Multifractal Detrended Fluctuation Analysis Method Research Articles

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.

Multifractality of stock markets based on cumulative distribution function and multiscale multifractal analysis

Considering the diverse application of multifractal techniques in natural scientific disciplines, this work underscores the versatility of multiscale multifractal detrended fluctuation analysis (MMA) method to investigate artificial and real-world data sets. The modified MMA method based on cumulative distribution function is proposed with the objective of quantifying the scaling exponent and multifractality of nonstationary time series. It is demonstrated that our approach can provide a more stable and faithful description of multifractal properties in comprehensive range rather than fixing the window length and slide length. Our analyzes based on CDF-MMA method reveal significant differences in the multifractal characteristics in the temporal dynamics between US and Chinese stock markets, suggesting that these two stock markets might be regulated by very different mechanism. The CDF-MMA method is important for evidencing the stable and fine structure of multiscale and multifractal scaling behaviors and can be useful to deepen and broaden our understanding of scaling exponents and multifractal characteristics.

MULTISCALE AND FRACTAL ANALYSIS OF SILICON CONTENT TIME SERIES OBSERVED IN BLAST FURNACE HOT METAL USING HURST EXPONENT CHAIN

Hurst exponent is an important measure of nonlinearity of dynamical time series. In this paper, using rescaled-range ([Formula: see text]/[Formula: see text]) analysis, multi-fractal detrended fluctuation analysis (MF-DFA) methods, the multiscale Hurst exponent (MHE) and the multiscale generalized Hurst exponent (MGHE) of coarse-grained silicon content ([Si]) time series in blast furnace (BF) hot metal were calculated. First, we collected these [Si] time series from No. 1 BF of Nanchang Iron and Steel Co. and No. 10 BF of Xinyu Iron and Steel Co. in Jiangxi Province, China. Then, we analyzed and compared the estimated Hurst exponents and the generalized Hurst exponent of these observed time series with some simulated time series. Our results show that the observed time series from these BFs have negative correlation with the Hurst exponent less than 0.5, the generalized Hurst exponent [Formula: see text] is a nonlinear function of [Formula: see text], and such negative correlation and local various structure persist in their moving averages of the observed time series up to lag 5 or 10.

Application of long-range correlation and multi-fractal analysis for the depiction of drought risk**Project supported by the National Basic Research Program of China (Grant No. 2012CB955901), the National Natural Science Foundation of China (Grant Nos. 41305056, 41175084, and 41375069), and the Special Scientific Research Fund of Meteorological Public Welfare Profession of China (Grant No. GYHY201506001).

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.

Fractal features of CdTe thin films grown by RF magnetron sputtering

Cadmium telluride (CdTe) thin films were prepared by RF magnetron sputtering on glass substrates at room temperature (RT). The film deposition was performed for 5, 10, and 15min at power of 30W with a frequency of 13.56MHz. The crystal structure of the prepared CdTe thin films was studied by X-ray diffraction (XRD) technique. XRD analyses indicate that the CdTe films are polycrystalline, having zinc blende structure of CdTe irrespective of their deposition time. All CdTe films showed a preferred orientation along (111) crystalline plane. The surface morphology characterization of the films was studied using atomic force microscopy (AFM). The quantitative AFM characterization shows that the RMS surface roughness of the prepared CdTe thin films increases with increasing the deposition time. The detrended fluctuation analysis (DFA) and also multifractal detrended fluctuation analysis (MFDFA) methods showed that prepared CdTe thin films have multifractal nature. The complexity, roughness of the CdTe thin films and strength of the multifractality increase as deposition time increases.

Multifractal and recurrence behaviors of continuum percolation-based financial price dynamics

The modeling of financial price fluctuations is always a hot research aspect that many researchers are concerned about. A financial price model based on two-dimensional continuum percolation system, which is one of the most important statistical physics systems, is introduced in this work. In the model, the fluctuations of stock price changes are assumed to be attributed to the market information interactions among the traders, and the percolation cluster is taken to represent the traders holding the same investment attitude. Then, multifractal detrended fluctuation analysis method is adopted to study the multifractal behaviors of simulation data with different parameter sets. Finally, the recurrence plot and recurrence quantification analysis techniques are applied to investigate the complex determinism dynamics hidden in the simulated stock returns from the price model, as well as in their different intrinsic mode functions (IMFs) decomposed from the empirical mode decomposition method. Abundant and distinctive recurrence behaviors can be observed among returns and IMFs time series. In the meanwhile, the corresponding behaviors of the Chinese Shanghai Composite Index is studied for comparison.

Multifractal nature of particulate matters (PMs) in Hong Kong urban air

In this study, we investigate the persistent variation and the multifractal nature of particulate matter (PM) concentrations from vehicle emissions at a typical traffic intersection of street canyon in Hong Kong. Six size groups of PMs are measured and collected during rush hour sessions on different dates respectively. A recently developed model, namely multifractal detrended fluctuation analysis (MF-DFA), is employed to decompose and analyze the collected database. Through estimating the scaling exponent, it is found that the PM levels from vehicular emissions display long-term correlation characters. By employing MF-DFA method to calculate the generalized Hurst exponent and discuss the multifractal spectrums of all size groups, it is noticed that the fine particulate matters in grain diameter of 0.3–0.499μm present strong multifractal nature, intensive oscillations of concentration variations, and long-term persistence. For fine particulate matters in the grain diameter ranges from 0.5μm to 4.99μm, their similar and weak multifractal natures reflect the self-similarity behaviors among these groups and the gradual decreases of the lasting effects. For large size particulate matters, i.e., grain diameter above 5μm, certain mono-fractal nature and sharp decay of long-term persistence are obtained, even for intermittent effects. It can therefore be concluded that the fine particulate matter diffuses anomaly and persists for a long time.

MULTIFRACTAL DETRENDED FLUCTUATION ANALYSIS OF ELECTRIC LOAD SERIES

Multifractal detrended fluctuation analysis (MF-DFA) method is applied to analyze the daily electric load time series. The results of the MF-DFA show that there are three crossover timescales at seven days, 15 days and 365 days approximately in the fluctuation function. Also we find that these fluctuations have multifractal nature with long range correlation behavior. The multifractal singularity spectrum of the daily electric load series has been fitted by the quadratic function model. Comparing the MF-DFA results of the original load series with those of shuffled and surrogate series, it concludes that the multifractal characteristics of the daily electric load time series is due to both broadness of the probability density function and long-range correlation, and the long-range correlation is dominant.

Extreme values in the Chinese and American stock markets based on detrended fluctuation analysis

This paper focuses on the comparative analysis of extreme values in the Chinese and American stock markets based on the detrended fluctuation analysis (DFA) algorithm using the daily data of Shanghai composite index and Dow Jones Industrial Average. The empirical results indicate that the multifractal detrended fluctuation analysis (MF-DFA) method is more objective than the traditional percentile method. The range of extreme value of Dow Jones Industrial Average is smaller than that of Shanghai composite index, and the extreme value of Dow Jones Industrial Average is more time clustering. The extreme value of the Chinese or American stock markets is concentrated in 2008, which is consistent with the financial crisis in 2008. Moreover, we investigate whether extreme events affect the cross-correlation between the Chinese and American stock markets using multifractal detrended cross-correlation analysis algorithm. The results show that extreme events have nothing to do with the cross-correlation between the Chinese and American stock markets.

Long-range memory and multifractality in gold markets

Long-range correlation and fluctuation in the gold market time series of the worldʼs two leading gold consuming countries, namely China and India, are studied. For both the market series during the period 1985–2013 we observe a long-range persistence of memory in the sequences of maxima (minima) of returns in successive time windows of fixed length, but the series, as a whole, are found to be uncorrelated. Multifractal analysis for these series as well as for the sequences of maxima (minima) is carried out in terms of the multifractal detrended fluctuation analysis (MF-DFA) method. We observe a weak multifractal structure for the original series that mainly originates from the fat-tailed probability distribution function of the values, and the multifractal nature of the original time series is enriched into their sequences of maximal (minimal) returns. A quantitative measure of multifractality is provided by using a set of ‘complexity parameters’.

Multifractal Processes and Self-Organized Criticality of PM2.5 during a Typical Haze Period in Chengdu, China

Scaling and multifractal properties of the hourly PM2.5 average concentration series at the four air monitoring stations of Chengdu (southwestern China) were explored by using a multifractal detrended fluctuation analysis method, during a typical haze episode (from 1 March to 17 March, 2013). Using shuffling procedure and phase randomization procedure, the major sources of multifractality in these PM2.5 series are studied. The results show that the multifractality nature of PM2.5 series is mainly due to long-range correlation. At the same time, the non-Gaussian probability distributions also partly contribute to the multifractal behaviour. The scale-free power laws behaviours are found to govern the cumulative distributions statistics for PM2.5 concentration fluctuations. The temporal evolutions of the multifractality were investigated by the approach of a sliding window. Further, we attempt to find the answers to the following questions: how does longrange correlation and power-law distribution in PM2.5 evolution emerge? It is inviting to do it in a self-organized criticality (SOC) framework, which was specially designed to model the dynamics of complex systems. A novel PM2.5 evolution model is developed on the bases of SOC theory. The model displays robust power law behaviour in certain dynamical region. The self-organized criticality properties of PM2.5 evolution are discussed. This SOC behaviour is related to a statistically steady state that implies the presence of long-range correlation and power-law distribution in PM2.5 evolution during the haze period. It is the stability of SOC that causes the haze period to be sustained for a long time in Chengdu.

Fractal property of sea clutter FRFT spectrum for small target detection

Because the detection of small target in the background of sea clutter is strongly influenced by the sea condition, this paper studies the fractal property of sea clutter in fractional Fourier transform (FRFT) domain and proposes the fractal detection method in single and high dimensions. The FRFT deduced from mathematical definition is not consistent with the self-similar properties in orders and scales. Multifractal detrended fluctuation analysis (MF-DFA) method is used to determine the fractal parameter H(q) and analyze the fractal property of the sea clutter in different situations, distances, and polarizations. In single dimension, the small target detection method is proposed based on an adaptive order. By comparing different factors of the multifractal parameters, the results show that the transform order method in sea clutter FRFT domain can detect small signals under complicated sea conditions. The detection threshold mostly increases above 200%, which is 26.3% higher than the method of time domain signal. H(q) has an obvious multifractal difference on high negative scale, the H(q)-q curve satisfies the arctangent distribution. The fitting amplitude ratios of pure sea clutter and target data are greater than 1.8 (HH) and 1.4 (VV), which provide the basis for the small target detection in sea clutter background.

Multifractal Flexibly Detrended Fluctuation Analysis

Multifractal time series analysis is a approach that shows the possible complexity of the system. Nowadays, one of the most popular and the best methods for determining multifractal characteristics is Multifractal Detrended Fluctuation Analysis (MFDFA). However, it has some drawback. One of its core elements is detrending of the series. In the classical MFDFA a trend is estimated by fitting a polynomial of degree $m$ where $m=const$. We propose that the degree $m$ of a polynomial was not constant ($m\neq const$) and its selection was ruled by an established criterion. Taking into account the above amendment, we examine the multifractal spectra both for artificial and real-world mono- and the multifractal time series. Unlike classical MFDFA method, obtained singularity spectra almost perfectly reflects the theoretical results and for real time series we observe a significant right side shift of the spectrum.

Non-stationary dynamics in the bouncing ball: a wavelet perspective.

The non-stationary dynamics of a bouncing ball, comprising both periodic as well as chaotic behavior, is studied through wavelet transform. The multi-scale characterization of the time series displays clear signatures of self-similarity, complex scaling behavior, and periodicity. Self-similar behavior is quantified by the generalized Hurst exponent, obtained through both wavelet based multi-fractal detrended fluctuation analysis and Fourier methods. The scale dependent variable window size of the wavelets aptly captures both the transients and non-stationary periodic behavior, including the phase synchronization of different modes. The optimal time-frequency localization of the continuous Morlet wavelet is found to delineate the scales corresponding to neutral turbulence, viscous dissipation regions, and different time varying periodic modulations.

Noise Effect on Multifractal Detrended Fluctuation Analysis Based on Element Enrichment Model

Used of the element enrichment model, the multifractal detrended fluctuation analysis (MF-DFA) is applied to analyze the multifractal property for sequences which were added stochastic noise and spike noise, and then we discuss the noise effect on the scaling exponent. The result shows that the scaling exponent is stable under stochastic noise and spike noise. Forq>0, the influence on the scaling exponent is rather small when the element enrichment model was added stochastic noise, and the difference is getting smaller with the increasing parameterp; When adding spike noise whose strength is from 1.5 times to 2.5 times, the corresponding influence is consistent, it indicates that MF-DFA method has a better noise immunity against stochastic noise and spike noise under given conditions.

Multiscaling Analysis of Monthly Runoff Series Using Improved MF-DFA Approach

An improved multifractal detrended fluctuation analysis(MF-DFA) method is applied to analyze the long-term monthly runoff records of a hydrological station in the Yangtze River with seasonal trend eliminated, through which the long-range correlation and the multifractal characteristics have been found. The multifractal spectrum has been fitted by a generalized expression of the multiplicative cascade model, and the results show that the monthly runoff series has strong multifractal characteristics. Comparing the results for the original runoff series with those of shuffled and surrogate series, it concludes that the multifractal characteristics of the monthly runoff time series is due to the broadness of both the probability density function and long-range correlation, and the broadness of the probability density function is dominant.

On multifractality and synchronization of nonlinear stellar pulsators

Simple models of nonlinear stellar pulsation, whose temporal behavior may reproduce some of the observed features of different classes of variable stars, have been studied. The temporal behavior of dynamical variables of these models exhibits a cascade of period doubling chaos, depending on the specific values of the various control parameters. A multifractal detrended fluctuation analysis (MFDFA) method is further used to identify the scaling behavior of such synthetic time series. The MFDFA of the considered time series, for various models of nonlinear stellar pulsation, shows that the observed multifractal nature is due to long-range correlations. The pulsating star with increased nonadiabaticity and the star with increased convective luminosity, as represented by the simulated data, is shown to possess a strange attractor with noninteger correlation dimension that lies between 2–3. Also the problem of synchronization in coupled nonlinear pulsation models has been investigated using permutation entropy—a complexity measure of the system. The computed order parameter, Γ, representing the correlation of computed permutation entropy for different segments of the simulated time series of displacement of two nonidentical oscillators, has been further used to find the critical coupling parameter for general synchronization of the oscillators.

Empirical properties of inter-cancellation durations in the Chinese stock market

Order cancellation process plays a crucial role in the dynamics of price formation in order-driven stock markets and is important in the construction and validation of computational finance models. Based on the order flow data of 18 liquid stocks traded on the Shenzhen Stock Exchange in 2003, we investigate the empirical statistical properties of inter-cancellation durations in units of events defined as the waiting times between two consecutive cancellations. The inter-cancellation durations for both buy and sell orders of all the stocks favor a $q$-exponential distribution when the maximum likelihood estimation method is adopted; In contrast, both cancelled buy orders of 6 stocks and cancelled sell orders of 3 stocks prefer Weibull distribution when the nonlinear least-square estimation is used. Applying detrended fluctuation analysis (DFA), centered detrending moving average (CDMA) and multifractal detrended fluctuation analysis (MF-DFA) methods, we unveil that the inter-cancellation duration time series process long memory and multifractal nature for both buy and sell cancellations of all the stocks. Our findings show that order cancellation processes exhibit long-range correlated bursty behaviors and are thus not Poissonian.

Multifractal detrended fluctuation analysis of the δ 18 O record of NGRIP ice core

The multifractal properties and scaling behaviors of the long-term and recent 2000-year δ18O records of NGRIP ice core are investigated by the multifractal detrended fluctuation analysis method. The generalized Hurst exponents, multifractal scaling exponents, and singularity spectrums of two δ18O records are derived to verify the multifractiality of two records. And the multifractal behaviors of two records are obviously different, which may reflect the climate change of the recent 2000-year time is quite different from one of the long-term time. In addition, the probability distribution analysis of two δ18O records is presented to manifest the different multifractality between two δ18O records of NGRIP ice core. Our results will be helpful to research the climate change.