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Nonlinear dynamics is useful for determining correlations in non-stationary of highly heterogeneous time series. In this work two computational methods derived from nonlinear dynamics were used to analyze tachograms of healthy subjects and patients with Congestive Heart Failure (CHF); such methods were the Detrended Fluctuation Analysis (DFA) and the Higuchi’s Fractal Dimension (HFD). First, both methods were applied separately. In DFA, marked differences could be observed in the obtained graphs and results from healthy subjects and CHF patients, a main difference was between the number of crossovers that led to a cumulative change in slopes (Δα), it was higher in the second ones which is explained by the increased presence of crossovers; in HFD case such differences were not very evident. With the obtained results from HFD new series were generated of the differences between each point and its corresponding point of the linear fit, then by plotting these series low-frequency oscillations were present, to characterize these oscillations DFA and HFD methods were used together. The obtained results let us infer that the use of these methods can help us to get more information from physiological signals (ECG for this work) and have a wider overview of a patient’s health state.

Sedimentation process as the form of iteration process has been studied using fractal statistics in the case of the change of Hurst exponent patterns. The similar profile of Hurst exponent can be related to the one depositional cycle, which can be explained that the fractal statistics might detect the sedimentation process in one interval time. Models that showed the effect of the chosen time lag have been presented. The time lag affects the effectively of fractal statistics in determining the stratification of sedimentary rocks. In this study, we modeled resistivity layer and 1/f noise is added to the resistivity values for each layer. We calculated Hurst exponent with different window length. We observed that the layer boundaries can be distinguished from the Hurst exponent. From our models, we obtained that the window length affects the fractal statistics in detection of top layer. Large window length hardly detects thin layer and accurately enough in detection of thick layers as shown in three isotropic layers model. Small window length is accurately enough in detecting the changes on layers although it also reflects the scaling behavior of the noise. The models show that when window length increases the correlation continues decreasing and becomes uncorrelated when window length infinite.

Detrended fluctuation analysis (DFA) was originally developed for the evaluation of DNA sequence and interval for heart rate variability (HRV), but it is now used to obtain various biological information. In this study we perform DFA on artificially generated data where we already know the relationship between signal and the physical event causing the signal. We generate artificial data using molecular dynamics. The Brownian motion of a polymer under an external force is investigated. In order to generate artificial fluctuation in the physical properties, we introduce obstacle pillars fixed to nanostructures. Using different conditions such as presence or absence of obstacles, external field, and the polymer length, we perform DFA on energies and positions of the polymer.

We suggest an algorithm for the estimation of the largest Lyapunov exponent for one-dimensional domains. We evaluate the largest Lyapunov exponent for one-dimensional domains, which present the surrogate long-range correlated stochastic time series with Hurst exponent H=0.1, H=0.9, H=0.5. It has been established that for an anticorrelated time series with the Hurst exponent H=0.1, the largest Lyapunov exponent is positive. For a correlated time series with Hurst exponent H=0.9, the largest Lyapunov exponent is negative. Also for a classical random walk with Hurst exponent H=0.5, the largest Lyapunov exponent is close to zero.

We report results concerning the pattern formation during the capillary rising of a fluid front into a dense dry granular media. The system consists in a modified Hele-Shaw cell filled with grains of different gradings and confined in a narrow gap between the glass plates. This assembly is vertically installed over a water reservoir to allow an ascending front of liquid to percolate into the granular media. We acquire digital images of the liquid/air front which are treated by means of imaging analysis techniques. Thus, we are able to assess the temporal evolution of the air/liquid boundary profiles. We measure the roughness of the profiles, using a detrended fluctuation analysis technique, to obtain their fractal dimension. We compare our results with similar experiments reported in literature considering fluid displacement into heterogeneous media and directed percolation theory. However, the range of values for the Hurst exponent obtained from our experiments are odd to the measured/predicted values in experiments or theory.

Many physiological systems have been in recent years quantitatively characterized using fractal analysis. We applied it to study heart variability of young subjects with metabolic syndrome (MS); we examined the RR time series (time between two R waves in ECG) with the detrended fluctuation analysis (DFA) method, the Higuchi's fractal dimension method and the multifractal analysis to detect the possible presence of heart problems. The results show that although the young persons have MS, the majority do not present alterations in the heart dynamics. However, there were cases where the fractal parameter values differed significantly from the healthy people values.

There are already‐established methods for investigating biological signals such as rhythmic heartbeats. We used detrended fluctuation analysis (DFA), originally developed by Peng et al. (1995) to check power‐law characteristics, because the method can quantify the heart condition numerically. In this article, we studied the heartbeat of sleeping subjects. Our purpose was to test whether DFA is useful to evaluate the subject’s wellness of both during being awake and sleeping. This is a challenge to measure sleep without complex/expensive machine, an electro encephalography (EEG). We conducted electrophysiological recording to measure heartbeats during sleep using electrocardiograph with three‐leads, one ground electrode and two active electrodes attached to chest. For good recording, a stable baseline must be maintained even when subjects move their body. We needed a tool to ensure long‐term steady recording. We thus invented a new electric‐circuit designed to produce this desired result. This gadget allowed us to perform heartbeat recording without any drifting baseline. We then were able to detect 100% of heartbeat peaks over the entire period of sleep. Here, we show a case study as empirical evidence that DFA is useful numerical method for quantifying sleep by using the scaling exponents.

We interrogated Swift Long GRBs using a Fast Wavelet technique to probe the observed variability for fractal (or self‐affine) behavior. Self‐affine behavior for a time‐series implies a statistical similarity after a particular rescaling transformation from the Affine Group, X(jt) = j−(α−1)/2X(t). It is straightforward to associate the slope parameter, α, with the Hurst exponent, H, and the fractal dimension, D. These measures may hold a key to a better understanding of GRB variability and provide constraints for theoretical models.

We investigate the scaling and fractal properties of the horizontal component of the geomagnetic field time series acquired by the Magnetic Data Acquisition System (MAGDAS) of the Space Environment Research Center (SERC) of Kyushu University in Japan. The data set covers the quiet period of geomagnetic activity in February 2007 at the near equatorial stations of Langkawi in Malaysia and Davao in the Philippines. These data were sampled every minute for 28 days giving a sample size of 40,320. The power spectra of the time series are obtained, showing dominant periodicities at 24, 12, 8 and 6 hours due to the effect of the sun on the geomagnetic field. The power spectra also indicate scaling with Hurst exponents of 0.50–0.77 in the period of 10 minutes to 6 hours. Then, rescaled range analysis and detrended fluctuation analysis are performed, producing similar ranges of Hurst exponents. Finally, these fractal methods are used to determine the Hurst exponents of similar sized data sets artificially generated...

We analyzed heartbeat‐intervals with our own program of detrended fluctuation analysis (DFA) to quantify the irregularity of the heartbeat. The present analysis revealed that normal healthy subjects have the scaling exponent of 1.0, and ischemic heart disease pushes the scaling exponent up to 1.2–1.5. We conclude that the scaling exponent, calculated by the DFA, reflects a risk for the “failing” heart. The scaling exponents could determine whether the subjects are under sick or in healthy conditions on the basis of cardiac physiology.

The time series of microwave flux density variations (2.7 and 8.1 GHz) from 24 quasars was analyzed in the methods of power spectral index, Higuchi’s fractal dimension, Hurst exponent, Correlation dimension and Kolmogorov entropy. The aim of study is to find cosmological meanings of analyzed results by mutually referring the indices varing with red shift of quasars. A systematic change was found among them, while yet unknown of discrimination between noise and dynamics of radio source systems.

We applied detrended fluctuation analysis (DFA) to characterize the intra‐beat scaling dynamics of electrocardiographic (ECG) recordings from the PhysioNet Sudden Cardiac Death Holter Database. The main finding of this contribution is that, in such recordings involving different types of arrhythmias; the ECG waveform, besides showing a less‐random intra‐beat dynamics, becomes more regular during bigeminy, ventricular tachycardia (VT) or even atrial fibrillation (AFIB) and ventricular fibrillation (VF) despite the appearance of erratic traces. Thus, notwithstanding that these cardiac rhythm abnormalities are generally considered as irregular and some of them generated by random impulses or wavefronts, the intra‐beat scaling properties suggest that regularity dominates the underlying mechanisms of arrhythmias. Among other explanations, this may result from shorted or restricted ‐less complex‐ pathways of conduction of the electrical activity within the ventricles.

The statistical study of large scale genomic data has turned out to be a very important tool in population genetics. Quantitative methods are essential to understand and implement association studies in the biomedical and health sciences. Nevertheless, the characterization of recently admixed populations has been an elusive problem due to the presence of a number of complex phenomena. For example, linkage disequilibrium structures are thought to be more complex than their non‐recently admixed population counterparts, presenting the so‐called ancestry blocks, admixed regions that are not yet smoothed by the effect of genetic recombination. In order to distinguish characteristic features for various populations we have implemented several methods, some of them borrowed or adapted from the analysis of nonlinear time series in statistical physics and quantitative physiology. We calculate the main fractal dimensions (Kolmogorov's capacity, information dimension and correlation dimension, usually named, D0, D1 and D2). We also have made detrended fluctuation analysis and information based similarity index calculations for the probability distribution of correlations of linkage disequilibrium coefficient of six recently admixed (mestizo) populations within the Mexican Genome Diversity Project [1] and for the non‐recently admixed populations in the International HapMap Project [2]. Nonlinear correlations showed up as a consequence of internal structure within the haplotype distributions. The analysis of these correlations as well as the scope and limitations of these procedures within the biomedical sciences are discussed.

Heartbeat fluctuations exhibit temporal structure with fractal and nonlinear features that reflect changes in the neuroautonomic control. In this work we have used the detrended fluctuation analysis (DFA) to analyze heartbeat (RR) intervals of 54 healthy subjects and 40 patients with congestive heart failure during 24 hours; we separate time series for sleep and wake phases. We observe long‐range correlations in time series of healthy persons and CHF patients. However, the correlations for CHF patients are weaker than the correlations for healthy persons; this fact has been reported by Ashkenazy et al. [1] but with a smaller group of subjects. In time series of CHF patients there is a crossover, it means that the correlations for high and low frequencies are different, but in time series of healthy persons there are not crossovers even if they are sleeping. These crossovers are more pronounced for CHF patients in the sleep phase. We decompose the heartbeat interval time series into magnitude and sign series, we know that these kinds of signals can exhibit different time organization for the magnitude and sign and the magnitude series relates to nonlinear properties of the original time series, while the sign series relates to the linear properties. Magnitude series are long‐range correlated, while the sign series are anticorrelated. Newly, the correlations for healthy persons are different that the correlations for CHF patients both for magnitude and sign time series. In the paper of Ashkenazy et al. they proposed the empirical relation: αsign≈1/2(αoriginal+αmagnitude) for the short‐range regime (high frequencies), however, we have found a different relation that in our calculations is valid for short and long‐range regime: αsign≈1/4(αoriginal+αmagnitude).

In this paper, the 1/fα fluctuation property of the botanic potential time series is investigated in terms of wavelet spectrum and fractal dimension. It is shown that the signal exhibits the fluctuation characteristics investigated and the index is about α = 1.145, in addition, the fractal dimension is about D = 1.352±0.084.

Detrended Fluctuation Analysis (DFA), suitable for the analysis of nonstationary time series, is used to investigate power law in some of the Bach's pitches series. Using DFA method, which also is a well-established method for the detection of long-range correlations, frequency series of Bach's pitches have been analyzed. In this view we find same Hurts exponents in the range (0.7-0.8) in his Inventions and sinfonia., Comment: 5 pages, 4 figures

Many physical and biological systems exhibit complex behavior characterized by long‐range power‐law correlations. Detrended fluctuation analysis (DFA) is a scaling analysis method that provides a scaling parameter to represent the correlation properties of a signal. The study of interbeat sequences with the DFA method has revealed the presence of crossovers associated with physiological aging and heart with failure; the hinges present in the crossover region from both the elderly healthy individuals and the patients with congestive heart failure (CHF) are in opposite directions. The interbeat sequences of healthy young persons do not show crossovers. In this paper we study interbeat time series of healthy young and elderly persons and patients with CHF. We use the DFA‐m method, where m refers to the order of the polynomial function used for the fitting. For instance, DFA‐2 filters linear trends and DFA‐3 filters quadratic trends. We found that the presence of the crossovers and the direction of the hinges...

We examine fractal scaling properties of heart rate variability using detrended fluctuation analysis (DFA), during physical activity in healthy subjects. We analyze 11 records of healthy subjects, which include both usual daily activity and experimental exercise. The subjects were asked to ride on a bicycle ergometer for 2.5 hours, and maintained a heartbeat interval of 500–600 ms. In order to estimate the long‐range correlation in the series of heartbeat intervals during controlled physical activity, we apply DFA to the data set with the third‐order polynomial trend removed. For all records during exercise, we observe a characteristic crossover phenomenon at ≈ 300 beats. The scaling exponent in the range > 300 beats (> 3 minutes) during exercise decreases and tends to be closer to white noise (≈ 0.5), which corresponds to uncorrelated behavior. The long‐range scaling exponent during exercise is significantly lower than that during daily activity in this range. Contrary to the currently held view, our results indicate a breakdown in long‐range correlations and 1/f‐like scaling, rather than the increase in the Hurst exponent characteristic of a (congestive) increase in afterload and observed, e.g., in congestive heart failure (CHF) patients. Further, our results suggest an increased load imbalance induced departure from critical‐like behavior, which has recently been reported in healthy human heart rate during daily activity.

We review a recently developed approach for analyzing time series with long-range cor- relations by decomposing the signal increment series into magnitude and sign series and analyzing their scaling properties. We show that time series with identical long-range correlations can exhibit different time organization for the magnitude and sign. We apply our approach to series of time intervals between consecutive heartbeats. Using the detrended fluctuation analysis method we find that the magnitude series is long-range correlated, while the sign series is anticorrelated and that both magnitude and sign series may have clinical applications. Further, we study the heartbeat mag- nitude and sign series during different sleep stages — light sleep, deep sleep, and REM sleep. For the heartbeat sign time series we find short-range anticorrelations, which are strong during deep sleep, weaker during light sleep and even weaker during REM sleep. In contrast, for the heartbeat magnitude time series we find long-range positive correlations, which are strong during REM sleep and weaker during light sleep. Thus, the sign and the magnitude series provide information which is also useful for distinguishing between different sleep stages. A broad class of physical and biological systems exhibits complex dynamics, asso- ciated with the presence of many components interacting over a wide range of time or space scales. These often-competing interactions may generate an output signal with fluctuations that appear "noisy" and "erratic" but reveal scale-invariant structure. One general approach to study these systems is to analyze the ways that such fluctuations obey scaling laws (1, 2, 3). We consider the time series formed by consecutive cardiac interbeat intervals (Fig. 1a) and focus on the correlations in the time increments between consecutive beats. This time series is of general interest, in part because it is the output of a complex integrated control system, including competing stimuli from the neuroautonomic nervous system (4). These stimuli modulate the rhythmicity of the heart's intrinsic pacemaker, leading to complex fluctuations. Previous reports indicate that these fluctuations exhibit scale- invariant properties (5, 6, 7), and are anticorrelated over a broad range of time scales (i.e., the power spectrum follows a power-law where the amplitudes of the higher frequencies are dominant) (8). By long-range anticorrelations we also mean that the root mean square fluctuations function of the integrated series is proportional to n α where n is the window scale and the scaling exponent α is smaller than 0.5. In contrast, for uncorrelated behavior α 0 5, while for correlated behavior α 0 5. The time series of the fluctuations in heartbeat intervals can be "decomposed" into

An alternative methodology to monitor the heartbeat is presented. The new technique is based on a low power laser beam focused reflected off by a living heart and detected with a linear response photodiode that serves to plot cardiac movements. In order to prove the laser technique an invertebrate, an oyster, was chosen. Three sets of measurements of the heartbeat were done with five oysters changing ambiental salinity, time of data acquisition, and sampling frequency. In all the experiments the temperature was (23±1) °C. Typical “laser-cardiograms” clearly show cardiac signals superposed to the motions of other organs like the gills and the valves. Analysis of the data is done using two methods: Traditional Fast Fourier Transformation to find the principal frequencies of motion and Detrended Fluctuation Analysis to study the nonlinear fluctuation of the interbeat. This algorithm developed handles very well the nonstationarities of biological signals and predicts the level of time scale invariance of the ...

A field experiment of ring‐wandering is executed on a wide playground. Blindfolded and stoppled subjects are observed to do ring‐wandering rather than random‐walking. This experiment simulates the phenomenon of ring‐wandering that climbers encounter in snowy mountains. 15 samples of walking for 13 subjects are reported. Their walking periods are about 40 minutes or 2 hours. The walking data are acquired every second, using a GPS receiver. The discrete velocity v(t) and discrete angular velocity ω(t) of the data are analyzed, using Hurst's R/S analysis and Fourier spectrum analysis. The Hurst exponents of v(t) show long‐range correlations. The Hurst exponents of ω(t) show anti‐correlations in short‐ranges and correlations in long‐ranges. These characteristics of the Hurst exponents in the present data in addition to previous data in this study series describe the ring‐wandering phenomena very well. Significant differences are not seen between 40‐minutes walking and 2‐hours walking.

The connection between soft Boson radiation and fractal energy probability distributions are explored, It is shown for a variety of impedance circuits that “(1/ω)ξ” noise can arise as a result of fractal Hurst exponents. Finally, a connection is made between between fractal energy probabilities and stretched exponentials for the decay of meta-stable quantum states.