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.

New seismic attribute:Fractal scaling exponent based on gray detrended fluctuation analysis

Seismic attributes have been widely used in oil and gas exploration and development. However, owing to the complexity of seismic wave propagation in subsurface media, the limitations of the seismic data acquisition system, and noise interference, seismic attributes for seismic data interpretation have uncertainties. Especially, the antinoise ability of seismic attributes directly affects the reliability of seismic interpretations. Gray system theory is used in time series to minimize data randomness and increase data regularity. Detrended fluctuation analysis （DFA） can effectively reduce extrinsic data tendencies. In this study, by combining gray system theory and DFA, we propose a new method called gray detrended fluctuation analysis （GDFA） for calculating the fractal scaling exponent. We consider nonlinear time series generated by the Weierstrass function and add random noise to actual seismic data. Moreover, we discuss the antinoise ability of the fractal scaling exponent based on GDFA. The results suggest that the fractal scaling exponent calculated using the proposed method has good antinoise ability. We apply the proposed method to 3D poststack migration seismic data from southern China and compare fractal scaling exponents calculated using DFA and GDFA. The results suggest that the use of the GDFA-calculated fractal scaling exponent as a seismic attribute can match the known distribution of sedimentary facies.

Detrended Fluctuation Analysis on Correlations of Complex Networks Under Attack and Repair Strategy

We analyze the correlation properties of the Erd6s-Rdnyi random graph （RG） and the Barabdsi-Albert scale-free network （SF） under the attack and repair strategy with detrended fluctuation analysis （DFA）. The maximum degree kmax, representing the local property of the system, shows similar scaling behaviors for random graphs and scale-free networks. The fluctuations are quite random at short time scales but display strong anticorrelation at longer time scales under the same system size N and different repair probability pre. The average degree 〈k〉, revealing the statistical property of the system, exhibits completely different scaling behaviors for random graphs and scale-free networks. Random graphs display long-range power-law correlations. Scale-free networks are uncorrelated at short time scales; while anticorrelated at longer time scales and the anticorrelation becoming stronger with the increase of pre.

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis （MF-DFA） is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis （DFA）, which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression r（q） = qh（q） - 1 stipulating the relationship between the multifractal exponent T（q） and the generalized Hurst exponent h（q）. As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as .t-（q） = qh（q） - qH - 1, where H is the nonconservation parameter in the universal multifractal formalism. The singular spectra, a and f（a）, are also derived according to this new relationship.

Correlation between detrended fluctuation analysis and the Lempel-Ziv complexity in nonlinear time series analysis

We study the correlation between detrended fluctuation analysis（DFA） and the Lempel-Ziv complexity（LZC） in nonlinear time series analysis in this paper.Typical dynamic systems including a logistic map and a Duffing model are investigated.Moreover,the influence of Gaussian random noise on both the DFA and LZC are analyzed.The results show a high correlation between the DFA and LZC,which can quantify the non-stationarity and the nonlinearity of the time series,respectively.With the enhancement of the random component,the exponent α and the normalized complexity index C show increasing trends.In addition,C is found to be more sensitive to the fluctuation in the nonlinear time series than α.Finally,the correlation between the DFA and LZC is applied to the extraction of vibration signals for a reciprocating compressor gas valve,and an effective fault diagnosis result is obtained.

A Multifractal Detrended Fluctuation Analysis of the Ising Financial Markets Model with Small World Topology

We present a multifractal detrended fluctuation analysis （MFDFA） of the time series of return generated by our recently-proposed Ising financial market model with underlying small world topology. The result of the MFDFA shows that there exists obvious multifractal scaling behavior in produced time series. We compare the MFDFA results for original time series with those for shuffled series, and find that its multifractal nature is due to two factors： broadness of probability density function of the series and different correlations in small- and large-scale fluctuations. This may provide new insight to the problem of the origin of multifractality in financial time series.

Detrended Fluctuation Analysis of the Human EEG during Listening to Emotional Music

A nonlinear method named detrended fluctuation analysis （DFA） was utilized to investigate the scaling behavior of the human electroencephalogram （EEG） in three emotional music conditions （fear, happiness, sadness） and a rest condition （eyes-closed）. The results showed that the EEG exhibited scaling behavior in two regions with two scaling exponents β1 and β2 which represented the complexity of higher and lower frequency activity besides α band respectively. As the emotional intensity decreased the value of β1 increased and the value of β2 decreased. The change of β1 was weakly correlated with the ＇approach-withdrawal＇ model of emotion and both of fear and sad music made certain differences compared with the eyes-closed rest condition. The study shows that music is a powerful elicitor of emotion and that using nonlinear method can potentially contribute to the investigation of emotion.

Crossover Phenomena in Detrended Fluctuation Analysis Used in Financial Markets

A systematic analysis of Shanghai and Japan stock indices for the period of Jan. 1984 to Dec. 2005 is performed. After stationarity is verified by ADF （Augmented Dickey-Fuller） test, the power spectrum of the data exhibits a power law decay as a whole characterized by 1/f^β processes with possible long range correlations. Subsequently, by using the method of detrended fluctuation analysis （DFA） of the general volatility in the stock markets, we find that the long-range correlations are occurred among the return series and the crossover phenomena exhibit in the results obviously.Further, Shanghai stock market shows long-range correlations in short time scale and shows short-range correlations in long time scale. Whereas, for Japan stock market, the data behaves oppositely absolutely. Last, we compare the varying of scale exponent in large volatility between two stock markets. All results obtained may indicate the possibility of characteristic of multifractal scaling behavior of the financial markets.

Detrended Fluctuation Analysis of Heart Rate and SaO_2 in Hypoxia

Detrended fluctuation analysis （DFA） is fit for studies on the long-range exponential correlation of non-stationary time serial. In this paper, in order to find a hy- poxia adaptability evaluation criterion, the heart rate and SaO2 signals are analyzed by this method. The demarcate exponent about fit-good-group and fit-bad-group in hy- poxia and normal air are calculated and compared. The result shows a is different in different situation, the α in hypoxia is much higher than α of breath in normal air. And α of fit-good-group is higher than fit-bad-group. It shows that DFA could be a good criterion to analyze hypoxia adaptability, which is useful in the analysis of hypoxia phys- iology signal.

Extracting a seizure intensity index from one-channel EEG signal using bispectral and detrended fluctuation analysis

Epilepsy is a medical condition that produces seizures affecting a variety of mental and physical functions. Seizures can last from a few seconds to a few minutes. They can have many symptoms, from convulsions and loss of consciousness to blank staring, lip smacking, or jerking movements of arms and legs. If early warning signals of an upcoming seizure (diagnosis of preictal period) are detected, proper treatment can be applied to the patient to help prevent the seizure. In this research, an epileptic disorder has been divided into three subsets: Normal, Preictal (just before the seizure), and Ictal (during seizure). By using Detrended Fluctuation Analysis (DFA), Bispectral Analysis (BIS), and Standard Deviation (SD) three features from single-channel EEG signals have been derived in the foresaid groups. A fuzzy classifier is used to separate the three groups which can successfully separate them with a separation degree of 100% and further a fuzzy inference engine is used to extract a Seizure Intensity Index (SII) from the Electroencephalogram (EEG) signals of the three different states. One can apparently see the distinction of SII amounts between the three states. It is more important when one remembers that these results are just from single-channel EEG signal.

A Sleep Scoring System Using EEG Combined Spectral and Detrended Fluctuation Analysis Features

Most of sleep disorders are diagnosed based on the sleep scoring and assessments. The purpose of this study is to combine detrended fluctuation analysis features and spectral features of single electroencephalograph (EEG) channel for the purpose of building an automated sleep staging system based on the hybrid prediction engine model. The testing results of the model were promising as the classification accuracies were 98.85%, 92.26%, 94.4%, 95.16% and 93.68% for the wake, non-rapid eye movement S1, non-rapid eye movement S2, non-rapid eye movement S3 and rapid eye movement sleep stages, respectively. The overall classification accuracy was 85.18%. We concluded that it might be possible to employ this approach to build an industrial sleep assessment system that reduced the number of channels that affected the sleep quality and the effort excreted by sleep specialists through the process of the sleep scoring.

A new protection scheme for PV-wind based DC-ring microgrid by using modified multifractal detrended fluctuation analysis

This paper presents fault detection,classification,and location for a PV-Wind-based DC ring microgrid in the MATLAB/SIMULINK platform.Initially,DC fault signals are collected from local measurements to examine the outcomes of the proposed system.Accurate detection is carried out for all faults,(i.e.,cable and arc faults)under two cases of fault resistance and distance variation,with the assistance of primary and secondary detection techniques,i.e.second-order differential current derivatived2I3 dt2and sliding mode window-based Pearson’s correlation coefficient.For fault classification a novel approach using modified multifractal detrended fluctuation analysis(M-MFDFA)is presented.The advantage of this method is its ability to estimate the local trends of any order polynomial function with the help of polynomial and trigonometric functions.It also doesn’t require any signal processing algorithm for decomposition resulting and this results in a reduction of computational burden.The detected fault signals are directly passed through the M-MFDFA classifier for fault type classification.To enhance the performance of the proposed classifier,statistical data is obtained from the M-MFDFA feature vectors,and the obtained data is plotted in 2-D and 3-D scatter plots for better visualization.Accurate fault distance estimation is carried out for all types of faults in the DC ring bus microgrid with the assistance of recursive least squares with a forgetting factor(FF-RLS).To verify the performance and superiority of the proposed classifier,it is compared with existing classifiers in terms of features,classification accuracy(CA),and relative computational time(RCT).

Multifractal Detrended Fluctuation Analysis of Interevent Time Series in a Modified OFC Model

We use multifractal detrended fluctuation analysis （MF-DFA） method to investigate the multifractal behavior of the interevent time series in a modified Olami-Feder-Christensen （OFC） earthquake model on assortative scale-free networks. We determine generalized Hurst exponent and singularity spectrum and find that these fluctuations have multifraetal nature. Comparing the MF-DFA results for the original interevent time series with those for shuffled and surrogate series, we conclude that the origin of multifractality is due to both the broadness of probability density function and long-range correlation.

INVESTIGATION OF THE SPATIAL AND TEMPORAL DISTRIBUTION OF EXTREME HIGH TEMPERATURE IN CHINA WITH DETRENDED FLUCTUATION AND PERMUTATION ENTROPY

With temperatures increasing as a result of global warming,extreme high temperatures are becoming more intense and more frequent on larger scale during summer in China.In recent years,a variety of researches have examined the high temperature distribution in China.However,it hardly considers the variation of temperature data and systems when defining the threshold of extreme high temperature.In order to discern the spatio-temporal distribution of extreme heat in China,we examined the daily maximum temperature data of 83 observation stations in China from 1950 to 2008.The objective of this study was to understand the distribution characteristics of extreme high temperature events defined by Detrended Fluctuation Analysis(DFA).The statistical methods of Permutation Entropy(PE)were also used in this study to analyze the temporal distribution.The results showed that the frequency of extreme high temperature events in China presented 3 periods of 7,10—13 and 16—20 years,respectively.The abrupt changes generally happened in the 1960s,the end of 1970s and early 1980s.It was also found that the maximum frequency occurred in the early 1950s,and the frequency decreased sharply until the late 1980s when an evidently increasing trend emerged.Furthermore,the annual averaged frequency of extreme high temperature events reveals a decreasing-increasing-decreasing trend from southwest to northeast China,but an increasing-decreasing trend from southeast to northwest China.And the frequency was higher in southern region than that in northern region.Besides,the maximum and minimum of frequencies were relatively concentrated spatially.Our results also shed light on the reasons for the periods and abrupt changes of the frequency of extreme high temperature events in China.

Multifractal Analysis of Human Heartbeat in Sleep

We study the dynamical properties of heart rate variability （HRV） in sleep by analysing the scaling behaviour with the multifractal detrended fluctuation analysis method. It is well known that heart rate is regulated by the interaction of two branches of the autonomic nervous system： the parasympathetic and sympathetic nervous systems. By investigating the multifractal properties of light, deep, rapid-eye-movement （REM） sleep and wake stages, we firstly find an increasing multifractal behaviour during REM sleep which may be caused by augmented sympathetic activities relative to non-REM sleep. In addition, the investigation of long-range correlations of HRV in sleep with second order detrended fluctuation analysis presents irregular phenomena. These findings may be helpful to understand the underlying regulating mechanism of heart rate by autonomic nervous system during wake-sleep transitions.

Long-range correlation analysis of urban traffic data

This paper investigates urban traffic data by analysing the long-range correlation with detrended fluctuation analysis. Through a large number of real data collected by the travel time detection system in Beijing, the variation of flow in different time periods and intersections is studied. According to the long-range correlation in different time scales, it mainly discuss the effect of intersection location in road net, people activity customs and special traffic controls on urban traffic flow. As demonstrated by obtained results, the urban traffic flow represents three-phase characters similar to highway traffic. Moreover, compared by the two groups of data obtained before and after the special traffic restrictions （vehicles with special numbered plates only run in a special workday） enforcement, it indicates that the rules not only reduce the flow but also avoid irregular fluctuation.

Application of long-range correlation and multi-fractal analysis for the depiction of drought risk

By using the multi-fractal detrended fluctuation analysis method, we analyze the nonlinear property of drought in southwestern China. The results indicate that the occurrence of drought in southwestern China is multi-fractal and long- range correlated, and these properties are indifferent to timescales. A power-law decay distribution well describes the return interval of drought events and the auto-correlation. Furthermore, a drought risk exponent based on the multi-fractal property and the long-range correlation is presented. This risk exponent can give useful information about whether the drought may or may not occur in future, and provide a guidance function for preventing disasters and reducing damage.

基于去趋势波动分析方法的直拉单晶硅拉速曲线研究

直拉单晶硅的生长是半导体工业中的关键步骤,其过程具有动态连续性。提拉速度的波动直接影响最终产品的成品率和质量。在直拉单晶硅的等径生长过程中,稳定的提拉速度有助于提高晶体的质量;而提拉速度波动过大会导致晶体中出现原生缺陷,甚至可能引发晶体直径控制的失效。采用去趋势波动分析法(DFA)对直拉单晶硅的拉速曲线进行处理,结合具体的单晶硅生长过程,分析了拉速出现较大波动的原因。结果表明,DFA对研究直拉单晶硅拉速曲线具有较好的适用性,并且结合工艺过程开展的研究对优化工艺和提高生产稳定性具有重要的指导意义。

Wave height statistical characteristic analysis

When exploring the temporal and spatial change law of ocean environment, the most common method used is using smaller-scale observed data to derive the change law for a larger-scale system. For instance, using 30-year observation data to derive 100-year return period design wave height. Therefore, the study of inherent self-similarity in ocean hydrological elements becomes increasingly important to the study of multi-year return period design wave height derivation. In this paper, we introduced multifractal to analyze the statistical characteristics of wave height series data observed from oceanic hydrological station. An improvement is made to address the existing problems of the multifractal detrended fluctuation analysis (MF-DFA) method, where trend function showed a discontinuity between intervals. The improved MFDFA method is based on signal mode decomposition, replacing piecewise polynomial fitting used in the original method. We applied the proposed method to the wave height data collected at Chaolian Island, Shandong, China, from 1963 to 1989 and was able to conclude the wave height sequence presented weak multi-fractality. This result provided strong support to the past research on the derivation of multi-year return period design wave height with observed data. Moreover, the new method proposed in this paper also provides a new perspective to explore the intrinsic characteristic of data.