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.

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 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 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.

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.

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.

Study on the Temporal-spatial Distribution Characteristics of Extreme Precipitation in Jiangsu Province

[Objective] The rasearch aimed to grasp the spatial and temporal distribution charactedstics of extreme precipitation in Jiangsu Prov- ince. [Methodl Threshold was determined by detrended fluctuation analysis （DFA） based on the data of daily precipitation over Jiangsu Province from 1961 to 2010, and the oscillation pedod and mutation were detected by the method of Morlet wavelet analysis and Mann-Kendall test respec- tively. [ Result~ The annual frequency of extreme precipitation in Jiangsu Province demonstrated oscillation characteristics with 8 -10 a, and the mutation mainly happened in 2006, In addition, the oscillation period of extreme precipitation during summer was 8 -10 a, and its mutation hap- pened in 1998. The oscillation periods of extreme precipitation, however, during autumn were 2 -3 and 5 -7 a. Extreme precipitation and total rain- fall had better consistency. They both presented the characteristics of more in the south and less in the north, and big-small-value zone was concentrated. [ Conclusion] The research could provide technical support for making defense policy of the related disaster for relevant departments.

Detection and Classification of Power Quality Disturbances in Distribution Networks Based on VMD and DFA

In this paper,a novel algorithm is proposed to detect and classify the power quality(PQ)disturbances for distribution networks with distributed generation.First,a distribution system with photovoltaic and wind power generation is built as a test platform.Then,nine types of power quality disturbances in the distribution network are decomposed by variational mode decomposition(VMD)and the noise is filtered.Meanwhile,the mode functions containing characteristic information are extracted as input signals of detrended fluctuation analysis(DFA).Power quality disturbances are classified from the view of their distributed energy operational status,and three types of windows are set up to deal with different frequency disturbances.The two-dimensional and three-dimensional scatter plots of each type under three windows are depicted,and the criteria are determined to distinguish the disturbances under the different operating conditions.The simulations show that the algorithm is simpler,more accurate and feasible.It provides an approach for online real-time detection of embedded systems.

Scaling Behavior of Bulk Freight Rate Volatility Before and After Noise Reduction

Analysis of freight rate volatility characteristics is a hot topic after year 2008 due to the effect of financial crisis in marine transportation.In this paper,we analyze the bulk freight rate index by detrended fluctuation analysis(DFA) method and discover the scaling behavior.Switching points(SPs),as the indicators of scaling behavior,can be eliminated after Vondr′ak noise reduction technique.Therefore,we conclude that highfrequency noise is the cause of SP.