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.

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.

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.

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).

Local multifractal detrended fluctuation analysis for tea breeds identification

In recent years, the popular multifractal detrended fluctuation analysis （MF-DFA） is extended to two-dimensional （2D） version, which has been applied in some field of image processing. In this paper, based on the 2D MF-DFA, a novel multifractal estimation method for images, which we called the local multifractal detrended fluctuation analysis （LMF-DFA）, is proposed to recognize and distinguish 20 types of tea breeds. A set of new multifractal descriptors, namely the local multifractal fluctuation exponents is defined to portray the local scaling properties of a surface. After collecting 10 tea leaves for each breed and photographing them to standard images, the LMF-DFA method is used to extract characteristic parameters for the images. Our analysis finds that there are significant differences among the different tea breeds＇ characteristic parameters by analysis of variance. Both the proposed LMF-DFA exponents and another classic parameter, namely the exponent based on capacity measure method have been used as features to distinguish the 20 tea breeds. The comparison results illustrate that the LMF-DFA estimation can differentiate the tea breeds more effectively and provide more satisfactory accuracy.

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.

Morphology and multifractal features of a guyot in specific topographic vicinity in the Caroline Ridge,West Pacific

Massive seamounts have been surveyed and documented in the last decades.However,the morphologies of seamounts are usually described in qualitative manners,yet few quantitative detections have been carried out.Here,based on the high-re solution multi-beam bathymetric data,we report a recentlysurveyed guy ot on the Caroline Ridge in the West Pacific,and the large-scale volcanic structures and smallscale erosive-depositional landforms in the guyot area have been identified.The multifractal features of the guyot are characterized for the first time by applying multifractal detrended fluctuation analysis on the surveyed bathymetric data.The results indicate that the multifractal spectrum parameters of the seafloor have strong spatial dependency on the fluctuations of local landforms.Both small-and large-scale components contribute to the degree of asymmetry of the multifractal spectrum(B),while the fluctuations of B are mostly attributed to the changes in small-scale roughness.The maximum singularity strength(α0)correlates well with the roughness of large-scale landforms and likely reflects the large-scale topographic irregularity.Comparing to traditional roughness parameters or monofractal exponents,multifractal spectra are able to depict not only the multiscale characteristics of submarine landforms,but also the spatial variations of scaling behaviors.Although more comparative works are required for various seamounts,we hope this study,as a case of quantifying geomorphological characters and multiscale behaviors of seamounts,can encourage further studies on seamounts concerning geomorphological processes,ocean bottom circulations,and seamount ecosystems.

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.

Multi-fractal Behaviors of Relative Humidity over China

The multi-fractal ity over China are studied behaviors of relative humid using the multi-fractal de trended fluctuation analysis （DFA） method. Three multi fractal parameters （the spectrum width Aa, the asymmetry Aaas, and the long-range correlation exponent a0） of the singularity spectrum are introduced to quantify the multi-fractal behaviors. The results show that multi-frac tality exists in daily humidity records over most stations in China and is mainly due to the broad distribution of the probability density of the sequence values. Strong multi fractal behaviors over some stations in the Yunnan, Guangdong, and Inner Mongolia provinces are obvious. These behaviors are mainly caused by different long range correlations between large and small fluctuations. The asymmetry of the singularity of relative humidity records is weak, except for a small number of stations in the far east and west of China, where the singularity spec trum is left-skewed. Finally, the long-range correlations in North China are stronger than those in South China, which indicates better predictability in North China. By studying the parameters of the multi-fractal spectrum, various data of long-range power law correlations of the relative humidity records are obtained, which may pro vide theoretical support for climate prediction.

Fractal-based weak target detection within sea clutter

The target on the sea surface is complex and difficult to detect due to the interference of backscattered returns from the sea surface illuminated by the radar pulse. Detrended fluctuation analysis （DFA） has been used successfully to extract the time-domain Hurst exponent of sea-clutter series. Since the frequency of the sea clutter mainly concentrates around Doppler center so that we consider to extract frequency-do- main fractal characterization and then detect a weak target within sea clutter by using the difference of frequency-domain fractal characterization. The generalized detrended fluctuation analysis （GDFA） is more flexible than traditional DFA owing to its smoothing action for the clutters. In this paper, we apply the GDFA to evaluate the generalized Hurst exponent of sea-clutter series in the frequency domain. The difference of generalized Hurst exponents between different sea-clutter range bins would be used to determine whether the target exists. Moreover, some simulations with the real IPIX radar data have also been demonstrated in order to suooort this conclusion.

Comparisons of MFDFA, EMD and WT by Neural Network, Mahalanobis Distance and SVM in Fault Diagnosis of Gearboxes

A method for gearbox fault diagnosis consists of feature extraction andfault identification. Many methods for feature extraction have beendevised for exposing nature of vibration data of a defective gearbox. Inaddition, features extracted from gearbox vibration data are identifiedby various classifiers. However, existing literatures leave much to bedesired in assessing performance of different combinatorial methods forgearbox fault diagnosis. To this end, this paper evaluated performance ofseveral typical combinatorial methods for gearbox fault diagnosis byassociating each of multifractal detrended fluctuation analysis (MFDFA),empirical mode decomposition (EMD) and wavelet transform (WT) witheach of neural network (NN), Mahalanobis distance decision rules(MDDR) and support vector machine (SVM). Following this,performance of different combinatorial methods was compared using agroup of gearbox vibration data containing slightly different faultpatterns. The results indicate that MFDFA performs better in featureextraction of gearbox vibration data and SVM does the same in faultidentification. Naturally, the method associating MFDFA with SVMshows huge potential for fault diagnosis of gearboxes. As a result, thispaper can provide some useful information on construction of a methodfor gearbox fault diagnosis.

Classification of low-density EEG for epileptic seizures by energy and fractal features based on EMD

We are here to present a new method for the classification of epileptic seizures from electroencephalogram(EEG) signals.It consists of applying empirical mode decomposition(EMD) to extract the most relevant intrinsic mode functions(IMFs) and subsequent computation of the Teager and instantaneous energy,Higuchi and Petrosian fractal dimension,and detrended fluctuation analysis(DFA) for each IMF.We validated the method using a public dataset of 24 subjects with EEG signals from 22 channels and showed that it is possible to classify the epileptic seizures,even with segments of six seconds and a smaller number of channels(e.g.,an accuracy of0.93 using five channels).We were able to create a general machine-learning-based model to detect epileptic seizures of new subjects using epileptic-seizure data from various subjects,after reducing the number of instances,based on the k-means algorithm.

Understanding long-term memory in global mean temperature:An attribution study based on model simulations

Long-term memory(LTM)in the climate system has been well recognized and applied in different research fields,but the origins of this property are still not clear.In this work,the authors contribute to this issue by studying model simulations under different scenarios.The global mean temperatures from pre-industrial control runs(pi Control),historical(all forcings)simulations,natural forcing only simulations(Historical Nat),greenhouse gas forcing only simulations(Historical GHG),etc.,are analyzed using the detrended fluctuation analysis.The authors find that the LTM already exists in the pi Control simulations,indicating the important roles of internal natural variability in producing the LTM.By comparing the results among different scenarios,the LTM from the piControl runs is further found to be strengthened by adding natural forcings such as the volcanic forcing and the solar forcing.Accordingly,the observed LTM in the climate system is suggested to be mainly controlled by both the‘internal’natural variability and the‘external’natural forcings.The anthropogenic forcings,however,may weaken the LTM.In the projections from RCP2.6 to RCP8.5,a weakening trend of the LTM strength is found.In view of the close relations between the climate memory and the climate predictability,a reduced predictability may be expected in a warming climate.

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.