Effects of quantization on detrended fluctuation analysis

Detrended fluctuation analysis(DFA) is a method foro estimating the long-range power-law correlation exponent in noisy signals.It has been used successfully in many different fields,especially in the research of physiological signals.As an inherent part of these studies,quantization of continuous signals is inevitable.In addition,coarse-graining,to transfer original signals into symbol series in symbolic dynamic analysis,can also be considered as a quantization-like operation.Therefore,it is worth considering whether the quantization of signal has any effect on the result of DFA and if so,how large the effect will be.In this paper we study how the quantized degrees for three types of noise series(anti-correlated,uncorrelated and long-range power-law correlated signals) affect the results of DFA and find that their effects are completely different.The conclusion has an essential value in choosing the resolution of data acquisition instrument and in the processing of coarse-graining of signals.     

Scaling characteristics of ocean wave height time series

Fluctuations in the significant wave height can be quantified by using scaling statistics. In this paper, the scaling properties of the significant wave height were explored by using a large data set of hourly series from 25 monitoring stations located off the west coast of the US. Detrended fluctuation analysis (DFA) was used to investigate the scaling properties of the series. DFA is a robust technique that can be used to detect long-range correlations in nonstationary time series. The significant wave height data was analyzed by using scales from hourly to monthly. It was found that a common scaling behavior can be observed for all stations. A breakpoint in the scaling region around 4-5 days was apparent. Spectral analysis confirms this result. This breakpoint divided the scaling region into two distinct parts. The first part was for finer scales (up to 4 days) which exhibited Brown noise characteristics, while the second one showed 1/f noise behavior at coarser scales (5 days to 1 month). The first order and the second order DFA (DFA1 and DFA2) were used to check the effect of seasonality. It was found that there were no differences between DFA1 and DFA2 results, indicating that there is no effect of trends in the wave height time series. The resulting scaling coefficients range from 0.696 to 0.890 indicating that the wave height exhibits long-term persistence. There were no coherent spatial variations in the scaling coefficients.     

Detrended fluctuation analysis of heart intrabeat dynamics

We investigate scaling properties of electrocardiogram (ECG) recordings of healthy subjects and heart failure patients based on detrended fluctuation analysis (DFA). While the vast majority of scaling analysis has focused on the characterization of the long-range correlations of interbeat (i.e., beat-to-beat) dynamics, in this work we consider instead the characterization of intrabeat dynamics. That is, here we use DFA to study correlations for time scales smaller than one heart beat period (about 0.75 s). Our results show that intrabeat dynamics of healthy subject are less correlated than for heart failure dynamics. As in the case of interbeat dynamics, the DFA scaling exponents can be used to discriminate healthy and pathological data. It is shown that 0.5 h recordings suffices to characterize the ECG correlation properties.     

Analysis of temporal fluctuations in Bach’s sinfonias

The correlation structures in 15 Bach’s sinfonias were analyzed. Each sinfonia is characterized by the superposition of three voices. Each voice is a sequence of pitches. Each voice was transformed in a time series, in which the sampling time was given by the smallest pitch duration in that voice. The scaling properties of the three voices of each sinfonia was quantified by means of the estimate of the scaling exponent, performed using the power spectral density (PSD) and the detrended fluctuation analysis (DFA). The results show that the voice time series are persistent. The DFA was applied not only to any single voice time series, but also to couples (2-DFA) of voices and to the triple (3-DFA) of voices. It was found that the first voice of each sinfonia modulates the scaling behavior of the whole sinfonia.     

Scaling analysis of paces of fetal breathing, gross-body and extremity movements

Using detrended fluctuation analysis (DFA), we studied the scaling properties of the time instances (occurrence) of the fetal breathing, gross-body, and extremity movements scored on a second by second basis from the recorded ultrasound measurements of 49 fetuses. The DFA exponent α of all the three movements of the fetuses varied between 0.63 and 1.1. We found an increase in α obtained for the movement due to breathing as a function of the gestational age while this trend was not observed for gross-body and extremity movements. This trend was argued as the indication of the maturation of lung and functional development of respiratory aspect of the fetal central nervous system. This result may be useful in discriminating normal fetuses from high-risk fetuses.     

Advection of passive magnetic field by the Gaussian velocity field with finite correlations in time and spatial parity violation

Using the field theoretic renormalization group technique the model of passively advected weak magnetic field by an incompressible isotropic helical turbulent flow is investigated up to the second order of the perturbation theory (two-loop approximation) in the framework of an extended Kazantsev-Kraichnan model of kinematic magnetohydrodynamics. Statistical fluctuations of the velocity field are taken in the form of a Gaussian distribution with zero mean and defined noise with finite correlations in time. The two-loop analysis of all possible scaling regimes is done and the influence of helicity on the stability of scaling regimes is discussed and shown in the plane of exponents ? ? η, where ? characterizes the energy spectrum of the velocity field in the inertial range E ∞ k ^{1 ? 2ε}, and η is related to the correlation time at the wave number k which is scaled as k ^{?2 + η}. It is shown that in non-helical case the scaling regimes of the present vector model are completely identical and have also the same properties as those obtained in the corresponding model of passively advected scalar field. Besides, it is also shown that when the turbulent environment under consideration is helical then the properties of the scaling regimes in models of passively advected scalar and vector (magnetic) fields are essentially different. The results demonstrate the importance of the presence of a symmetry breaking in a given turbulent environment for investigation of the influence of an internal tensor structure of the advected field on the inertial range scaling properties of the model under consideration and will be used in the analysis of the influence of helicity on the anomalous scaling of correlation functions of passively advected magnetic field.     

Multifractal Detrended Cross-Correlation Analysis of sunspot numbers and river flow fluctuations

We use the Detrended Cross-Correlation Analysis (DCCA) to investigate the influence of sun activity represented by sunspot numbers on one of the climate indicators, specifically rivers, represented by river flow fluctuation for Daugava, Holston, Nolichucky and French Broad rivers. The Multifractal Detrended Cross-Correlation Analysis (MF-DXA) shows that there exist some crossovers in the cross-correlation fluctuation function versus time scale of the river flow and sunspot series. One of these crossovers corresponds to the well-known cycle of solar activity demonstrating a universal property of the mentioned rivers. The scaling exponent given by DCCA for original series at intermediate time scale, , is λ=1.17±0.04 which is almost similar for all underlying rivers at 1σ confidence interval showing the second universal behavior of river runoffs. To remove the sinusoidal trends embedded in data sets, we apply the Singular Value Decomposition (SVD) method. Our results show that there exists a long-range cross-correlation between the sunspot numbers and the underlying streamflow records. The magnitude of the scaling exponent and the corresponding cross-correlation exponent are λ∈(0.76,0.85) and γ_×∈(0.30,0.48), respectively. Different values for scaling and cross-correlation exponents may be related to local and external factors such as topography, drainage network morphology, human activity and so on. Multifractal cross-correlation analysis demonstrates that all underlying fluctuations have almost weak multifractal nature which is also a universal property for data series. In addition the empirical relation between scaling exponent derived by DCCA and Detrended Fluctuation Analysis (DFA), is confirmed.     

Long range correlation and possible electron conduction through DNA sequences

Long range correlation analysis and charge conductivity investigation are applied to sequences in 16 chromosomes in the Saccharomyces cerevisiae genome. DNA sequence data are analyzed via Hurst’s analysis and Detrended Fluctuation Analysis (DFA) analysis. Super diffusive nature of mapping sequences are evident with the measured Hurst exponent H to be around the value of 0.60 for all sequences in the 16 chromosomes. The DFA result is consistent with the result from the Hurst analysis. Tight binding models are applied for the investigation of charge conduction through DNA sequences. The overall averaged transmission coefficients, 〈T_N〉_av, calculated from sixteen chromosomes are shown to be significantly different from values calculated from random as well as periodic sequences. Sequences from the S. cerevisiae genome promise better charge conduction ability than random sequences. Finally, delocalized electronic wave function patterns are also shown through calculations using the tight binging model. Slightly delocalized electronic wavefunctions are seen on sequences in sixteen chromosomes, as compared with those obtained from random sequences on the same eigenenergies.     

Levels of complexity in turbulent time series for weakly and high Reynolds number

We use the detrended fluctuation analysis (DFA), the detrended cross correlation analysis (DCCA) and the magnitude and sign decomposition analysis to study the fluctuations in the turbulent time series and to probe long-term nonlinear levels of complexity in weakly and high turbulent flow. The DFA analysis indicate that there is a time scaling region in the fluctuation function, segregating regimes with different scaling exponents. We discuss that this time scaling region is related to inertial range in turbulent flows. The DCCA exponent implies the presence of power-law cross correlations. In addition, we conclude its multifractality for high Reynold’s number in inertial range. Further, we find that turbulent time series exhibit complex features by magnitude and sign scaling exponents.     

Atomic force microscopy study of growth kinetics: Scaling in TiN-TiB2 nanocomposite films on Si(1 0 0)

We used the reactive unbalanced close-field dc-magnetron sputtering growth of TiN-TiB₂ on Si(1 0 0) at room temperature to determine if scaling theory provides insight into the kinetic mechanisms of two-phase nanocomposite thin films. Scaling analyses along with height-difference correlation functions of measured atomic force microscopy (AFM) images have shown that the TiN-TiB₂ nanocomposite films with thickness ranging from 70 to 950 nm exhibit a kinetic surface roughening with the roughness increasing with thickness exponentially. The roughness exponent α and growth exponent β are determined to be ∼0.93 and ∼0.25, respectively. The value of dynamic exponent z, calculated by measurement of the lateral correlation length ξ, is ∼3.70, agreeing well with the ratio of α to β. These results indicate that the surface growth behavior of sputter-deposited TiN-TiB₂ thin films follows the classical Family-Vicseck scaling and can be reasonably described by the noisy Mullins diffusion model, at which surface diffusion serves as the smoothing effect and shot noise as the roughening mechanism.     

含关联噪声的空间分数阶随机生长方程的动力学标度行为研究

为探讨含关联噪声的空间分数阶随机生长方程的动力学标度行为,本文利用Riesz分数阶导数和Grümwald-Letnikov分数阶导数定义方法研究了关联噪声驱动下的空间分数阶Edwards-Wilkinson (SFEW)方程在1+1维情况下的数值解,得到了不同噪声关联因子和分数阶数时的生长指数、粗糙度指数、动力学指数等,所求出的临界指数均与标度分析方法的结果相符合.研究表明噪声关联因子和分数阶数均影响到SFEW方程的动力学标度行为,且表现为连续变化的普适类.     

Chaotic and fractal properties of laboratory-generated surface water waves

Summary We report the results of four laboratory experiments on surface water waves generated with the Pierson-Moskowitz power spectrum, and characterized by different values of the ratiof _p/f _N and of the water depthh. The scope of the experiments was to test the dependence of the chaotic and fractal properties of the data on the parameterf _p/f _N, which has been indicated as determinant by previous numerical studies; the different water depths are used to induce different levels of non-linearity in the records. The analysis indicates that the Grassberger and Procaccia correlation integrals, the largest Lyapunov exponent and the scaling exponent of the data sets considered herein are completely assimilable to those of numerically generated linear time series; the algorithms used are insensitive to the presence of non-linearities because they sample essentially the high-frequency components.     

Concurrent sympathetic activation and vagal withdrawal in hyperthyroidism: Evidence from detrended fluctuation analysis of heart rate variability

Despite many previous studies on the association between hyperthyroidism and the hyperadrenergic state, controversies still exist. Detrended fluctuation analysis (DFA) is a well recognized method in the nonlinear analysis of heart rate variability (HRV), and it has physiological significance related to the autonomic nervous system. In particular, an increased short-term scaling exponent α1 calculated from DFA is associated with both increased sympathetic activity and decreased vagal activity. No study has investigated the DFA of HRV in hyperthyroidism. This study was designed to assess the sympathovagal balance in hyperthyroidism. We performed the DFA along with the linear analysis of HRV in 36 hyperthyroid Graves’ disease patients (32 females and 4 males; age 30 ± 1 years, means ± SE) and 36 normal controls matched by sex, age and body mass index. Compared with the normal controls, the hyperthyroid patients revealed a significant increase (P<0.001) in α1 (hyperthyroid 1.28±0.04 versus control 0.91±0.02), long-term scaling exponent , overall scaling exponent , low frequency power in normalized units (LF%) and the ratio of low frequency power to high frequency power (LF/HF); and a significant decrease (P<0.001) in the standard deviation of the R-R intervals (SDNN) and high frequency power (HF). In conclusion, hyperthyroidism is characterized by concurrent sympathetic activation and vagal withdrawal. This sympathovagal imbalance state in hyperthyroidism helps to explain the higher prevalence of atrial fibrillation and exercise intolerance among hyperthyroid patients.     

Scaling properties of image textures: A detrending fluctuation analysis approach

The aim of this paper is to explore the application of detrended fluctuation analysis (DFA) to study roughness features of images. Unidimensional sequences at different image orientations are extracted and their average scaling exponent is estimated. In this form, the existence of anisotropies can be detected when considerable variations in the scaling exponent at different image orientation are observed. Different images from grass to solar granulation are analyzed and the underlying physics of such results is briefly commented.     

Temporal variations of long-term correlations in seismic activity fluctuations

The scaling behavior of the 1998-2009 seismicity in Guerrero, southern Mexico, was studied by means of the detrended fluctuation analysis (DFA). We found that inter-seismic periods are correlated with a transition in the scaling behavior at about 200 seismic events. Correlations are relatively weak for small time scales. However, for large time scales, correlations are associated with a 1/f fractional process, indicating that the seismicity pattern emerges from a self-organized critical state. Temporal variations of the scaling exponent along years computed from the DFA indicate the presence of a quasi-biennial cycle in the seismicity correlations. This cyclic behavior was apparently triggered by the large 2001-2002 slow slip event in the Guerrero seismic gap. Besides, the significant seismic events (M_w>5) originate, on the average, at deeper regions in each cycle.     

Long-range correlations in two-dimensional spatio-temporal seismic fluctuations

We analysed the scaling behaviour of the two-dimensional (2-D) sequence (Δs, Δt) of the 1981–1998 southern California seismicity, where Δs is the distance between two consecutive earthquakes (jump) and Δt is their interevent interval. The 2-D seismic spatio-temporal fluctuations were investigated by means of the detrended fluctuation analysis (DFA), well-known methodology used to detect scaling behaviour in observational time series possibly affected by nonstationarities. The estimated scaling exponents α_DFA, larger than 0.5, indicate the presence of persistent long-range correlations in the 2-D sequence analysed. The variation of the scaling exponent with the increase of threshold magnitude shows a two-fold behaviour: in the range between 1.5 (the completeness magnitude of the catalog) and 3.0, the scaling exponent is quite constant and denoting a flicker-noise dynamics; while for magnitudes larger than 3.0 it decreases with the increase of magnitude, indicating a tendency toward a 2-D space–time Poissonian process for large events.     

Lyapunov exponent of the random Schrödinger operator with short-range correlated noise potential

We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent γ which we determine explicitly as a function of the noise intensity σ and the frequency ω. We find uniform two-parameter asymptotic expressions for γ which allow us to evaluate γ for different relations between σ and ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart.     

Power law and multiscaling properties of the Chinese stock market

We investigate the cumulative probability density function (PDF) and the multiscaling properties of the returns in the Chinese stock market. By using returns data adjusted for thin trading, we find that the distribution has power-law tails at shorter microscopic timescales or lags. However, the distribution follows an exponential law for longer timescales. Furthermore, we investigate the long-range correlation and multifractality of the returns in the Chinese stock market by the DFA and MFDFA methods. We find that all the scaling exponents are between 0.5 and 1 by DFA method, which exhibits the long-range power-law correlations in the Chinese stock market. Moreover, we find, by MFDFA method, that the generalized Hurst exponents h(q) are not constants, which shows the multifractality in the Chinese stock market. We also find that the correlation of Shenzhen stock market is stronger than that of Shanghai stock market.     

Correlations in a Mozart's music score (K-73x) with palindromic and upside-down structure

In this work, we study long-range correlations in a “Scherzo-Duetto di Mozart” score (K-73x) for two violins. This is a fascinating piece, as the second violin part is upside down on the same sheet below the first violin, and some parts are like a palindrome. Given such ingenious structure, it is expected the existence of long-range correlations in the score structure. In order to quantify long-range correlations, we considered the music score as a sequence of integer numbers, each of them corresponding to last common denominator units of note. By using detrended fluctuation analysis (DFA), correlations are quantified by means of the scaling exponent that reflects the type of correlations for a given distance between neighbors note. The following conclusions can be drawn from the analysis: (a) For about 10-25 neighbor note distances, correlations are similar to 1/f-noise. This is an interesting finding since it has been shown that pleasant sounds for humans display a behavior similar to 1/f noise. (b) As the neighbor note distance increases, the long-range correlations decays continuously. For some score sections, the music score behaves like non-correlated (i.e., purely random) noise. Summing up, the results show that the studied Mozart's score contains a certain degree of correlation for relatively small note distances, and becomes close to non-correlated behavior for long note distances. We considered also the sequence constructed by considering the distance between the simultaneously played notes of the two violins. Interestingly, for relatively small neighbor note distances, a scaling behavior similar to that found for individual violins is also displayed. In some sense, this is an expression of the specific structure (palindromes plus upside down construction) used by Mozart in the composition of this music score. Although we focused on a particular high-art music score, our results suggest that modern methods borrowed from statistical physics can be useful for the systematic study of music composition techniques.