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.

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

<正> We analyze the correlation properties of the Erd(o|¨)s-Rényi random graph(RG)and the Barabási-Albertscale-free network(SF)under the attack and repair strategy with detrended fluctuation analysis(DFA).The maximumdegree k_(max),representing the local property of the system,shows similar scaling behaviors for random graphs andscale-free networks.The fluctuations are quite random at short time scales but display strong anticorrelation at longertime scales under the same system size N and different repair probability p_(re).The average degree〈k〉,revealing thestatistical property of the system,exhibits completely different scaling behaviors for random graphs and scale-freenetworks.Random graphs display long-range power-law correlations.Scale-free networks are uncorrelated at short timescales;while anticorrelated at longer time scales and the anticorrelation becoming stronger with the increase of p_(re·)

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

为到 2005 年 12 月的 1984 年 1 月的时期的上海和日本股票索引的系统的分析被执行。在 stationarity 被 ADF (扩充女性衬衫的胸襟福勒) 验证以后，测试，整个的幂定律腐烂由 1/f β 描绘了的数据展览的力量光谱与可能的长期的关联处理。由在证券市场使用一般轻快的 detrended 变化分析(DFA ) 的方法，随后，我们发现远程的关联显然在结果在回来系列和转线路现象展览之中被发生。进一步，上海证券市场证明在短时间的远程的关联可伸缩并且出现短期在长时间的关联可伸缩。而为日本证券市场，数据绝对相对地表现。最后，我们比较在在二个证券市场之间的大轻快变化规模代表。获得的所有结果可以显示金融市场的多分数维的可伸缩的行为的特征的可能性。

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 hypoxia 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 hypoxia and normal air are calculated and compared. The result shows αis different in different situation, the αin hypoxia is much higher than αof breath in normal air. And αof 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 physiology signal.

基于改进DFA和LDA的永磁同步电机机械故障检测

为提高故障检测的精度,研究了变转速工况下永磁同步电机的机械故障检测方法.首先,分析了电机轴承、转子偏心及其复合故障的振动特性;其次,采用Vold-Kalman算法对故障特征分量进行跟踪提取,并通过信号重构消除转速变化对故障特征分量的影响;提出一种基于改进去趋势波动分析和线性判别式分析的机械故障检测方法,实现对重构信号的故障特征提取和故障检测;最后,对所提出故障检测方法的有效性进行实验验证.实验结果表明文中所提出方法的故障检测精度为88%.

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MFDFA的PEMFC水淹和膜干故障诊断

为延长质子交换膜燃料电池(PEMFC)的使用寿命并提升其可靠性,采用基于信号处理的多重分形去趋势波动分析(MFDFA)方法对PEMFC的水管理故障进行诊断。基于MFDFA方法分别对正常、水淹和膜干3种状态下的电压信号进行分析。结果表明:故障时Gr(t)波动函数对数图相对分散、标度指数α在0~2之间、分形维数约为1.6与1.9,正常状态下Gr(t)波动函数对数图收敛、标度指数α在20~40之间、分形维数约为2.15。该方法通过以上3种指标结合,实现了PEMFC水淹和膜干故障诊断。

基于EEMD-MFDFA的镇江港PM_(2.5)演化特征分析

为定量分析镇江港PM_(2.5)浓度时间序列的内在演化特征,以镇江港2019—2020年的205667组PM_(2.5)时均浓度数据为基础,采用基于集合经验模态分解(EEMD)的多重分形去趋势波动分析(MFDFA)算法进行研究.结果表明:各监测点PM_(2.5)日均浓度演化均具有显著的长期持续性特征,现阶段PM_(2.5)浓度升高可能会导致未来一段时间内PM_(2.5)浓度的持续升高,不同监测点PM_(2.5)浓度的多重分形谱参数存在差异性,多重分形特征强的区域应侧重即时性防治,反之则应注重长期性监管;运用自组织临界性理论,发现15个监测点PM_(2.5)浓度波动呈现幂律分布特征,符合自组织临界性特征,港口系统存在发生大气高污染的风险.

基于替代数据法和MF-DFA的合意工业产能利用率区间估计方法及应用

为实现工业产能过剩的精准判别与调控,提出了一种基于替代数据法与多重分形去趋势波动分析的合意工业产能利用率区间估计方法。首先判断原始时序的长程相关性和多重分形特征;然后采用替代数据法与多重分形去趋势波动分析计算出所有重排序列的Hurst指数,据此确定指数序列收敛情况和合意产能利用率区间;最后以煤炭行业为例验证了模型的有效性。结果表明:该方法能够从数据自身演化规律中自适应确定阈值,有效克服了传统统计与经验方法的主观性和缺乏理论依据的局限性;我国煤炭行业的合意产能利用率区间为73.73%~86.23%。该研究为工业产能过剩风险监测与判别提供了量化分析工具,为深化煤炭产能过剩治理提供了决策依据。

基于ICEEMDAN-MFDFA的广东日降水序列的多重分形特征

【目的】降水时间序列是典型的非平稳、非线性时间序列,在全球变暖的背景下呈现出明显的复杂波动特征。通过全面考察广东日降水序列的分形特征,为有效应对区域水旱灾害风险提供参考。【方法】引入改进的自适应噪声完全集合经验模态分解(ICEEMDAN)方法改善传统多重分形去趋势波动分析(MFDFA)方法在去趋势过程中的不足,利用ICEEMDAN-MFDFA方法得到1973—2021年广东日降水序列的多重分形结果。【结果】结果显示:广东日降水序列具有复杂的多重分形特征,多重分形谱表现为右偏、左勾状,Hurst指数小于0.5;各项多重分形谱参数呈现阶段性上下波动特征,谱宽、不对称指数、谱端高差和Holder指数的极值分别达到1.00、-0.72、1.01和0.12;降水序列的多重分形谱宽与谱端高差、不对称指数与Holder指数两两之间有着密切的联系。【结论】结果表明:降水序列内部演变的波动行为具有长程相关性,表现为反持续性;日降水对小幅度的局部气候波动较敏感,且有增大的趋势;当降水复杂性越强时对应着日降水增大的态势越明显,当多重分形谱的右偏程度越弱时其精细结构越丰富;与传统方法相比,ICEEMDAN-MFDFA方法可以有效提高降水序列多重分形结果的可靠性和稳定性。

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。