signals. Five minutes of recording were acquired with a 148-channel whole-head magnetometer in 15 patients with probable AD and 15 control subjects. Both DFA and BDMA exhibited two scaling regions with different slopes. Significant differences between both groups were found in the second region of DFA and in the first region of BDMA (p < 0.01, Student's t-test). Using receiver operating characteristic curves, accuracies of 83.33% with DFA and of 80% with BDMA were reached. Our findings show the usefulness of these scaling analysis methods to increase our insight into AD.

Human EEG shows long-range temporal correlations of oscillation amplitude in Theta, Alpha and Beta bands across a wide age range

OBJECTIVE

Long-range temporal correlations (LRTC) of EEG amplitude fluctuations in adults reveal power-law statistics and have been interpreted within the framework of self-organized criticality (SOC). In physical systems states of self-organized criticality showing power-law statistics take time to develop. In this paper we have sought evidence for the idea that brain development tends towards SOC through examining the hypothesis that during normal human development a power law behaviour of EEG oscillations is approached with increasing chronological age.

METHODS

We examined EEGs from central and parietal electrodes in 36 subjects aged between 0 and 660months during performance of a steady wrist extension task with their dominant hand and applied spectral and detrended fluctuation analysis in 36 subjects to assess long-range temporal correlations of oscillation amplitude in the Theta, Alpha and Beta frequency bands.

RESULTS

Our data indicate that at all subject ages power-law statistics dominate the records at Alpha, Beta and Theta frequencies. Small consistent effects of chronological age were detected for amplitude fluctuations at Theta and Beta frequencies.

CONCLUSIONS

The data suggest that the scale-free nature of EEG LRTCs is a feature from early childhood through to maturity but that there are changes in the magnitude of these effects with age.

SIGNIFICANCE

This study is the first to have explored long-range temporal correlations over a wide range of chronological age.

An empirical examination of detrended fluctuation analysis for gait data

Stride interval series exhibit statistical persistence, and detrended fluctuation analysis (DFA) is a routinely employed technique for describing this behavior. However, the implementation of DFA to gait data varies considerably between studies. We empirically examine two practical aspects of DFA which significantly affect the analysis outcome: the box size range and the stride interval series length. We conduct an analysis of their effect using stride intervals from 16 able-bodied adults, for overground walking, treadmill walking while holding a handrail, and treadmill walking without using a handrail. Our goal is to provide general guidelines for these two choices, with the aim of standardizing the application of DFA and facilitating inter-study comparisons. Based on the results of our analysis, we propose the use of box sizes from 16 to N/9, where N is the number of stride intervals. Moreover, for differentiating between normal and pathological walking with reasonable accuracy, we recommend a minimum of 600 stride intervals.

Effect of extreme data loss on long-range correlated and anticorrelated signals quantified by detrended fluctuation analysis

Detrended fluctuation analysis (DFA) is an improved method of classical fluctuation analysis for nonstationary signals where embedded polynomial trends mask the intrinsic correlation properties of the fluctuations. To better identify the intrinsic correlation properties of real-world signals where a large amount of data is missing or removed due to artifacts, we investigate how extreme data loss affects the scaling behavior of long-range power-law correlated and anticorrelated signals. We introduce a segmentation approach to generate surrogate signals by randomly removing data segments from stationary signals with different types of long-range correlations. The surrogate signals we generate are characterized by four parameters: (i) the DFA scaling exponent alpha of the original correlated signal u(i) , (ii) the percentage p of the data removed from u(i) , (iii) the average length mu of the removed (or remaining) data segments, and (iv) the functional form P(l) of the distribution of the length l of the removed (or remaining) data segments. We find that the global scaling exponent of positively correlated signals remains practically unchanged even for extreme data loss of up to 90%. In contrast, the global scaling of anticorrelated signals changes to uncorrelated behavior even when a very small fraction of the data is lost. These observations are confirmed on two examples of real-world signals: human gait and commodity price fluctuations. We further systematically study the local scaling behavior of surrogate signals with missing data to reveal subtle deviations across scales. We find that for anticorrelated signals even 10% of data loss leads to significant monotonic deviations in the local scaling at large scales from the original anticorrelated to uncorrelated behavior. In contrast, positively correlated signals show no observable changes in the local scaling for up to 65% of data loss, while for larger percentage of data loss, the local scaling shows overestimated regions (with higher local exponent) at small scales, followed by underestimated regions (with lower local exponent) at large scales. Finally, we investigate how the scaling is affected by the average length, probability distribution, and percentage of the remaining data segments in comparison to the removed segments. We find that the average length mu_{r} of the remaining segments is the key parameter which determines the scales at which the local scaling exponent has a maximum deviation from its original value. Interestingly, the scales where the maximum deviation occurs follow a power-law relationship with mu_{r} . Whereas the percentage of data loss determines the extent of the deviation. The results presented in this paper are useful to correctly interpret the scaling properties obtained from signals with extreme data loss.

Detrended fluctuation analysis predicts successful defibrillation for out-of-hospital ventricular fibrillation cardiac arrest

AIMS

Repeated failed shocks for ventricular fibrillation (VF) in out-of-hospital cardiac arrest (OOHCA) can worsen the outcome. It is very important to rapidly distinguish between early and late VF. We hypothesised that VF waveform analysis based on detrended fluctuation analysis (DFA) can help predict successful defibrillation.

METHODS

Electrocardiogram (ECG) recordings of VF signals from automated external defibrillators (AEDs) were obtained for subjects with OOHCA in Taipei city. To examine the time effect on DFA, we also analysed VF signals in subjects who experienced sudden cardiac death during Holter study from PhysioNet, a publicly accessible database. Waveform parameters including root-mean-squared (RMS) amplitude, mean amplitude, amplitude spectrum analysis (AMSA), frequency analysis as well as fractal measurements including scaling exponent (SE) and DFA were calculated. A defibrillation was regarded as successful when VF was converted to an organised rhythm within 5s after each defibrillation.

RESULTS

A total of 155 OOHCA subjects (37 successful and 118 unsuccessful defibrillations) with VF were included for analysis. Among the VF waveform parameters, only AMSA (7.61+/-3.30 vs. 6.30+/-3.13, P=0.028) and DFAalpha2 (0.38+/-0.24 vs. 0.49+/-0.24, P=0.013) showed significant difference between subjects with successful and unsuccessful defibrillation. The area under the curves (AUCs) for AMSA and DFAalpha2 was 0.63 (95% confidence interval (CI)=0.52-0.73) and 0.65 (95% CI=0.54-0.75), respectively. Among the waveform parameters, only DFAalpha2, SE and dominant frequency showed significant time effect.

CONCLUSIONS

The VF waveform analysis based on DFA could help predict first-shock defibrillation success in patients with OOHCA. The clinical utility of the approach deserves further investigation.

An investigation of stride interval stationarity in a paediatric population

Fluctuations in the stride interval of human gait have been found to exhibit statistical persistence over hundreds of strides, the extent of which changes with age, pathology, and speed-constrained walking. Thus, recent investigations have focused on quantifying this scaling behavior in order to gain insight into locomotor control. While the ability of a given analysis technique to provide an accurate scaling estimate depends largely on the stationary properties of the given series, direct investigation of stride interval stationarity has been largely overlooked. In the present study we test the stride interval time series obtained from able-bodied children for weak stationarity. Specifically, we analyze signals obtained during three distinct modes of self-paced locomotion: (i) overground walking, (ii) unsupported (hands-free) treadmill walking, and (iii) handrail-supported treadmill walking. Using the reverse arrangements test, we identify non-stationary signals in all three walking conditions and find the major known cause to be due to time-varying first and second moments. We further discuss our findings in terms of locomotor control and the differences between the locomotor modalities investigated. Overall, our results advocate against scaling analysis techniques that assume stationarity.

Cross-correlations between volume change and price change

In finance, one usually deals not with prices but with growth rates R, defined as the difference in logarithm between two consecutive prices. Here we consider not the trading volume, but rather the volume growth rate R, the difference in logarithm between two consecutive values of trading volume. To this end, we use several methods to analyze the properties of volume changes |R|, and their relationship to price changes |R|. We analyze 14,981 daily recordings of the Standard and Poor's (S & P) 500 Index over the 59-year period 1950-2009, and find power-law cross-correlations between |R| and |R| by using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these cross-correlations. Motivated by the relationship between |R| and |R|, we estimate the tail exponent alpha of the probability density function P(|R|) approximately |R|(-1-alpha) for both the S & P 500 Index as well as the collection of 1819 constituents of the New York Stock Exchange Composite Index on 17 July 2009. As a new method to estimate alpha, we calculate the time intervals tau(q) between events where R > q. We demonstrate that tau(q), the average of tau(q), obeys tau(q) approximately q(alpha). We find alpha approximately 3. Furthermore, by aggregating all tau(q) values of 28 global financial indices, we also observe an approximate inverse cubic law.

Effect of treadmill walking on the stride interval dynamics of human gait

Metronomic walking has been found to diminish the statistical persistence intrinsic to the stride interval time series of human gait. Since treadmill walking (TW) possesses a similar form of external pacing, we proposed to study the disruptions in the natural neuromuscular rhythms of gait during TW. Treadmill walking is a widespread rehabilitative tool, however, its effect on an individual's stride dynamics is not well understood. To better elucidate potential effects, we tested the hypothesis that TW without handrails would diminish the statistical persistence in an individual's stride interval time series. The scaling exponent (alpha) was employed in this study as a measure of the statistical persistence of the stride interval time series. Sixteen able-bodied young adults (mean age: 23.3+/-3.3 years) were instructed to walk at a self-selected comfortable pace for 15 min in three different conditions in a randomized order: (1) overground walking, (2) TW without holding a handrail (NoRail) and (3) TW while holding a front handrail (Rail). The alpha did not differ significantly between the overground and NoRail conditions (P>0.5). However, the alpha of the Rail condition (alpha=0.92+/-0.10) differed significantly from both the overground (alpha=0.83+/-0.06; P<0.015) and NoRail conditions (alpha=0.82+/-0.08; P<0.01). In contrast, stride interval variability did not change between walking conditions (P>0.5). These findings indicate that comfortable-paced TW does not diminish the intrinsic stride dynamics of human gait.

Local scale exponents of blood pressure and heart rate variability by detrended fluctuation analysis: effects of posture, exercise, and aging

Heart rate self-affinity is often assessed by detrended fluctuations analysis, obtaining two coefficients only: a short-term (alpha(1)) exponent and a long-term (alpha(2)) exponent. Our aim is to show the limits of this approach and alternatively propose the estimation of the whole spectrum of local exponents alpha(n) for heart rate and blood pressure. To illustrate the advantages of this approach, we assess the effects of autonomic activations and age on alpha(n). We measured ECG and arterial pressure in 60 volunteers for 10 min, considering three conditions at increasing sympathetic activation: supine rest, sitting, and sitting during exercise. We computed alpha(n) of R-R intervals and systolic, mean, and diastolic blood pressures, as the slope of the detrended fluctuations function in a log-log plot. Volunteers were divided into age groups and compared. Results indicate that: 1) alpha(1) cannot be defined because short-term coefficients decrease with n, while alpha(2) cannot be defined only for blood pressure during supine rest; 2) heart rate and blood pressure scaling structures differ during supine rest but not during exercise; and 3) age effects appear mainly in supine rest, explaining discrepant results in literature. In conclusion, we recommend estimating the whole alpha(n) spectrum before possibly providing the "two-exponent" description only.

Detrended fluctuation analysis of intracranial pressure predicts outcome following traumatic brain injury

Detrended fluctuation analysis (DFA) is a recently developed technique suitable for describing scaling behavior of variability in physiological signals. The purpose of this study is to explore applicability of DFA methods to intracranial pressure (ICP) signals recorded in patients with traumatic brain injury (TBI). In addition to establishing the degree of fit of the power-law scaling model of detrended fluctuations of ICP in TBI patients, we also examined the relationship of DFA coefficients (scaling exponent and intercept) to: 1) measures of initial neurological functioning; 2) measures of functional outcome at six month follow-up; and 3) measures of outcome, controlling for patient characteristics, and initial neurological status. In a sample of 147 moderate-to-severely injured TBI patients, we found that a higher DFA scaling exponent is significantly associated with poorer initial neurological functioning, and that lower DFA intercept and higher DFA scaling exponent jointly predict poorer functional outcome at six month follow-up, even after statistical control for covariates reflecting initial neurological condition. DFA describes properties of ICP signal in TBI patients that are associated with both initial neurological condition and outcome at six months postinjury.

Long-range correlations in rectal temperature fluctuations of healthy infants during maturation

Background

Control of breathing, heart rate, and body temperature are interdependent in infants, where instabilities in thermoregulation can contribute to apneas or even life-threatening events. Identifying abnormalities in thermoregulation is particularly important in the first 6 months of life, where autonomic regulation undergoes critical development. Fluctuations in body temperature have been shown to be sensitive to maturational stage as well as system failure in critically ill patients. We thus aimed to investigate the existence of fractal-like long-range correlations, indicative of temperature control, in night time rectal temperature (T_rec) patterns in maturing infants.

Methodology/Principal Findings

We measured T_rec fluctuations in infants every 4 weeks from 4 to 20 weeks of age and before and after immunization. Long-range correlations in the temperature series were quantified by the correlation exponent, α using detrended fluctuation analysis. The effects of maturation, room temperature, and immunization on the strength of correlation were investigated. We found that T_rec fluctuations exhibit fractal long-range correlations with a mean (SD) α of 1.51 (0.11), indicating that T_rec is regulated in a highly correlated and hence deterministic manner. A significant increase in α with age from 1.42 (0.07) at 4 weeks to 1.58 (0.04) at 20 weeks reflects a change in long-range correlation behavior with maturation towards a smoother and more deterministic temperature regulation, potentially due to the decrease in surface area to body weight ratio in the maturing infant. α was not associated with mean room temperature or influenced by immunization

Conclusions

This study shows that the quantification of long-range correlations using α derived from detrended fluctuation analysis is an observer-independent tool which can distinguish developmental stages of night time T_rec pattern in young infants, reflective of maturation of the autonomic system. Detrended fluctuation analysis may prove useful for characterizing thermoregulation in premature and other infants at risk for life-threatening events.

Eliminating finite-size effects and detecting the amount of white noise in short records with long-term memory

Long-term memory is ubiquitous in nature and has important consequences for the occurrence of natural hazards, but its detection often is complicated by the short length of the considered records and additive white noise in the data. Here we study synthetic Gaussian distributed records x_{i} of length N that consist of a long-term correlated component (1-a)y_{i} characterized by a correlation exponent gamma , 0<gamma<1 , and a white-noise component aeta_{i} , 0< or =a< or =1 . We show that the autocorrelation function C_{N}(s) has the general form C_{N}(s)=[C_{infinity}(s)-E_{a}]/(1-E_{a}) , where C_{infinity}(0)=1 , C_{infinity}(s>0)=B_{a}s;{-gamma} , and E_{a}={2B_{a}/[(2-gamma)(1-gamma)]}N;{-gamma}+O(N;{-1}) . The finite-size parameter E_{a} also occurs in related quantities, for example, in the variance Delta_{N};{2}(s) of the local mean in time windows of length s : Delta_{N};{2}(s)=[Delta_{infinity};{2}(s)-E_{a}]/(1-E_{a}) . For purely long-term correlated data B_{0} congruent with(2-gamma)(1-gamma)/2 yielding E_{0} congruent withN;{-gamma} , and thus C_{N}(s)=[(2-gamma)(1-gamma)/2s;{-gamma}-N;{-gamma}]/[1-N;{-gamma}] and Delta_{N};{2}(s)=[s;{-gamma}-N;{-gamma}]/[1-N;{-gamma}] . We show how to estimate E_{a} and C_{infinity}(s) from a given data set and thus how to obtain accurately the exponent gamma and the amount of white noise a .

Detrended fluctuation analysis of the magnetic and electric field variations that precede rupture

Magnetic field variations are detected before rupture in the form of "spikes" of alternating sign. The distinction of these spikes from random noise is of major practical importance since it is easier to conduct magnetic field measurements than electric field ones. Applying detrended fluctuation analysis (DFA), these spikes look to be random at short time lags. On the other hand, long-range correlations prevail at time lags larger than the average time interval between consecutive spikes with a scaling exponent alpha around 0.9. In addition, DFA is applied to recent preseismic electric field variations in long duration (several hours to a couple of days) and reveals a scale invariant feature with an exponent alpha approximately 1 over all scales available (around five orders of magnitude).

Gait dynamics in Parkinson's disease: common and distinct behavior among stride length, gait variability, and fractal-like scaling

Parkinson's disease (PD) is a common, debilitating neurodegenerative disease. Gait disturbances are a frequent cause of disability and impairment for patients with PD. This article provides a brief introduction to PD and describes the gait changes typically seen in patients with this disease. A major focus of this report is an update on the study of the fractal properties of gait in PD, the relationship between this feature of gait and stride length and gait variability, and the effects of different experimental conditions on these three gait properties. Implications of these findings are also briefly described. This update highlights the idea that while stride length, gait variability, and fractal scaling of gait are all impaired in PD, distinct mechanisms likely contribute to and are responsible for the regulation of these disparate gait properties.

Statistical analysis of the overnight and daytime return

We investigate the two components of the total daily return (close-to-close), the overnight return (close-to-open), and the daytime return (open-to-close), as well as the corresponding volatilities of the 2215 New York Stock Exchange stocks for the 20 year period from 1988 to 2007. The tail distribution of the volatility, the long-term memory in the sequence, and the cross correlation between different returns are analyzed. Our results suggest that (i) the two component returns and volatilities have features similar to that of the total return and volatility. The tail distribution follows a power law for all volatilities, and long-term correlations exist in the volatility sequences but not in the return sequences. (ii) The daytime return contributes more to the total return. Both the tail distribution and the long-term memory of the daytime volatility are more similar to that of the total volatility, compared to the overnight records. In addition, the cross correlation between the daytime return and the total return is also stronger. (iii) The two component returns tend to be anticorrelated. Moreover, we find that the cross correlations between the three different returns (total, overnight, and daytime) are quite stable over the entire 20 year period.

Stratification pattern of static and scale-invariant dynamic measures of heartbeat fluctuations across sleep stages in young and elderly

Cardiac dynamics exhibit complex variability characterized by scale-invariant and nonlinear temporal organization related to the mechanism of neuroautonomic control, which changes with physiologic states and pathologic conditions. Changes in sleep regulation during sleep stages are also related to fluctuations in autonomic nervous activity. However, the interaction between sleep regulation and cardiac autonomic control remains not well understood. Even less is known how this interaction changes with age, as aspects of both cardiac dynamics and sleep regulation differ in healthy elderly compared to young subjects. We hypothesize that because of the neuroautonomic responsiveness in young subjects, fractal and nonlinear features of cardiac dynamics exhibit a pronounced stratification pattern across sleep stages, while in elderly these features will remain unchanged due to age-related loss of cardiac variability and decline of neuroautonomic responsiveness. We analyze the variability and the temporal fractal organization of heartbeat fluctuations across sleep stages in both young and elderly. We find that independent linear and nonlinear measures of cardiac control consistently exhibit the same ordering in their values across sleep stages, forming a robust stratification pattern. Despite changes in sleep architecture and reduced heart rate variability in elderly subjects, this stratification surprisingly does not break down with advanced age. Moreover, the difference between sleep stages for some linear, fractal, and nonlinear measures exceeds the difference between young and elderly, suggesting that the effect of sleep regulation on cardiac dynamics is significantly stronger than the effect of healthy aging. Quantifying changes in this stratification pattern may provide insights into how alterations in sleep regulation contribute to increased cardiac risk.

Levels of complexity in scale-invariant neural signals

Many physical and physiological signals exhibit complex scale-invariant features characterized by 1/f scaling and long-range power-law correlations, indicating a possibly common control mechanism. Specifically, it has been suggested that dynamical processes, influenced by inputs and feedback on multiple time scales, may be sufficient to give rise to 1/f scaling and scale invariance. Two examples of physiologic signals that are the output of hierarchical multiscale physiologic systems under neural control are the human heartbeat and human gait. Here we show that while both cardiac interbeat interval and gait interstride interval time series under healthy conditions have comparable 1/f scaling, they still may belong to different complexity classes. Our analysis of the multifractal scaling exponents of the fluctuations in these two signals demonstrates that in contrast to the multifractal behavior found in healthy heartbeat dynamics, gait time series exhibit less complex, close to monofractal behavior. Further, we find strong anticorrelations in the sign and close to random behavior for the magnitude of gait fluctuations at short and intermediate time scales, in contrast to weak anticorrelations in the sign and strong positive correlation for the magnitude of heartbeat interval fluctuations-suggesting that the neural mechanisms of cardiac and gait control exhibit different linear and nonlinear features. These findings are of interest because they underscore the limitations of traditional two-point correlation methods in fully characterizing physiological and physical dynamics. In addition, these results suggest that different mechanisms of control may be responsible for varying levels of complexity observed in physiological systems under neural regulation and in physical systems that possess similar 1/f scaling.

Gender and breathing route modulate cardio-respiratory variability in humans

During spontaneous breathing, there is an intrinsic scaling of respiratory variability and a correlation between respiratory and heart rate variabilities. To identify the effect of breathing route on respiratory and heart rate variabilities, breath-to-breath interval (BBI) and heartbeat-to-heartbeat interval (RRI) were recorded from 12 female and 12 male adult subjects breathing through the nose or mouth. Temporal scaling within the BBI and RRI was quantified with detrended fluctuation analysis (DFA). We identified a significant gender-based breathing route interaction in the short-term scaling of BBI (p=0.007), a decrease in the short-term scaling of RRI during nose breathing (p=0.026), and a significant interdependence of short-term scaling of BBI and RRI in female subjects. We conclude that there is a gender-based differential effect of breathing route on the control of respiration and an increase in the random behavior of RRI associated with nasal breathing. These data also suggest the presence cardio-respiratory coupling of scaling behavior in female subjects.

Reduction of scale invariance of activity fluctuations with aging and Alzheimer's disease: Involvement of the circadian pacemaker

Human motor control systems orchestrate complex scale-invariant patterns of activity over a wide range of time scales (minutes to hours). The neural mechanisms underlying scale-invariance are unknown in humans. In rats, the master circadian pacemaker [suprachiasmatic nucleus (SCN)] is crucially involved in scale-invariant activity fluctuations over multiple time scales from minutes to 24 h. Aging and Alzheimer's disease (AD) are associated with progressive dysfunction of the SCN. Thus, if the SCN is responsible for the scale-invariant activity fluctuations in humans, we predict disturbances of scale-invariant activity fluctuations in elderly humans and even more pronounced disturbances in elderly humans with AD. To test these hypotheses, we studied spontaneous daytime activity patterns in 13 young adults (mean +/- SD: 25.5 +/- 6.1 y); 13 elderly people with early-stage AD (68.5 +/- 6.1 y) matched with 13 elderly controls (68.6 +/- 6.1 y); and 14 very old people with late-stage AD (83.9 +/- 6.7 y) matched with 12 very old controls (80.8 +/- 8.6 y). In young adults, activity exhibited robust scale-invariant correlations across all tested time scales (minutes to 8 h). The scale-invariant correlations at 1.5-8 h declined with age (P = 0.01) and were significantly reduced in the elderly (P = 0.04) and very old controls (P = 0.02). Remarkably, an age-independent AD effect further reduced the scale-invariant correlations at 1.5-8 h (P = 0.04), leading to the greatest reduction of the scale-invariant correlations in very old people with late-stage AD-resembling closely the loss of correlations at large time scales in SCN-lesioned animals. Thus, aging and AD significantly attenuate the scale invariance of activity fluctuations over multiple time scales. This attenuation may reflect functional changes of the SCN.

Scaling behaviour of the global tropopause

Abstract. Detrended fluctuation analysis is applied to the time series of the global tropopause height derived from the 1980–2004 daily radiosonde data, in order to detect long-range correlations in its time evolution. Global tropopause height fluctuations in small time-intervals are found to be positively correlated to those in larger time intervals in a power-law fashion. The exponent of this dependence is larger in the tropics than in the middle and high latitudes in both hemispheres. Greater persistence is observed in the tropopause of the Northern than in the Southern Hemisphere. A plausible physical explanation of the fact that long-range correlations in tropopause variability decreases with increasing latitude is that the column ozone fluctuations (that are closely related with the tropopause ones) exhibit long range correlations, which are larger in tropics than in the middle and high latitudes at long time scales. This finding for the tropopause height variability should reduce the existing uncertainties in assessing the climatic characteristics. More specifically the reliably modelled values of a climatic variable (i.e. past and future simulations) must exhibit the same scaling behaviour with that possibly existing in the real observations of the variable under consideration. An effort has been made to this end by applying the detrended fluctuation analysis to the global mean monthly land and sea surface temperature anomalies during the period January 1850–August 2008. The result obtained supports the findings presented above, notably: the correlations between the fluctuations in the global mean monthly land and sea surface temperature display scaling behaviour which must characterizes any projection.

How main-chains of proteins explore the free-energy landscape in native states

Understanding how a single native protein diffuses on its free-energy landscape is essential to understand protein kinetics and function. The dynamics of a protein is complex, with multiple relaxation times reflecting a hierarchical free-energy landscape. Using all-atom molecular dynamics simulations of an alpha/beta protein (crambin) and a beta-sheet polypeptide (BS2) in their "native" states, we demonstrate that the mean-square displacement of dihedral angles, defined by 4 successive C(alpha) atoms, increases as a power law of time, t(alpha), with an exponent alpha between 0.08 and 0.39 (alpha = 1 corresponds to Brownian diffusion), at 300 K. Residues with low exponents are located mainly in well-defined secondary elements and adopt 1 conformational substate. Residues with high exponents are found in loops/turns and chain ends and exist in multiple conformational substates, i.e., they move on multiple-minima free-energy profiles.