Multifractal Detrended Fluctuation Analysis Method Research Articles

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 MF-DFA 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.

Extraction of Coal and Gangue Geometric Features with Multifractal Detrending Fluctuation Analysis

The separation of coal and gangue is an important process of the coal preparation technology. The conventional way of manual selection and separation of gangue from the raw coal can be replaced by computer vision technology. In the literature, research on image recognition and classification of coal and gangue is mainly based on the grayscale and texture features of the coal and gangue. However, there are few studies on characteristics of coal and gangue from the perspective of their outline differences. Therefore, the multifractal detrended fluctuation analysis (MFDFA) method is introduced in this paper to extract the geometric features of coal and gangue. Firstly, the outline curves of coal and gangue in polar coordinates are detected and achieved along the centroid, thereby the multifractal characteristics of the series are analyzed and compared. Subsequently, the modified local singular spectrum widths Δ h of the outline curve series are extracted as the characteristic variables of the coal and gangue for pattern recognition. Finally, the extracted geometric features by MFDFA combined with the grayscale and texture features of the images are compared with other methods, indicating that the recognition rate of coal gangue images can be increased by introducing the geometric features.

Feature extraction method based on VMD and MFDFA for fault diagnosis of reciprocating compressor valve

Aiming at the nonlinearity, nonstationarity and multi-component coupling characteristics of reciprocating compressor vibration signals, an integrated feature extraction method based on the variational mode decomposition (VMD) and multi-fractal detrended fluctuation analysis (MFDFA) is proposed for a fault diagnosis for a reciprocating compressor valve. Firstly, to eliminate the noise interference, a novel VMD method with superior anti-interference performance was utilized to obtain several components of the quasi-orthogonal band-limited intrinsic mode function (BLIMF) from a strong non-stationarity vibration signal, and a consistent number K of BLIMFs was selected based on a novel criterion for all fault states. Secondly, the MFDFA method, which can describe the multi-fractal structure feature of non-stationary time series, was applied to analyze each BLIMF component, and the parameters of MFDFA were employed as the eigenvectors to reflect the structure characteristics and local scale behavior of the vibration signal. Then, the principal component analysis (PCA) was introduced to refine the eigenvectors for a higher recognition efficiency and accuracy. Finally, the vibration signals of four types of reciprocating compressor valve faults were analyzed by this method, and the faults were identified correctly by pattern classifiers of BTSVM and CNN. Further results comparison with other feature extraction methods verifies the superiority of the proposed method.

Financial liberalization and stock market cross-correlation: MF-DCCA analysis based on Shanghai-Hong Kong Stock Connect

Based on the implementation of Shanghai-Hong Kong Stock Connect in China, this paper examines the effects of financial liberalization on stock market comovement using both multifractal detrended fluctuation analysis (MF-DFA) and multifractal detrended cross-correlation analysis (MF-DCCA) methods. Results based on MF-DFA confirm the multifractality of Shanghai and Hong Kong stock markets, and the market efficiency of Shanghai stock market increased after the implementation of this connect program. Besides, analysis based on MF-DCCA has verified the existence of persistent cross-correlation between Shanghai and Hong Kong stock markets, and the cross-correlation gets stronger after the launch of this liberalization program. Finally, we find that fat-tail distribution is the main source of multifractality in the cross-correlations before the stock connect program, while long-range correlation contributes to the multifractality after this program.

Multifractal property of Chinese stock market in the CSI 800 index based on MF-DFA approach

CSI 800 index consists of CSI 500 index and CSI 300 index, aiming to reflect the performance of stocks with large, mid and small size of China A share market. In this paper we analyze the multifractal structure of Chinese stock market in the CSI 800 index based on the multifractal detrended fluctuation analysis (MF-DFA) method. We find that the fluctuation of the closing logarithmic returns have multifractal properties, the shape and width of multifractal spectrum are depended on the weighing order q. More interestingly, we observe a bigger market crash in June–August 2015 than the one in 2008 based on the local Hurst exponents. The result provides important information for further study on dynamic mechanism of return fluctuation and whether it would trigger a new financial crisis.

Multivariate multifractal detrended fluctuation analysis of 3D wind field signals

Characterizing the dynamic behavior underlying wind field from experimental multivariate signals is a challenging problem of continuous interest. In this work, we propose the multivariate multifractal detrended fluctuation analysis (MV-MFDFA) method to directly study the fractal dynamics of multichannel data in a complex system. By conducting several simulations on synthetic multivariate series, the validity of the proposed MV-MFDFA is illustrated. Then we utilize MV-MFDFA to analyze the 3D wind field signals collected at two different airflow environments, i.e., indoor and outdoor environments. Results show that the indoor and outdoor three wind vectors show multifractal properties, and the multifractal degrees of outdoor three wind vectors are stronger than those of corresponding indoor three wind vectors. By analyzing the indoor and outdoor multivariate wind speed, we find that the indoor and outdoor multivariate wind speed are antipersistent long-range correlation, and the indoor multivariate wind speed exhibits weaker multifractal properties than that of outdoor multivariate wind speed. Moreover, the multifractality of indoor multivariate wind speed depends mainly on the large fluctuations, while the multifractality of outdoor multivariate wind speed depends mainly on the small fluctuations. These findings indicate that the MV-MFDFA allows better understanding the dynamical mechanisms governing 3D wind variability.

Multifractal Detrended Fluctuation Analysis on Friction Coefficient During the Friction Process

The dynamic evolutionary law and tribological behavior of tribopair AISI 52100-AISI 1045 were studied via the multifractal method. Friction experiment was performed on a ring-on-disk tribometer under lubrication, and the multifractal detrended fluctuation analysis (MF-DFA) method was adapted to characterize the multifractality of the friction coefficient. The multifractal spectra first exhibited a left-hook, then right-hook, and left-hook, respectively, during the friction stages. The multifractal spectrum width W decreases in running-in friction process, maintains at small values in steady friction process, and increases rapidly in increasing friction process. Corresponding shuffled series was analyzed to distinguish that the multifractality of friction coefficient is due to the long-range correlation of the fluctuations. The results inform quantitative interpretations of friction system's tribological behavior and friction process identification.

Multifractals of investor behavior in stock market

In this paper, we analyze the nonlinear properties of investor activity using the multifractal detrended fluctuation analysis (MF-DFA) method. Using the aggregated trading volumes of buying, selling, and normalized net investor trading (NIT) to quantify the characteristics of trader behavior in the KOSPI market, we find that the cumulative distribution functions of all NIT time series, except for individual traders, follow a power-law distribution with an exponent in the range of 2.92 ≤ γ ≤ 3.87. To observe the nonlinear features of investor activity, we also calculate the multifractal spectra for the buyer, seller, and NIT data sets and find that a multifractal structure exists in all of the data, regardless of the investor type studied.

A multifractal detrended fluctuation analysis of financial market efficiency: Comparison using Dow Jones sector ETF indices

This study challenges the efficient market hypothesis, relying on the Dow Jones sector Exchange-Traded Fund (ETF) indices. For this purpose, we use the generalized Hurst exponent and multifractal detrended fluctuation analysis (MF-DFA) methods, using daily data over the timespan from 2000 to 2015. We compare the sector ETF indices in terms of market efficiency between short- and long-run horizons, small and large fluctuations, and before and after the global financial crisis (GFC). Our findings can be summarized as follows. First, there is clear evidence that the sector ETF markets are multifractal in nature. We also find a crossover in the multifractality of sector ETF market dynamics. Second, the utilities and consumer goods sector ETF markets are more efficient compared with the financial and telecommunications sector ETF markets, in terms of price prediction. Third, there are noteworthy discrepancies in terms of market efficiency, between the short- and long-term horizons. Fourth, the ETF market efficiency is considerably diminished after the global financial crisis.

Dynamical behaviors of multifractal strengths in meteorological factors

In this paper, we study the multifractal properties of cloud observation time-series data in Daegwanryung, Korea, containing two meteorological factors, the effective radius of a cloud droplet and the average particle size of a raindrop. We simulate and analyze the generalized Hurst exponent, the Renyi exponent, the spectrum, and the multifractal strength by using the multifractal detrended fluctuation analysis method. The result obtained may have a useful and effective influences on determining the observation location.

Long memory and scaling behavior study of bulk freight rate volatility with structural breaks

ABSTRACTDue to the effect of credit crunch and slowdown in marine transportation, analysis of freight rate volatility characters under structural breaks is of great importance after year 2008. In this paper, the presence of structural break points in bulk freight rate index time series is investigated through the Iterated Cumulative Sum of Squares algorithm. Furthermore, long memory features of different vessel sizes, Supramax, Panamax and Capesize bulk carriers are also accomplished, and emperical results show that the daily returns of Baltic Supramax Index inclined to have stronger long-range correlation. Multifractal detrended fluctuation analysis (MF-DFA) method is applied with the influence of seasonality taken into consideration and the origin of multifractality is revealed. For policy-makers and investors with different trading horizons, structural break is an important indicator of different long memory characters of freight rate index time series in different time ranges. Hence, structural breaks in MF-DFA results are picked out and conclusions of long memory feature are achieved.

Investigation of multifractal nature of floating potential fluctuations obtained from a dc glow discharge magnetized plasma

Abstract In this paper, multifractal detrended fluctuation analysis (MF-DFA) has been used to analyze the floating potential fluctuations obtained with a Langmuir probe from a dc glow discharge magnetized plasma device. The generalized Hurst exponents ( h ( q ) ) , local fluctuation function ( F q ( s ) ) , the Renyi exponents ( τ ( q ) ) and the multifractal spectrum F ( α ) have been calculated by applying the MF-DFA method. The result of the MF-DFA shows the multifractal nature of these fluctuations. We have investigated the influence of magnetic field on the multifractal nature of the fluctuations and it is seen that degree of multifractality is reduced with the increase in the magnetic field strength. The values of h ( q ) have been restricted between 0.7 and 1 for the magnetized fluctuations. This result is evidence of the existence of long-range correlations in the fluctuations. Furthermore, we employed shuffle and surrogate approaches to analyze the origins of multifractality. Comparing the MF-DFA results for the data set to those for shuffled and surrogate series, we have found that its multifractal nature is due to the existence of significant long-term correlation.

Temporal and spatial characteristics of extreme precipitation events in the Midwest of Jilin Province based on multifractal detrended fluctuation analysis method and copula functions

Environmental changes have brought about significant changes and challenges to water resources and management in the world; these include increasing climate variability, land use change, intensive agriculture, and rapid urbanization and industrial development, especially much more frequency extreme precipitation events. All of which greatly affect water resource and the development of social economy. In this study, we take extreme precipitation events in the Midwest of Jilin Province as an example; daily precipitation data during 1960–2014 are used. The threshold of extreme precipitation events is defined by multifractal detrended fluctuation analysis (MF-DFA) method. Extreme precipitation (EP), extreme precipitation ratio (EPR), and intensity of extreme precipitation (EPI) are selected as the extreme precipitation indicators, and then the Kolmogorov-Smirnov (K-S) test is employed to determine the optimal probability distribution function of extreme precipitation indicators. On this basis, copulas connect nonparametric estimation method and the Akaike Information Criterion (AIC) method is adopted to determine the bivariate copula function. Finally, we analyze the characteristics of single variable extremum and bivariate joint probability distribution of the extreme precipitation events. The results show that the threshold of extreme precipitation events in semi-arid areas is far less than that in subhumid areas. The extreme precipitation frequency shows a significant decline while the extreme precipitation intensity shows a trend of growth; there are significant differences in spatiotemporal of extreme precipitation events. The spatial variation trend of the joint return period gets shorter from the west to the east. The spatial distribution of co-occurrence return period takes on contrary changes and it is longer than the joint return period.

Multifractal characterization of mechanical vibration signals through improved empirical mode decomposition-based detrended fluctuation analysis

A novel multifractal detrended fluctuation analysis based on improved empirical mode decomposition for the non-linear and non-stationary vibration signal of machinery is proposed. As the intrinsic mode functions selection and Kolmogorov–Smirnov test are utilized in the detrending procedure, the present approach is quite available for contaminated data sets. The intrinsic mode functions selection is employed to deal with the undesired intrinsic mode functions named pseudocomponents, and the two-sample Kolmogorov–Smirnov test works on each intrinsic mode function and Gaussian noise to detect the noise-like intrinsic mode functions. The proposed method is adaptive to the signal and weakens the effect of noise, which makes this approach work well for vibration signals collected from poor working conditions. We assess the performance of the proposed procedure through the classic multiplicative cascading process. For the pure simulation signal, our results agree with the theoretical results, and for the contaminated time series, the proposed method outperforms the traditional multifractal detrended fluctuation analysis methods. In addition, we analyze the vibration signals of rolling bearing with different fault types, and the presence of multifractality is confirmed.

Dam structural behavior identification and prediction by using variable dimension fractal model and iterated function system

According to the observations of dam structural health monitoring, iterated function system is adopted to implement the analysis and forecast for dam structural behavior. Firstly, the multifractal detrended fluctuation analysis (MF-DFA) method is employed to identify the fractal characteristics in the measured data series of dam structural behavior. Secondly, the iterated function system algorithm is studied to build the fitting model. The ways to determine the interpolating points (position and number) and vertical scaling factors are given in detail. Thirdly, the variable dimension fractal model and iterated function system are combined to forecast the dam structural behavior. Lastly, the displacement behavior of one concrete gravity dam is analyzed and predicted by the proposed approach. It is shown that the whole trend and detail characteristics of dam structural behavior observed can be described well, and the prediction precision can be improved.

Fluctuation analysis of high frequency electric power load in the Czech Republic

We analyze the electric power load in the Czech Republic (CR) which exhibits a seasonality as well as other oscillations typical for European countries. Moreover, we detect 1/f noise property of electrical power load with extra additional peaks that allows to separate it into a deterministic and stochastic part. We then focus on the analysis of the stochastic part using improved Multi-fractal Detrended Fluctuation Analysis method (MFDFA) to investigate power load datasets with a minute resolution. Extracting the noise part of the signal by using Fourier transform allows us to apply this method to obtain the fluctuation function and to estimate the generalized Hurst exponent together with the correlated Hurst exponent, its improvement for the non-Gaussian datasets. The results exhibit a strong presence of persistent behaviour and the dataset is characterized by a non-Gaussian skewed distribution. There are also indications for the presence of the probability distribution that has heavier tail than the Gaussian distribution.

Multi-fractal scaling comparison of the Air Temperature and the Surface Temperature over China

Abstract The spatial and temporal multi-scaling behaviors between the daily Air Temperature (AT) and the Surface Temperature (ST) over China are compared in about 60-yr observations by Multi-fractal Detrended Fluctuation Analysis (MF-DFA) method. The different fractal phenomena and diversity features in the geographic distribution are found for the AT and ST series using MF-DFA. There are more multi-fractal features for the AT records but less for ST. The respective geographic sites show important scaling differences when compared to the multi-fractal signatures of AT with ST. An interval threshold for 95% confidence level is obtained by shuffling the AT records and the ST records. For the AT records, 93% of all observed stations shows the strong multi-fractal behaviors. In addition, the multi-fractal characteristics decrease with increasing latitude in South China and are obviously strong along the coast. The multi-fractal behaviors of the AT records between the Yangtze River and Yellow River basin and in most regions of Northwest China seem to be weak and not significant, even single mono-fractal features. However, for the ST records, the geographical distributions of multi-fractal phenomenon seem to be in disorder which account for 81% of the stations. The weak multi-fractal behaviors of the ST records are concentrated in North China, most regions of Northeast China.

Speech, Music and Multifractality

Audio signal categorization is one of the rudimentary steps in applications like content-based audio information retrieval, audio indexing, speaker identification, and so on. In this work, a rigorous, non-stationary methodology capable of categorization among speech and various music signals is proposed. Multifractal detrended fluctuation analysis method is used to analyse the internal dynamics of the acoustics of digitized audio signal. The test data include speech (nonmusical), drone (periodically musical) and music samples of Rāgas (having different musicality) from Indian classical music (INDIC). It is found that the degree of complexity and multifractality (measured by width of the multifractal spectrum) changes from the start towards the end of each audio sample. However, the range of this variation is the smallest for speech and drone. The normalized value of the width of the multifractal spectrum is strikingly different for speech and drone. Experimental results show that this parameter can effectively classify speech and drone signals. Further, we have experimented with a number of clips of INDIC Rāgas with a range of variation in musicality and mood content. The results show that the width of the multifractal spectrum of the signals can categorize different music signals. In contrast with the conventional stationary techniques for audio signal analysis, we have used the method of complexity analysis without converting the non-stationary audio signals in frequency domain. We have used basic waveforms of the audio signals after de-noising them.

Comparative Multifractal Detrended Fluctuation Analysis of Heavy Ion Interactions at a Few GeV to a Few Hundred GeV

We have studied the multifractality of pion emission process in16O-AgBr interactions at 2.1 AGeV and 60 AGeV,12C-AgBr and 24Mg-AgBr interactions at 4.5 AGeV, and32S-AgBr interactions at 200 AGeV using Multifractal Detrended Fluctuation Analysis (MFDFA) method which is capable of extracting the actual multifractal property filtering out the average trend of fluctuation. The analysis reveals that the pseudorapidity distribution of the shower particles is multifractal in nature for all the interactions; that is, pion production mechanism has inbuilt multiscale self-similarity property. We have employed MFDFA method for randomly generated events for32S-AgBr interactions at 200 AGeV. Comparison of expt. results with those obtained from randomly generated data set reveals that the source of multifractality in our data is the presence of long range correlation. Comparing the results obtained from different interactions, it may be concluded that strength of multifractality decreases with projectile mass for the same projectile energy and for a particular projectile it increases with energy. The values of ordinary Hurst exponent suggest that there is long range correlation present in our data for all the interactions.

Dual-tree complex wavelet transform based multifractal detrended fluctuation analysis for nonstationary time series

Multifractal detrended fluctuation analysis is an effective tool for dealing with the non-uniformity and singularity of nonstationary time series. For the serious issues of the trend extraction and the inefficient computation in the traditional polynomial fitting based multifractal detrended fluctuation analysis, based on the dual-tree complex wavelet transform, a novel multifractal analysis is proposed. To begin with, as the dual-tree complex wavelet transform has the anti-aliasing and nearly shift-invariance, it is first utilized to decompose the signal through the pyramid algorithm, and the scale-dependent trends and the fluctuations are extracted from the wavelet coefficients. Then, using the wavelet coefficients, the length of the non-overlapping segment on a corresponding time scale is computed through the Hilbert transform, and each of the extracted fluctuations is divided into a series of non-overlapping segments whose sizes are identical. Next, on each scale, the detrended fluctuation function for each segment is calculated, and the overall fluctuation function can be obtained by averaging all segments with different orders. Finally, the generalized Hurst index and scaling exponent spectrum are determined from the logarithmic relations between the overall detrended fluctuation function and the time scale and the standard partition function, respectively, and then the multifractal singularity spectrum is calculated with the help of Legendre transform. We assess the performance of the dual-tree-complex wavelet transform based multifractal detrended fluctuation analysis (MFDFA) procedure through the classic multiplicative cascading process and the fractional Brownian motions, which have the theoretical fractal measures. For the multiplicative cascading process, compared with the traditional polynomial fitting based MFDFA methods, the proposed multifractal approach defines the trends and the length of non-overlapping segments adaptively and obtains a more precise result, while for the traditional MFDFA method, for the negative orders, no matter the generalized Hurst index, scaling exponents spectrum, or the multifractal singularity spectrum, the acquired results each have a significant deviation from the theoretical one. For the time series with different sizes, the proposed method can also give a stable result. Compared with the other adaptive method such as maximal overlap discrete wavelet transform based MFDFA and the discrete wavelet transfrom based MFDFA, the proposed approach obtains a very accurate result and has a fast calculation speed. For another time series of fractional Brownian motions with different Hurst indexes of 0.4, 0.5 and 0.6, which represent the anticorrelated, uncorrelated, correlated process, respectively, the results of the proposed method are consistent with those analytical results, while the results of the polynomial fitting based MFDFA methods are most greatly affected by the order of the fitting polynomial. The method in this article provides a valuable reference for how to use the dual-tree complex wavelet transform to realize the multifractal detrended fluctuation analysis, and we can benefit from the signal self-adaptive trend extraction and the high computation efficiency.