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.

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.

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.

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.

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.

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.

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.

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.

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.

基于DFA法的江苏省极端降水时空分布特征研究

为进一步掌握江苏省极端降水的时空分布特征,基于该省1961-2010年均一性较好的逐日降水数据,利用去趋势波动分析法确定了全省13个站点的极端降水阈值,并通过Morlet小波及Mann-Kendall法分析了江苏省极端降水频数的振荡周期及其突变。结果表明,江苏省极端降水年频数和夏季极端降水均呈现8～10 a的变化周期,且1998年和2006年分别为其突变增加年,而秋季极端降水主要呈2～3 a与5～7 a的变化周期;极端降水与降水总量的空间分布具有较好的一致性,均呈南部大、北部小的特征。

基于去趋势波动分析(DFA)的脑卒中后抑郁症静息脑电特征提取与识别

为考察脑卒中后抑郁症(PSD)患者静息态脑电(EEG)特异性,通过去趋势波动分析(DFA),提取16导联EEG信号波动函数F(s)和区间长度s函数关系在双对数坐标中线性拟合斜率之标度指数α值,作为EEG信号的长程幂函数相关性特征参数。将3类人群(健康正常组10人、脑卒中后无抑郁症组4人、脑卒中后抑郁症组7人)的16导联EEG信号α值作为样本进行独立样本t检验,结果显示,健康人与脑卒中患者在顶叶、颞叶以及枕叶处α值存在显著性差异(P<0.05);利用支持向量机(SVM)分类器,将EEG信号DFA的α值放入16维特征空间在脑卒中患者组(含抑郁症与无抑郁症)内进行模式识别,可获得最高90.9%的分类正确率,有望为临床PSD提供客观有效的辅助诊断新手段。

MF-DFA在大坝安全监测序列分析和整体性态识别中的应用

为更好地分析监测序列的波动特征进而识别大坝整体性态演变规律,将多重分形消除趋势波动分析(MF-DFA)方法应用到大坝监测序列分析,并以一混凝土坝垂线监测序列为例,通过单个测点、环境量的长程相关性和多重分形特征分析及3个不同坝段测点比较,结果表明:坝体变形序列具有较强的多重分形特征,大波动受温度和库水位的周期性变化影响,小波动则受水位的频繁涨落控制;同时,溢流坝段和挡水坝段又因其环境荷载条件和工作机理不同波动特征略有差异;总体看,变形序列受环境量控制,大坝整体性态正常。实例分析结果表明MF-DFA可以从局部和整体两个层面评价大坝的工作性态及其演变规律。

驾驶员疲劳状态生理信号的DFA

通过去趋势波动分析(DFA)研究驾驶员疲劳状态的心率变异性信号和脑电信号特征。22名被试在驾驶模拟器上进行模拟驾驶作业,采集驾驶过程中的心率变异性和脑电信号并进行离线分析。Wilcoxon符号秩检验用于分析清醒和疲劳两种状态间DFA标度指数的差异,受试者工作曲线(ROC)分析用于确定DFA区分疲劳驾驶和清醒驾驶的能力。结果表明:疲劳状态时,心率变异性和脑电信号的标度指数显著增加(Wilcoxon符号秩检验,p<0.01);心率变异性和脑电信号标度指数的ROC下面积最大分别为0.75和0.78。DFA的标度指数具有应用于驾驶疲劳监测系统的前景。

成都市一次重雾霾期间PM_(10)自组织演化的分形特征及DFA分析

运用计盒维数法和消除趋势波动分析方法,研究了成都市2013年1月8—20日一次重度雾霾期间,草堂寺、金牛坝、三道堰和十里店4个监测站点连续PM10小时平均质量浓度序列的时间演化规律。雾霾期间4个监测站点的PM10质量浓度序列在某一临界质量浓度以下均表现出标度不变分形结构和长期持续性非线性特征,且这两种物理特征具有很好的动力学同步性;但一旦超过临界质量浓度,PM10演化的长期持续性特征就会消失,PM10演化的标度不变分形结构也随之消失。进一步根据自组织临界(SOC)理论,探讨了导致此次成都重度雾霾期间PM10污染演化呈现分形结构和长期持续性特征的内在动力学机制,并将雾霾天气大气污染系统的动态特征与自组织临界性的判断标准进行了对比分析。结果表明,大气PM10污染演化具有自组织临界性复杂系统的基本特征,此次成都重度雾霾期间PM10在临界质量浓度以下的波动演化主要受系统自组织内禀动力机制主导控制。在该机制的影响下,即使在雾霾期间,污染源排放相对平稳,城市大气系统也相对稳定,但PM10污染演化仍会出现非线性波动,从而导致极高PM10质量浓度的涌现。这一特征可能是由SOC特有的内禀敏感性和不可预测性决定的,也正是由于SOC系统的稳定性使得此次成都重度雾霾污染持续时间很长。

基于改进MF-DFA和SSM-FCM的液压泵退化状态识别方法

针对液压泵振动信号通常具有非线性强和信噪比低的特点,提出了一种基于改进多重分形去趋势波动分析(MF-DFA)和半监督马氏距离模糊C均值(SSM-FCM)的液压泵退化状态识别方法。该方法首先引入滑动窗口技术改进传统MF-DFA方法在时间序列数据分割过程中存在的不足,提高MF-DFA方法的计算精度;然后利用改进MF-DFA方法计算液压泵多重分形谱参数,并分析了不同分形谱参数对液压泵退化状态的反映能力,选取奇异指数α_0和广义Hurst指数波动均值Δh(q)作为退化特征量;最后利用半监督马氏距离模糊C均值方法实现了液压泵退化状态识别,并以液压泵实测数据为例验证本文所提方法的有效性。

元谋干热河谷近50a降水量时间序列的DFA分析

干热河谷是一类特殊的退化生态系统区域，降水对河谷景观格局的变化起着十分重要的作用。基于消除趋势波动分析法（DFA）分析了元谋地区近50a降雨量变化趋势，同时也对1978～2006年进行了滑移分析。结果表明，元谋地区年均降水量序列至少在50a的时间尺度上具有微弱的Hurst效应，即具有较弱的长期持续性及较强的随机性。区域植被覆盖率的变化能对该区域降雨量的动力学变化趋势做出有效的解释。