Detrended Fluctuation Analysis Method Research Articles

MULTIFRACTAL ANALYSIS OF SOLAR RADIATION IN THE NORTHEASTERN REGION OF BRAZIL

In this work, we used hourly data of high frequency of solar radiation from the entire Northeast region of Brazil. We used the Multifractal Detrended Fluctuation Analysis (MFDFA) method to analyze the characteristics of the solar radiation series in 137 meteorological stations from 2010 to 2022. For all analyzed series, the parameter [Formula: see text] characterizes persistent series. The values of [Formula: see text] reveal asymmetry to the right, indicating that large fluctuations contributed to the multifractality process. The states of Maranhão and Bahia presented the highest values of spectrum width [Formula: see text], indicating greater complexity. We found that long-range correlations are the leading cause of multifractality observed in the dynamics of the series of solar radiation anomalies.

Multifractal analysis of model fractal and multifractal signals

One of the topical directions of modern fractal radio physics is the multifractal analysis of signals and processes of various origins. A set of deterministic and stochastic models of monofractal and multifractal signals and processes in the time domain is proposed. New multifractal signal characteristics, namely, the coefficient of asymmetry of the multifractal spectrum function, the relative width of the multifractal spectrum and the dimension of the multifractal support, are introduced, the necessity of their use is demonstrated on examples. Using Wavelet Transform Modulus Maxima Method and Multi-Fractal Detrended Fluctuation Analysis Method, a detailed multifractal analysis of model signals is performed. The features of multifractal analysis of monofractal, multifractal and non-fractal signals are established, the appropriate recommendations for practitioners are formulated. Convenient formats for presenting analysis results have been developed. It was found that during the transition of the multifractal signal to the monofractal regime, the function of the multifractal spectrum of the physical fractal does not collapse into a point, as it should happen in theory for a mathematical fractal. Threshold values of multifractal characteristics, which are indicators of the appearance of the monofractal, are proposed. It has been shown that multifractal analysis of non-fractal signals leads to the appearance of multifractal spectra with anomalous values of multifractal characteristics. The correction function method is modified for the methods of multifractal analysis of signals and processes. It is proved that its usage makes it possible to reduce the deviation of the obtained estimate of the generalized Hurst exponent from the true known value of the Hölder exponent of the analyzed signal from 5 – 90% to 3 – 8%.

Homology groups of embedded fractional Brownian motion.

A well-known class of nonstationary self-similar time series is the fractional Brownian motion (fBm) considered to model ubiquitous stochastic processes in nature. Due to noise and trends superimposed on data and even sample size and irregularity impacts, the well-known computational algorithm to compute the Hurst exponent (H) has encountered superior results. Motivated by this discrepancy, we examine the homology groups of high-dimensional point cloud data (PCD), a subset of the unit D-dimensional cube, constructed from synthetic fBm data as a pipeline to compute the H exponent. We compute topological measures for embedded PCD as a function of the associated Hurst exponent for different embedding dimensions, time delays, and amount of irregularity existing in the dataset in various scales. Our results show that for a regular synthetic fBm, the higher value of the embedding dimension leads to increasing the H dependency of topological measures based on zeroth and first homology groups. To achieve a reliable classification of fBm, we should consider the small value of time delay irrespective of the irregularity presented in the data. More interestingly, the value of the scale for which the PCD to be path connected and the postloopless regime scale are more robust concerning irregularity for distinguishing the fBm signal. Such robustness becomes less for the higher value of the embedding dimension. Finally, the associated Hurst exponents for our topological feature vector for the S&P500 were computed, and the results are consistent with the detrended fluctuation analysis method.

Detrended fluctuation analysis based on best-fit polynomial

Detrended fluctuation analysis (DFA) can quantify long-range correlation (LRC) and fractal scaling behavior of signal. We compared the results of variant DFA methods by varying the order of the polynomial and found that the order of 6 was relatively better than the others when both the accuracy and computational cost were taken into account. An alternative DFA method is proposed to quantify the LRC exponent by using best-fit polynomial algorithm in each segment instead of the polynomial of the same order in all of segments. In this study, the best-fit polynomial algorithm with the maximum order of 6 is used to fit the local trend in each segment to detrend the trend of a time series, and then the revised DFA is used to quantify the LRC in the time series. A series of numerical studies demonstrate that the best-fit DFA performs better than regular DFA, especially for the time series with scaling exponent smaller than 0.5. This may be attributed to the improvement of the fitted trend at the end of each segment. The estimation results of variant DFA methods reach stable when the time series length is greater than 1,000.

A Driving Fatigue Feature Detection Method Based on Multifractal Theory

Driving fatigue seriously threatens traffic safety. In our work, the multifractal detrended fluctuation analysis (MF-DFA) method is proposed to detect driver fatigue caused by driving for a long time. First, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula> (4 ~ 7 Hz) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> (14 ~ 32 Hz) subbands of subjects’ electroencephalogram (EEG) signals were extracted. Furthermore, the multifractal spectrum characteristic indexes, the fluctuation function, mass exponent, Hurst exponent, spectral width values, and symmetry, were analyzed. Finally, the characteristics of different driving states of subjects were compared and analyzed. The results show that there are significant differences in the Hurst exponent width values of the subjects’ EEG signals corresponding to different driving times, and the spectrum width values and symmetry of the multifractal spectrum. In addition, compared with several typical fatigue detection methods, the absolute value of the slope of the fit straight line for fatigue feature extraction by MF-DFA is larger. It means that the MF-DFA method is more effective in detecting driving fatigue.

Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

In the article, a comparative analysis is performed regarding the accuracy parameter in determining the degree of self-similarity of fractal processes between the following methods: Variance-Time plot, Rescaled Range (R/S), Wavelet-based, Detrended Fluctuation Analysis (DFA) and Multifractal Detrended Fluctuation Analysis (MFDFA). To evaluate the methods, fractal processes based of Fractional Gaussian Noise were simulated and the dependence between the length of the simulated process and the degree of self-similarity was investigated by calculating the Hurst exponent (H &gt; 0.5). It was found that the Wavelet-based, DFA and MFDFA methods, with a process length greater than 214 points, have a relative error of the Hurst exponent is less than 1%. A methodology for the Wavelet-based method related to determining the size of the scale and the wavelet algorithm was proposed, and it was investigated in terms of the exact determination of the Hurst exponent of two algorithms: Haar and Daubechies with different number of coefficients and different values of the scale. Based on the analysis, it was determined that the Daubechies algorithm with 10 coefficients and scale (i = 2, j = 10) has a relative error of less than 0.5%. The three most accurate methods are applied to the study of real cardiac signals of two groups of people: healthy and unhealthy (arrhythmia) subjects. The results of the statistical analysis, using the t-test, show that the proposed methods can distinguish the two studied groups and can be used for diagnostic purposes.

Cavitation detection in a Kaplan turbine based on multifractal detrended fluctuation analysis of vibration signals

Cavitation generally causes a decrease in efficiency and vibrations to a Kaplan turbine, but effectively identifying cavitation state is quite difficult. In this paper, a comprehensive test system consisting of a high-speed camera and a vibration testing system has been constructed based on a high-precision universal hydraulic machinery test stand. The vibration signals of the runner and the draft tube and cavitation images on the runner blade have been obtained for a Kaplan turbine under different cavitation states based on the system. In addition, the multifractal detrended fluctuation analysis (MF-DFA) method has been introduced to analyse the obtained vibration signals, in order to investigate the influence of runner blade cavitation on turbine vibration. Results show that the vibration signals of both the runner and the draft tube have multifractal characteristics. The Hurst exponent, the scaling exponent and the multifractal spectrum by MF-DFA are strongly dependent on turbine cavitation state. It is found that cavitation states including the incipient cavitation of the turbine could be effectively identified by using characteristic parameters extracted from the multifractal spectrum obtained by the MF-DFA of vibration signals. This work provides an effective method for cavitation identification of Kaplan turbines.

Multifractality of global sea level heights in the satellite altimeter-era

Sea level rise is the major concern of climate change research. Beside the positive trends, global sea levels also demonstrate a complex scaling behavior. In this study, multifractal analyses based on the multifractal detrended fluctuation analysis (MFDFA) method are conducted to the global sea level anomaly (SLA) time series in the altimeter-era since 1993. By eliminating the multifractality caused by finite-size effect, the effective multifractality is estimated as the degree of multifractality of SLA time series. The results of show that global SLA time series are multifractals rather than monofractals. Long range dependence was also heuristically demonstrated by both autocorrelation functions and multifractal spectrums, respectively. The complex spatial patterns of sea level changes and multifractality could be partly explained by different mechanisms such as thermal expansion, the shift of ocean currents, and river injections, which are all associated with global climate change.

Spatial Characteristics Analysis for Coupling Strength among Air Pollutants during a Severe Haze Period in Zhengzhou, China.

This paper investigates the multifractal characteristics of six air pollutants using the coupling detrended fluctuation analysis method. The results show that coupling correlations exist among the air pollutants and have multifractal characteristics. The sources of multifractality are identified using the chi square test. The coupling strengths between different pollutants are quantified. In addition, the coupling contribution of a series in the haze system is calculated, and SO2, as the main pollutant, plays a key role in the pollution system. Moreover, the Kriging interpolation method is used to analyze the spatial characteristic on coupling contribution of SO2. The spatial analysis of coupling strength for air pollutants will provide an effective approach for pollution control.

NONLINEAR ANALYSIS AND PREDICTION OF BITCOIN RETURN’S VOLATILITY

This paper mainly studies the market nonlinearity and the prediction model based on the intrinsic generation mechanism (chaos) of Bitcoin’s daily return’s volatility from June 27, 2013 to November 7, 2019 with an econophysics perspective, so as to avoid the forecasting model misspecification. Firstly, this paper studies the multifractal and chaotic nonlinear characteristics of Bitcoin volatility by using multifractal detrended fluctuation analysis (MFDFA) and largest Lyapunov exponent (LLE) methods. Then, from the perspective of nonlinearity, the measured values of multifractal and chaos show that the volatility of Bitcoin has short-term predictability. The study of chaos and multifractal dynamics in nonlinear systems is very important in terms of their predictability. The chaos signals may have short-term predictability, while multifractals and self-similarity can increase the likelihood of accurately predicting future sequences of these signals. Finally, we constructed a number of chaotic artificial neural network models to forecast the Bitcoin return’s volatility avoiding the model misspecification. The results show that chaotic artificial neural network models have good prediction effect by comparing these models with the existing Artificial Neural Network (ANN) models. This is because the chaotic artificial neural network models can extract hidden patterns and accurately model time series from potential signals, while the benchmark ANN models are based on Gaussian kernel local approximation of non-stationary signals, so they cannot approach the global model with chaotic characteristics. At the same time, the multifractal parameters are further mined to obtain more market information to guide financial practice. These above findings matter for investors (especially for investors in quantitative trading) as well as effective supervision of financial institutions by government.

DETRENDED FLUCTUATION ANALYSIS OF THREE-DIMENSIONAL DATA: APPLICATION TO SOIL X-RAY CT SCANS

In this work, we compare three distinct schemes for detrended fluctuation analysis (DFA) of three-dimensional (3D) [Formula: see text] data sets: (i) an [Formula: see text] set of one-dimensional (1D) DFA runs for each vertical line of length [Formula: see text], (ii) a set of [Formula: see text] two-dimensional (2D) DFA runs for each of the [Formula: see text] planes along the vertical direction, and (iii) a single 3D DFA run. The objective of this work is two-fold: (i) to compare the performance of 1D, 2D, and 3D DFA methods in terms of compatibility as well as the richness of obtained results, and (ii) to demonstrate how these three complementary approaches shed new light on the complexity of the structure of the two real soil samples, through the analysis of grayscale high-resolution X-ray computed tomography scan images.

Long-Term Sustainability and Multifractal Characteristics of Air Pollution Evolution

Air pollution is a complex phenomenon caused by humans causing certain substances to enter the atmosphere during production activities or natural processes. Its formation and evolution have long-term sustainability, macroscopicity, and integrity. The concentration of each pollutant is high or low, and it will last for a period of time. So, what are the macro and overall characteristics of the self-evolution of the concentration of each pollutant? This article will use the detrended fluctuation analysis (DFA) method to analyze the long-term sustainability of each pollutant concentration sequence at nine sites. At the same time, in order to describe the nonlinear characteristics of each pollutant concentration sequence in more detail, use the multifractal detrended fluctuation analysis (MF-DFA) method to analyze the internal local structure. The MF-DFA method can describe the unique mode of the pollution process during the haze period, record the detailed information of the pollutants on different time scales during the haze period, provide probability estimates for the pollutant concentration, and display the pollutant concentration. The MF-DFA method can also describe the characteristics of time series in a more detailed, precise, and comprehensive manner and quantitatively describe the long-term sustainability of time series evolution. The experimental analysis results of the MF-DFA method on the concentration of each pollutant at nine monitoring points during the haze period have achieved extraordinary results.

Study on Multiple Fractal Analysis and Response Characteristics of Acoustic Emission Signals from Goaf Rock Bodies.

Based on the actual monitoring data of the acoustic emission (AE) ground pressure monitoring and positioning system, this paper introduces fractal theory and the multifractal detrended fluctuation analysis (MF-DFA) method to estimate the waveform multifractal spectrum of goaf rock acoustic emission signals and investigate multifractal time-varying response characteristics. The research results show that the wavelet hard thresholding method has the best noise reduction effect on the original signal, and the box counting dimension has a strong waveform identification effect. Before deformation damage occurs, fractal spectral width Δα shows an increase and then decrease and the fluctuation scale factor Δf(α) decreases and then increases. When damage occurs, fractal spectral width Δα decreases and then stabilizes and concentrates. Simultaneously, the fluctuation scale factor Δf(α) keeps decreasing until the lowest point, and then shows an increasing trend until it reaches a stable state. This study is of great significance to the stability evaluation and disaster early warning of mine goaf.

Classification of melanoma images using 2D multifractal detrended cross-correlation analysis

Skin cancer is a common human malignant tumor and melanoma is one of the most fatal diseases in skin cancer. There exists a high degree of visual similarity between melanoma and non-melanoma. In addition, the acquisition and labeling of skin cancer images need relevant medical knowledge, therefore, it is difficult to obtain natural images. These problems make it difficult to distinguish between melanoma and non-melanoma. How to extract the high-dimensional features of skin cancer images is the main problem to improve the classification performance of skin cancer images. For this purpose, we propose a new computer-aided classification system with a two-dimensional (2D) multifractal detrended fluctuation analysis (MF-DCCA) method and classifier combination. The proposed 2D MF-DCCA is expanded by the 2D MF-DFA and we aim to change the distribution of the generalized Hurst exponents calculated by the original 2D MF-DFA, making the image features represented by the generalized Hurst exponents calculated by 2D MF-DCCA are more significant. Therefore, we take the generalized Hurst exponents calculated by the two methods into the classifier and evaluate their classification performance. The classification metrics, such as Accuracy (Acc), Sensitivity (Sen), and Specificity (Spe) of two classifiers, show that the features extracted by MF-DCCA are better than those by MF-DFA. In addition, among the two classifiers such as the SVM and k-NN, the k-NN performs best in Acc and Spe with [Formula: see text] and [Formula: see text], respectively. The k-NN performs better than SVM in the classification Sen with [Formula: see text].

A novel Bayesian approach to estimate long memory parameter

The property of long memory and its parameter estimate are important in financial econometrics. This paper proposes a novel Bayesian approach to estimate the fractional differencing order and compares its performance with rescaled range analysis (R/S), detrended fluctuation analysis (DFA) and detrended moving average (DMA), based on the fractal Brownian motion sequence simulated by the ARFIMA model. The results indicate that the estimation accuracy of the Bayesian approach is significantly improved and is less volatile than R/S and DFA methods. We adopt mean absolute error and standard deviation as the criteria for measuring the finite sample performance of the estimators. To check robust results, four different methods (i.e. the Gelman–Rubin diagnostic, the Geweke diagnostic, the Heidelberger–Welch diagnostic, the Raftery–Lewis diagnostic) are simultaneously implemented to monitor the convergence of the posterior distribution in the Bayesian methodology. Therefore, the long memory estimator from the Bayesian model is preferred according to the results in this paper.

Multifractal analysis of social media use in financial markets.

We analyze the nonlinear properties of social media activity(SMA) using the multifractal detrended fluctuation analysis (MF-DFA) method. Social media data related to the stock market are gathered from social media platforms. Using data on over 2000 firms in the Korean stock market for 2018–2020, we study social media activity and its differences to evaluate associated nonlinear and statistical properties. We find that the cumulative distribution function of SMA follows a stretched exponential distribution with beta = 0.85. The Hurst exponent of SMA for three datasets (2018, 2019, 2020 year) is larger than 0.9, whereas the Hurst exponents of shuffled time series have values of approximately 0.5. In particular, we find a multifractal structure in both SMA and SMA difference results irrespective of the period and degree of multifractality defined as alpha _mathrm{max} - alpha _mathrm{min}, which reaches a maximum value during the COVID-19 pandemic as a financial crisis.

Multifractality during upside/downside trends in the MENA stock markets: the effects of the global financial crisis, oil crash and COVID-19 pandemic

Purpose This paper examines asymmetric multifractality (A-MF) in the leading Middle East and North Africa (MENA) stock markets under different turbulent periods (global financial crisis [GFC] and European sovereign debt crisis [ESDC], oil price crash and COVID-19 pandemic).Design/methodology/approach This study applies the asymmetric multifractal detrended fluctuation analysis (A-MF-DFA) method of Cao et al. (2013) to identify A-MF and MENA stock market efficiency during the COVID-19 pandemic.Findings The results show strong evidence of different patterns of MF during upward and downward trends. Inefficiency is higher during upward trends than during downward trends in most of the stock markets in the whole sample period, and the opposite is true during financial crises. The Turkish stock market is the least inefficient during upward and downward trends. A-MF intensifies with an increase in scales. The evolution of excessive A-MF for MENA stock returns is heterogeneous. Most of the stock markets are more inefficient during a pandemic crisis than during an oil crash and other financial crises. However, the inefficiency of the Saudi Arabia and Qatar stock markets is highly sensitive to oil price crashes. Overall, the level of inefficiency varies across market trends, scales and stock markets and over time. The findings of this study provide investors and policymakers with valuable insights into efficient investment strategies, risk management and financial stability.Originality/value This paper first explores A-MF in the MENA emerging stock markets. The A-MF analysis provides useful information to investors regarding asset allocation, portfolio risk management and investment strategies during bullish and bearish market states. In addition, this paper examines A-MF under different turbulent periods, such as the GFC, the ESDC, the 2014–2016 oil crash and the COVID-19 pandemic.

A study of cross-correlations between PM[formula omitted] and O3 based on Copula and Multifractal methods

As typical nonlinear dynamic system, the interactions between PM2.5 and O3 have complexity characteristics at different spatiotemporal scales. The deep understanding of the multi-scale cross-correlation between PM2.5 and O3 is helpful to the achievement of the target on two pollutants coordinated control in environmental management. Based on copula model and Multifractal Detrended Fluctuation Analysis (MFDFA) method, this paper studies the multi-scale cross-correlation between PM2.5 and O3 in three metropolises in China (Beijing, Shanghai and Guangzhou). Furthermore, a new multifractal index is established to evaluate the difficulty degree of PM2.5-O3 coordinated control based on enlarged window method. The hourly PM2.5 and O3 concentrations in these three metropolises from 1 January 2015 to 31 December 2018 are chosen as the research objects in order to analyze the effect of Air Pollution Prevention and Control Action Plan on air quality. The results show that PM2.5 and O3 have negative correlation in January and positive correlation in July. However, compared with Beijing and Shanghai, PM2.5 and O3 have upper tail dependence in Guangzhou in July. Furthermore, the strength of multifractality for PM2.5 is the weakest and that for O3 is the strongest in Guangzhou, which may be due to the special climatic conditions in Guangzhou. Then, the difficulty degree of PM2.5-O3 coordinated control in three metropolises is quantified based on new multifractal index. Moreover, it discusses the influence of meteorological factors on PM2.5-O3 coordinated control in different metropolises. The comparative analysis confirms the availability of the multifractal index. This index can evaluate the difficulty degree of PM2.5-O3 coordinated control of metropolis according to field observation, which reduces the influence of personal subjective factors. The new method maybe contribute to the new evaluate index of atmospheric environmental management.

Long-range correlations of the wind speed in a northeast region of Brazil

The objective of this work is to analyze the scaling behavior of wind speed in the region of the state of Bahia, northeastern Brazil, in search of long-range correlations and other information about the crossover phenomena. Thus, data from 41 automatic surface stations were used for a period of five years—between 2015 and 2020—for onshore reading. For offshore readings, data from a surface station located in the Abrolhos Archipelago were used. To achieve this goal, the DFA (detrended fluctuation analysis) technique was used in the analysis of measured data at the stations, along with numerical simulations using the WRF (weather research and forecasting) mesoscale model. The results of the analysis of hourly average wind speed from the measured and simulated data show the existence of scale behavior with the appearance, in most cases, of a double crossover—onshore and offshore. This suggests the phenomenon's dependence on the time period of the analyzed data, and also on the geographic location, showing a strong correlation with the Atlantic and Pacific oscillations (La Niña and El Niño), indicating the influence of local, mesoscale, and macroscale effects in the region of study. For the offshore case, the measured data and simulations presented a subdiffusive behavior (α≥1) before the first crossover, and persistence (0.5<α<1) for the other two scales. For the onshore case, the results showed different behaviors, with some stations and simulations showing subdiffusive behavior and others showing persistence before and after the first and second crossovers, but most showed persistence between the first and second crossovers. The results of the analysis of daily averages of station data and simulations confirm the existence of only one crossover as a reflection of global effects (macroscale), since local effects have a daily cycle (mesoscale). The DFA method does not detect fluctuations in the measured and simulated data at all locations—especially in regions with extreme slopes, or in lowland plains—as double crossover, and in a few cases even simple crossovers are not recorded. From a practical point of view, unlike other methods to detect wind persistence, the methodology considered in this study reveal the multifractality in the wind speed data, showing that fluctuations in wind speed can be dominated by atmospheric phenomena governed by a local or regional meteorological system, while fluctuations on long-range time scales in some locations can be influenced by global weather patterns, providing new perspectives about the best locations for wind energy.

Non-Linear Impact of Chinese Treasury Bond Futures on the Information Content of IRS

This paper investigates the impact of Chinese Treasury bond (CTB) futures on the information content of interest rate swap (IRS) from a multifractality perspective. We first use multifractal detrended fluctuation analysis (MF-DFA) method and show that the swap rate and the CTB yield exhibit strong multifractality. In addition, employing multifractal detrended cross-correlation analysis (MF-DCCA) method, we find that cross-correlations between the swap rate and the CTB yield are multifractally persistent. Moreover, after the reintroduction of Treasury bond futures, the persistence of cross-correlation between the series is weaker. Our results indicate that the information content of IRS decreased after the re-launch of CTB futures.