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Motivation The quantification of long-range correlation of electroencephalogram (EEG) signals is an important research direction for its relevance in helping understanding the brain activity. Epileptic seizures have been studied in the past years where different non-linear statistical approaches have been employed to understand the relationship between the EEG signal and the epileptic discharge. One of the most widely used method for to analyse long memory processes is the detrended fluctuation analysis (DFA). However, no objective and pragmatic methods have been developed to detect crossover points and reference channels in DFA. Results In this article, we propose: (i) two automatic approaches that successfully detect crossover points in DFA related methods on the log–log plot and (ii) a criteria to choose the reference channel for the log-amplitude function. Moreover, the DFA is applied to EEG signals of 10 epileptic patients collected from the CHB-MIT database, being the log-amplitude function used to compare the different brain hemispheres by making use of the methodology proposed in the article. The existence of long-range power-law correlations is demonstrated and indicates that the EEG signals of epileptic patients present three well-defined regions with the first region showing a 1/f noise (pink noise) for seven subjects and a random walk behaviour for three subjects. The second and third regions show anti-persistence behaviour. Moreover, the results of the log-amplitude function were divided in two groups: the first, including seven subjects, where the increase in the scales results in an increase in the fluctuation in the frontal channels and the second, included three subjects, where the fluctuation for large scales are greater for the parietal channels. Availability and implementation The functions used in this article are available in the R package DFA (Mesquita et al., 2020). Supplementary information Supplementary data are available at Bioinformatics online.

Electrocardiogram (ECG) is a critical biological signal, which usually carries a great deal of essential information about patients. The high quality ECG signals are always required for a proper diagnosis of cardiac disorders. However, the raw ECG signals are highly noisy in nature. In the paper, we propose a hybrid denoising scheme to enhance ECG signals by combining high-order synchrosqueezing transform (FSSTH) with non-local means (NLM). With this method, a noisy ECG signal is first decomposed into an ensemble of intrinsic mode functions (IMFs) by FSSTH. Then, some noise is removed by eliminating a set of noisy IMFs that are determined by a scaling exponent obtained by the detrended fluctuation analysis (DFA); while the remaining IMFs are filtered by NLM. Finally, the denoised ECG signal is obtained by reconstructing the processed IMFs. Experiments are carried out using the simulated ECG signals and real ones from the MIT-BIH database, and the denoising performances are evaluated in terms of signal to noise ratio (SNR), root mean squared error (RMSE) and percent root mean square difference (PRD). Results show that the hybrid denoising scheme involving both FSSTH and NLM is able to suppress complex noise from ECG signals more effectively while preserving the details well.

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The process of reading a specific text is considered complex and little known in neuroscience, since it involves the vision, memory, motor control, learning, among others. In this sense, an excellent possibility to study the brain activity in the reading task can be achieved by the analysis of the multi-channel Electroencephalogram (EEG) and also with new statistical methods, like the detrended fluctuation analysis method (DFA). In this paper it will be proposed a model to analyze the brain activity in the reading task, performed by two subjects using a 22-channels EEG (NEUROMAP® model E Q S A 260 ). In order to test our model, two adults subjects (graduates) were tested here. These subjects were arranged in a chair facing a panel with the specific text, excluding involuntary movements that activated regions of the brain that were not being stimulated by reading. For the first subject, chosen at random, the text was presented before the task for understanding and some memorization. For the other subject the text was presented at the time of task. For the signal processing we chose 11 bio-electrodes located at the frontal, parietal, temporal and occipital regions of the brain. Therefore, to treat these non-stationary bio-signals we must apply robust and modern statistical techniques. With this objective, DFA method was applied in order to analyze the F D F A ( n ) fluctuation function in multi-channel EEG bio-sensors, more specifically the difference of its logarithm, i.e., Δ l o g F D F A . The results show that the use of this new function can be useful for brain activities. This paper, as we shall see here, is an initial contribution for EEG data analyze, that would be of medical interest, mainly in neuroscience area.

Among the sports fans beliefs about "hot hands" and "winning streaks" are widely spread, while the scientific debate about these effects is still ongoing. Recently in a paper by P. Ferreira [Physica A 500: 92-96] detrended fluctuation analysis was applied to the NBA teams' win records. It was shown that 28 considered NBA teams exhibit persistence in the win record time series. In this paper we take the same data set and compare the obtained results against various random models. We find that the empirical results are consistent with the results obtained from various simple random models., Comment: 10 pages, 4 figures, 3 tables

By employing Detrended Fluctuation Analysis (DFA), which has been established as a standard method to investigate long range correlations in nonstationary time series, we study the temporal correlations between the magnitudes of the earthquakes that occurred before the recent deadly Mexico M8.2 earthquake on 7 September 2017 in Chiapas region. Our aim here is to shed light on the origin of the following precursory phenomenon found in our previous publication (Sarlis et al., 2018): Upon considering the analysis of seismicity in the new time domain termed natural time, it was shown that the entropy change of seismicity under time reversal exhibited an important minimum almost 3 months before this major earthquake. Here the application of DFA to earthquake magnitude time series reveals that this minimum of the entropy change of seismicity is preceded as well as followed by characteristic changes of temporal correlations between earthquake magnitudes, which are quantified by the DFA exponent α . In particular, before this minimum the long range correlations breakdown to an almost random behavior possibly turning to anticorrelation ( α ≤ 0 . 5 ) while after the minimum (and before the major M8.2 earthquake) long range correlations develop with an exponent α between 0.6 and 0.7.

This paper adopts multifractal methods to analyze the multifractal characteristics of polluted time series in Beijing, Zhengzhou, and Jinan. The results of the multifractal detrended fluctuation analysis (MF-DFA) show that each of three air quality index (AQI) time series has multifractal characteristics, which provides evidence that the multifractal method but the traditional linear or monofractal method is feasible. The sources of multifractality in three AQI time series are all from long-range correlations and fat tail distributions, although the highlighted sources are different in three cities. The results of the coupling detrended fluctuation analysis (CDFA) show that SO2 has a vital impact on air quality in each of three cities. O 3 has an important impact on air quality in Beijing and Jinan city and NO2 has an important impact on air quality in Zhengzhou city. It is noted that PM 2.5 has not as large an impact as we imagine in three cities. It has a relatively large impact on air quality of Zhengzhou city, while it has a small impact on air quality of Beijing and Jinan city. The chi square χ 2 test further confirms the above results.

An experimental investigation on the self-excited spiky oscillations in a glow discharge plasma device is presented. The system showed a continuous transition from chaotic bursting to chaotic spiking states with the increase in the applied discharge voltage at a particular pressure. Below a threshold discharge voltage, the system exhibits a fixed point dynamical behaviour and with the increase in the discharge voltage bursting type oscillations appear which are identified as “fold/fold circle” type. The occurrence of the relaxation type spiky oscillations is attributed to the formation and disruption of anode double layer. The burst phases occurred intermittently and the frequency of their occurrence increased with increase in the discharge voltage and finally the oscillation turned into spiking behaviour. The intermittent oscillations manifested the scaling law of the average laminar phase with respect to the change in the discharge voltage. The scaling law of the intermittent oscillations was similar to type III intermittency. During the transition, the system changes its behaviour from long range correlation to long range anti correlation. Detrended fluctuation analysis technique has been used to identify the nature of long range correlations. We found that bursting oscillations show scale invariant, long-range correlation behaviour whereas spiking oscillations show scale invariant, long range anti correlation behaviour.

Some physiological series, like the cardiovascular signals, show multifractal structures that depend on the temporal scale. Thus, their fractal nature is better assessed by a Detrended Fluctuation Analysis (DFA) approach that provides multifractal coefficients scale by scale. Our aim is to compare two estimators of the degree of scale-by-scale multifractality based on the width of the singularity spectrum or on the statistical dispersion of the DFA coefficients. We synthesized 1000 series of white noise (monofractal and monoscale), of autoregressive noise (monofractal and multiscale) and of Cauchy-distributed noise (multifractal and monoscale) comparing the two estimators at scales between 8 and 228 samples. We found that the two estimators provide similar scale-by-scale profiles of multifractality. However, the statistical-dispersion estimator better distinguishes multifractal from monofractal noises at all the scales, thus appearing more suitable than the singularity-spectrum width to describe the fractal structure of physiological time series.

Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter.

Music elicits a wide range of effects on human physiology. The diverse effects of musical stimuli on cardiovascular and autonomic nervous-system dynamics are characterized by the analysis of Electrocardiography (ECG) signals. ECG reflects the electrical activity of the heart. Studies suggest that human heart and ECG is non-linear and chaotic in nature, therefore needs to be studied using non-linear methods. In this paper, monofractal Detrended Fluctuation Analysis or DFA method will be first used, followed by subsequent multifractal method - Multifractal Detrended Fluctuation Analysis (MFDFA). Using these methods, we will be studying the non-linear characteristics of ECG while listening to Indian Classical Music. Very few studies have been performed to characterize how Indian Classical Music affects the non-linear nature of ECG signals using multifractal tools. In our experiment, the presence of long-range scale-invariant structure in ECG signals is initially studied using DFA. The scaling behaviour of ECG signals is characterized by more than one scaling exponent, suggesting the multifractal nature of ECG. MFDFA is performed to study the multifractal nature of ECG. Width of multifractal spectrum while listening to music is compared with resting condition. Results show that the range of multifractality increases while listening to music.

Recently, a multifractal-multiscale approach to detrended fluctuation analysis (DFA) was proposed to evaluate the cardiovascular fractal dynamics providing a surface of self-similarity coefficients &alpha, (q,&tau, ), function of the scale &tau, and moment order q. We hypothesize that this versatile DFA approach may reflect the cardiocirculatory adaptations in complexity and nonlinearity occurring during the day/night cycle. Our aim is, therefore, to quantify how &alpha, (q, &tau, ) surfaces of cardiovascular series differ between daytime and night-time. We estimated &alpha, ) with &minus, 5 &le, q &le, 5 and 8 &le, &tau, &le, 2048 s for heart rate and blood pressure beat-to-beat series over periods of few hours during daytime wake and night-time sleep in 14 healthy participants. From the &alpha, ) surfaces, we estimated short-term (&lt, 16 s) and long-term (from 16 to 512 s) multifractal coefficients. Generating phase-shuffled surrogate series, we evaluated short-term and long-term indices of nonlinearity for each q. We found a long-term night/day modulation of &alpha, ) between 128 and 256 s affecting heart rate and blood pressure similarly, and multifractal short-term modulations at q &lt, 0 for the heart rate and at q &gt, 0 for the blood pressure. Consistent nonlinearity appeared at the shorter scales at night excluding q = 2. Long-term circadian modulations of the heart rate DFA were previously associated with the cardiac vulnerability period and our results may improve the risk stratification indicating the more relevant &alpha, ) area reflecting this rhythm. Furthermore, nonlinear components in the nocturnal &alpha, ) at q &ne, 2 suggest that DFA may effectively integrate the linear spectral information with complexity-domain information, possibly improving the monitoring of cardiac interventions and protocols of rehabilitation medicine.

The possibility of distinguishing between different types of complex oscillations using datasets contaminated with measurement noise is studied based on multiresolution wavelet analysis (MWA). Unlike the conventional approach, which characterizes the differences in terms of standard deviations of detail wavelet coefficients at independent resolution levels, we consider ways to improve the separation between complex motions by applying several measures for the decomposition coefficients. We show that MWA’s capabilities in diagnosing dynamics can be expanded by applying detrended fluctuation analysis (DFA) to sets of detail wavelet coefficients or by computing the excess of the probability density function of these sets.

Based on the detrended fluctuation analysis (DFA), we address the problem of recognizing the electrical activity of the brain associated with movements of the right arm and the imagination of this procedure. In experiments with untrained volunteers, we show the possibility to distinguish between the related EEG patterns. We also demonstrate significant distinctions in the scaling features between background records and EEG signals associated with mental intentions that can be transformed to control commands for brain–computer interfaces. This effect is observed in many EEG channels, although individual features of subjects provided various distributions of brain areas with the most prominent differences.

Investigating the dynamics of long-term memory in online rating behaviors is significant for understanding the evolution mechanism of collective behaviors and trust formation for online social networks. Since users are allowed to deliver ratings in many online systems, ratings can well reflect the user’s opinions. In this paper, we empirically investigate the long-term memory, measured by the Detrended Fluctuation Analysis, in collective rating behaviors before and after the trust formation. The results for the Epinions data set show that, comparing with the null model generated by the reshuffle process, the Hurst exponent of trustors (trustees) decreases 7.12% (9.05%) before and increases 7.36% (9.20%) after trust formation, which stably remains close to 0.5 in null model I and 0.6 in null model II, suggesting that the collective rating behavior plays an important role for the trust formation. Furthermore, we divide users into 8 groups according to the user degree and find that the correlation of the user degree and the variation of Hurst exponent, measured by the Pearson Correlation Coefficient, is 0.8629 and 0.8620 before and after trust formation respectively, reflecting a significant correlation between user degrees and collective rating behavior patterns. Finally, we select the users without creating other trust relations around the trust formation time and the result suggests that the collective rating behaviors indeed change for the trust formation. This work helps deeply understand the intrinsic feedback effects between collective behaviors and trust relationship.

The purpose of this study was to explore the extent of multifractality in unperturbed and constrained locomotion, and to determine if multifractality predicted gait adaptability. Young, healthy participants (n = 15) walked at preferred and slow speeds, as well as asymmetrically (one leg at half speed) on a split-belt treadmill. Stride time multifractality was assessed via local detrended fluctuation analysis, which evaluates the evolution of fluctuations both spatially and temporally. Unperturbed walking exhibited monofractal behavior. Asymmetric walking displayed greater multifractality in the faster moving limb, indicating more intermittent periods of extreme high or low variance. Multifractality was not associated with adaptation to asymmetric walking. These findings further suggest that unperturbed locomotion is monofractal and establish that perturbed walking yields multifractal behavior.

Detrended fluctuation analysis (DFA) is a standard method to evaluate long-range correlations embedded in non-stationary time series. To obtain a reliable estimation of scaling behavior, it requires the length of a time series is long enough (at least ∼ 10 , 000 ), which is not always the case in reality. How to evaluate long-range correlation behavior in a very short time series is still an open problem. In the present paper, we propose an improvement of DFA by correcting the bias in estimation of variance, called Unbiased Detrended Fluctuation Analysis (UDFA). Extensive calculations show its high-performance. For instance, from a fractional Brownian motion (fBm) series with length 500 the estimated long-range correlation exponent has negligible bias and acceptable confidence region (standard deviation less than 0.05). As a typical example, the proposed method is used to monitor evolution of fractal gait rhythm of a volunteer. Rich patterns are found in the evolutionary process.

This study empirically compares the development of China’s interest rate market in the past decade. The yield of Shibor market from 2006 to 2016 showed anti-persistence, and the later five-year trend was stronger than that in the previous half period, indicating a significantly growing trend of China’s interest rate market. We perform a comparative analysis of detrended fluctuation analysis (DFA), detrending moving average (DMA) algorithm, and the maximum overlap wavelet transform (DFA-MODWT) using grey and original sequences. The autoregressive integrated moving average model is adopted to generate set-point H sequences. Results illustrated that 1) for original sequences, both the DFA-MODWT and DMA ( θ = 0 . 5 ) shows outstanding performances when the set Hurst exponents are 0.6 and 0.7 and 0.8 and 0.9, respectively. 2) For grey sequences, G-DFA performs best among the three methods, followed by G-DMA and G-DFA-MODWT. 3) The long-range correlation of the original sequences is slightly lower than the set values. On the contrary, the long-range correlation of grey sequences is much higher than the set values. Thus, the grey sequence exerts a strong aggregation effect as it can accumulate the trend of the sequence. Our results show that the grey sequence can determine the regular pattern from the messy sequence. However, grey sequences tend to be overly analyzed and misleading when large volumes of data are involved. Empirical studies confirm that China’s interest rate market remains an inefficient market.

Although the analysis of dependence in financial markets started a century ago, there is still room for new work, both because statistical methods continue to de developed, allowing stronger and more robust analysis, and because more and more data is available. In this context, we propose to make a deep analysis of the Swiss stock market, one of the most important financial centres in the world, studying the main index and also 19 of its 20 components. We use detrended fluctuation analysis, which allows us to analyse the existence of long-term dependence in a given variable. As our objective is to analyse the evolution of that dependence over time, we use a sliding windows approach. The results show that several of the analysed stocks have a behaviour which is not consistent with the absence of dependence, which could be informative for actual and potential investors.

Autoregressive models are commonly used for modeling time-series from nature, economics and finance. This work explored simple autoregressive AR ( p ) models to remove long-term trends in detrended fluctuation analysis (DFA). Crude oil prices and bitcoin exchange rate were considered, with the former corresponding to a mature market and the latter to an emergent market. Results showed that AR ( p ) -based DFA performs similar to traditional DFA. However, the former DFA provides information on stability of long-term trends, which is valuable for understanding and quantifying the dynamics of complex time series from financial systems.

The wind speed persistence is an important factor in the assessment of wind energy potential. In this paper, we explore the persistence of Mean Wind Speed (MWS) with many years of record using Detrended Fluctuation Analysis (DFA) over China. The results illustrate that there exist irregular high-frequency fluctuations for daily MWS anomaly records. Long-term persistence of MWS is found for all meteorological observed sites. We also make some numerical tests in order to verify the significance of long-term persistence by shuffling the data records many times. These facts prove that the MWS anomaly records have long-term persistence over all the stations in China. The mean value 0.64 in DFA-exponents for all stations over China is also obviously higher than the value 0.53 according to interval threshold of 95% confidence level after shuffling the MWS records many times. In addition, the values of scaling exponent vary from station to station over China. Long-term persistence of MWS in spatial distributions seems to be downward trends from east to west China. Many factors may affect long-term persistence of MWS such as southwest monsoon, Tibetan Plateau landform and atmosphere–ocean–land interaction and so on. Possible physical mechanism need further analysis in the future.

Basketball is one of the favourite sports in the United States and NBA the most popular basketball league in the world, followed not only by Americans but by fans everywhere. At present, people are increasingly interested in betting on sports in general and on basketball in particular. So analysis of trends in the results could be important for gamblers. In this paper, detrended fluctuation analysis is applied to 28 NBA teams, in order to understand if their results have (or not) any kind of memory. Our results show that all the teams are persistent in their results. Nevertheless, some teams have higher persistence than others, which could be important for gamblers in deciding how to bet.

Detrended fluctuation analysis (DFA) is an effective method to accurately quantify long-term correlations embedded in a nonstationary time series. In this paper, we study the effect of signal filtering of a signal on the DFA method. The theoretical and simulated results show that the signal filtering will affect the range of scale in DFA. Moreover, this effect is different for fractal Gaussian noise series and fractal Brown movement series. Our study is meaningful for improving accuracy and efficiency of DFA method in theory and practice.

We study auto-correlations and cross-correlations of IBovespa index and its constituent companies. We use Detrended Fluctuation Analysis (DFA) to quantify auto-correlations and Detrended Cross-Correlation Analysis (DCCA) to quantify cross-correlations in absolute returns of daily closing prices of IBovespa and the individual companies. We find persistent long-term correlations and cross-correlations which are weaker than those found for USA market. Our results indicate that market indices of developing markets exhibit weaker coupling with its constituents than for mature developed markets.

This work studies long-range correlations and informational efficiency of the Bitcoin market for the period from June 30, 2013 to June 3rd, 2017. To this end, the detrended fluctuation analysis (DFA) was implemented over sliding windows to estimate long-range correlations for price returns. It was found that the Bitcoin market exhibits periods of efficiency alternating with periods where the price dynamics are driven by anti-persistence. The pattern is replicated by prices samples at day, hour and second frequencies. The Bitcoin market also presents asymmetric correlations with respect to increasing and decreasing price trending, with the former trend linked to anti-persistence of returns dynamics.

The purpose of this paper is to propose a new estimator of Hurst exponent based on the combined information of the conventional rescaled range methods. We demonstrate the superiority of the proposed estimator by Monte Carlo simulations, and the applications in estimating the Hurst exponent of daily volatility series in Chinese stock market. Moreover, we indicate the impact of the type of estimator and structural break on the estimating results of Hurst exponent.

Heart rate asymmetry is a phenomenon by which the accelerations and decelerations of heart rate behave differently, and this difference is consistent and unidirectional, i.e. in most of the analyzed recordings the inequalities have the same directions. So far, it has been established for variance and runs based types of descriptors of R R intervals time series. In this paper we apply the newly developed method of Asymmetric Detrended Fluctuation Analysis, which so far has mainly been used with economic time series, to the set of 420 stationary 30 min time series of R R intervals from young, healthy individuals aged between 20 and 40. This asymmetric approach introduces separate scaling exponents for rising and falling trends. We systematically study the presence of asymmetry in both global and local versions of this method. In this study global means “applying to the whole time series” and local means “applying to windows jumping along the recording”. It is found that the correlation structure of the fluctuations left over after detrending in physiological time series shows strong asymmetric features in both magnitude, with α + α − , where α + is related to heart rate decelerations and α − to heart rate accelerations, and the proportion of the signal in which the above inequality holds. A very similar effect is observed if asymmetric noise is added to a symmetric self-affine function. No such phenomena are observed in the same physiological data after shuffling or with a group of symmetric synthetic time series.

This paper presents the parametric modified limited penetrable visibility graph (PMLPVG) algorithm for constructing complex networks from time series. We modify the penetrable visibility criterion of limited penetrable visibility graph (LPVG) in order to improve the rationality of the original penetrable visibility and preserve the dynamic characteristics of the time series. The addition of view angle provides a new approach to characterize the dynamic structure of the time series that is invisible in the previous algorithm. The reliability of the PMLPVG algorithm is verified by applying it to three types of artificial data as well as the actual data of natural gas prices in different regions. The empirical results indicate that PMLPVG algorithm can distinguish the different time series from each other. Meanwhile, the analysis results of natural gas prices data using PMLPVG are consistent with the detrended fluctuation analysis (DFA). The results imply that the PMLPVG algorithm may be a reasonable and significant tool for identifying various time series in different fields.

Long-range temporal dependence is an important research perspective for modelling of traffic flow time series. Various methods have been proposed to depict the long-range temporal dependence, including autocorrelation function analysis, spectral analysis and fractal analysis. However, few researches have studied the daily temporal dependence (i.e. the similarity between different daily traffic flow time series), which can help us better understand the long-range temporal dependence, such as the origin of crossover phenomenon. Moreover, considering both types of dependence contributes to establishing more accurate model and depicting the properties of traffic flow time series. In this paper, we study the properties of daily temporal dependence by simple average method and Principal Component Analysis (PCA) based method. Meanwhile, we also study the long-range temporal dependence by Detrended Fluctuation Analysis (DFA) and Multifractal Detrended Fluctuation Analysis (MFDFA). The results show that both the daily and long-range temporal dependence exert considerable influence on the traffic flow series. The DFA results reveal that the daily temporal dependence creates crossover phenomenon when estimating the Hurst exponent which depicts the long-range temporal dependence. Furthermore, through the comparison of the DFA test, PCA-based method turns out to be a better method to extract the daily temporal dependence especially when the difference between days is significant.

Detrended Fluctuation Analysis (DFA) has become a widely-used tool to examine the correlation structure of a time series and provided insights into neuromuscular health and disease states. As the popularity of utilizing DFA in the human behavioral sciences has grown, understanding its limitations and how to properly determine parameters is becoming increasingly important. DFA examines the correlation structure of variability in a time series by computing α , the slope of the log S D – log n diffusion plot. When using the traditional DFA algorithm, the timescales, n , are often selected as a set of integers between a minimum and maximum length based on the number of data points in the time series. This produces non-uniformly distributed values of n in logarithmic scale, which influences the estimation of α due to a disproportionate weighting of the long-timescale regions of the diffusion plot. Recently, the evenly spaced DFA and evenly spaced average DFA algorithms were introduced. Both algorithms compute α by selecting k points for the diffusion plot based on the minimum and maximum timescales of interest and improve the consistency of α estimates for simulated fractional Gaussian noise and fractional Brownian motion time series. Two issues that remain unaddressed are (1) how to select k and (2) whether the evenly-spaced DFA algorithms show similar benefits when assessing human behavioral data. We manipulated k and examined its effects on the accuracy, consistency, and confidence limits of α in simulated and experimental time series. We demonstrate that the accuracy and consistency of α are relatively unaffected by the selection of k . However, the confidence limits of α narrow as k increases, dramatically reducing measurement uncertainty for single trials. We provide guidelines for selecting k and discuss potential uses of the evenly spaced DFA algorithms when assessing human behavioral data.

In this paper, a generalized method of detrended fluctuation analysis (DFA) is proposed as a new measure to assess the complexity of a complex dynamical system such as stock market. We extend DFA and local scaling DFA to higher moments such as skewness and kurtosis (labeled SMDFA and KMDFA), so as to investigate the volatility scaling property of financial time series. Simulations are conducted over synthetic and financial data for providing the comparative study. We further report the results of volatility behaviors in three American countries, three Chinese and three European stock markets by using DFA and LSDFA method based on higher moments. They demonstrate the dynamics behaviors of time series in different aspects, which can quantify the changes of complexity for stock market data and provide us with more meaningful information than single exponent. And the results reveal some higher moments volatility and higher moments multiscale volatility details that cannot be obtained using the traditional DFA method.

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Brain electrical activity recorded as electroencephalogram data provides relevant information that can contribute to a better understanding of pathologies and human behaviour. This study explores extant electroencephalogram (EEG) signals in search of patterns that could differentiate subjects undertaking mental tasks and reveals insights on said data. We estimated the power spectral density of the signals and found that the subjects showed stronger gamma brain waves during activity while presenting alpha waves at rest. We also found that subjects who performed better in those tasks seemed to present less power density in high-frequency ranges, which could imply decreased brain activity during tasks. In a time-domain analysis, we used Hall–Wood and Robust–Genton estimators along with the Hurst exponent by means of a detrented fluctuation analysis and found that the first two fractal measures are capable of better differentiating signals between the rest and activity datasets. The statistical results indicated that the brain region corresponding to Fp channels might be more suitable for analysing EEG data from patients conducting arithmetic tasks. In summary, both frequency- and time-based methods employed in the study provided useful insights and should be preferably used together in EEG analysis.

In this paper, we have modified the Detrended Fluctuation Analysis (DFA) using the ternary Cantor set. We propose a modification of the DFA algorithm, Cantor DFA (CDFA), which uses the Cantor set theory of base 3 as a scale for segment sizes in the DFA algorithm. An investigation of the phenomena generated from the proof using real-world time series based on the theory of the Cantor set is also conducted. This new approach helps reduce the overestimation problem of the Hurst exponent of DFA by comparing it with its inverse relationship with α of the Truncated Lévy Flight (TLF). CDFA is also able to correctly predict the memory behavior of time series.

In this paper, we analyzed price–volume multifractal cross-correlations of Moroccan Stock Exchange. We chose the period from January 1st 2000 to January 20th 2017 to investigate the multifractal behavior of price change and volume change series. Then, we used multifractal detrended cross-correlations analysis method (MF-DCCA) and multifractal detrended fluctuation analysis (MF-DFA) to analyze the series. We computed bivariate generalized Hurst exponent, Renyi exponent and spectrum of singularity for each pair of indices to measure quantitatively cross-correlations. Furthermore, we used detrended cross-correlations coefficient (DCCA) and cross-correlation test ( Q ( m ) ) to analyze cross-correlation quantitatively and qualitatively. By analyzing results, we found existence of price–volume multifractal cross-correlations. The spectrum width has a strong multifractal cross-correlation. We remarked that volume change series is anti-persistent when we analyzed the generalized Hurst exponent for all moments q . The cross-correlation test showed the presence of a significant cross-correlation. However, DCCA coefficient had a small positive value, which means that the level of correlation is not very significant. Finally, we analyzed sources of multifractality and their degree of contribution in the series.

The aim of this study is to investigate the scaling exponent properties of mountainous river flow fluctuations by detrended fluctuation analysis (DFA). Streamflow data were collected continuously using Fluvial Acoustic Tomography System (FATS), which is a novel system for measuring continuous streamflow at high-frequency scales. The results revealed that river discharge fluctuations have two scaling regimes and scaling break. In contrast to the Ranting Curve method (RC), the small-scale exponent detected by the FATS is estimated to be 1.02 ± 0.42% less than that estimated by RC. More importantly, the crossover times evaluated from the FATS delayed approximately by 42 ± 21 hr ≈ 2–3 days than their counterparts estimated by RC. The power spectral density analysis assists our findings. We found that scaling characteristics information evaluated for a river using flux data obtained by RC approach might not be accurately detected, because this classical method assumes that flow in river is steady and depends on constructing a relationship between discharge and water level, while the discharge obtained by the FATS decomposes velocity and depth into two ratings according to the continuity equation. Generally, this work highlights the performance of FATS as a powerful and effective approach for continuous streamflow measurements at high-frequency levels.

The heart rate variability (HRV) is an important physiological signal used either to assess the risk of cardiac death or to model the cardiovascular regulatory dynamics. Asymmetries in HRV data have been observed using 2D Poincare plots, which have been linked to a non-equilibrium operation of the cardiac autonomic system. This work further explores the presence of asymmetries but in the serial correlations of the dynamics of HRV data. To this end, detrended fluctuation analysis (DFA) was used to estimate the Hurst exponent both when the heart rate is accelerating and when it is decelerating. The analysis is conducted using data collected from subjects under normal sinus rhythm (NSR), congestive heart failure (CHF) and atrial fibrillation (AF) . For the NSR cases, it was found that correlations are stronger ( p 0 . 05 ) when the heart rate is accelerating than when it is decelerating over different scales in the range 20–40 beats. In contrast, the opposite behavior was detected for the CHF and AF patients. Possible links between asymmetric correlations in the dynamics and the mechanisms controlling the operation of the heart rate are discussed, as well as their implications for modeling the cardiovascular regulatory dynamics.

We investigate the self-affinity of a fluid velocity time series in turbulent regime. We use the CFD OpenFoam library to simulate the flow of fluid in a duct with two obstacles injecting vorticity downstream. We calculated the fluid velocity in three sections in of the structure. The simulation presented more turbulences in sections further away from obstacles. We assess the self-affinity of properties of those sections by using the Detrended Fluctuation Analysis method. This scaling method presented two behaviors. First, a persistent behavior after passing through obstacles. Second, a sub-diffusive behavior in turbulent sections, presenting a long-range correlation α exponent ranging between 1.0 and 1.5. The findings suggest that chaotic states tend to follow a sub-diffusive process.

The purpose of this study is to examine existence of fractality and chaos in returns and volatilities of family business companies listed on the Casablanca Stock Exchange (CSE) in Morocco, and also in returns and volatility of the CSE market index. Detrended fluctuation analysis based Hurst exponent and fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) model are used to quantify fractality in returns and volatility time series respectively. Besides, the largest Lyapunov exponent is employed to quantify chaos in both time series. The empirical results from sixteen family business companies follow. For return series, fractality analysis show that most of family business returns listed on CSE exhibit anti-persistent dynamics, whilst market returns have persistent dynamics. Besides, chaos tests show that business family stock returns are not chaotic while market returns exhibit evidence of chaotic behaviour. For volatility series, fractality analysis shows that most of family business stocks and market index exhibit long memory in volatility. Furthermore, results from chaos tests show that volatility of family business returns is not chaotic, whilst volatility of market index is chaotic. These results may help understanding irregularities patterns in Moroccan family business stock returns and volatility, and how they are different from market dynamics.

Mexico City is a megalopolis with severe pollution problems caused by vehicles and industrial activity. This condition imposes important risks to human health and economic activity. Based on hourly-sampled data during the last decade, in a recent work (Meraz et al., 2015) we showed that the pollutant dynamics in Mexico City exhibits long-term and scale-dependent persistence effects resulting from the combination of pollutants generation by vehicles and removal by advection mechanisms. In this work, we analyzed the dynamics of ozone, a key component reflecting the degree of atmospheric contamination, to determine if its long-term correlations are asymmetric in relation to the actual concentration trend (increasing or decreasing). The analysis is conducted with detrended fluctuation analysis. The results showed that the average ozone dynamics is uncorrelated when the concentration is increasing. In contrast, the ozone dynamics shows long-term anti-persistence effects when the concentration is decreasing.

This paper uses the coupling detrended fluctuation analysis (CDFA) method to investigate the multifractal characteristics of four Asian stock markets using three stock indices: stock price returns, trading volumes and the composite index. The results show that coupled correlations exist among the four stock markets and the coupled correlations have multifractal characteristics. We then use the chi square ( χ 2 ) test to identify the sources of multifractality. For the different stock indices, the contributions of a single series to multifractality are different. In other words, the contributions of each country to coupled correlations are different. The comparative analysis shows that the research on the combine effect of stock price returns and trading volumes may be more comprehensive than on an individual index. By comparing the strength of multifractality for original data with the residual errors of the vector autoregression (VAR) model, we find that the VAR model could not be used to describe the dynamics of the coupled correlations among four financial time series.

There are a number of situations in which several signals are simultaneously recorded in complex systems, which exhibit long-term power-law cross-correlations. This paper presents two-dimensional multifractal cross-correlation analysis based on the partition function (2D-MFXPF), two-dimensional multifractal cross-correlation analysis based on the detrended fluctuation analysis (2D-MFXDFA) and two-dimensional multifractal cross-correlation analysis based on the detrended moving average analysis (2D-MFXDMA). We apply these methods to pairs of two-dimensional multiplicative cascades (2D-MC) to do a comparative study. Then, we apply the two-dimensional multifractal cross-correlation analysis based on the detrended fluctuation analysis (2D-MFXDFA) to real images and unveil intriguing multifractality in the cross correlations of the material structures. At last, we give the main conclusions and provide a valuable reference on how to choose the multifractal algorithms in the potential applications in the field of SAR image classification and detection.

A new method of fractal analysis of time series for estimating the chaoticity of behaviour of open stochastic dynamical systems is developed. The method is a modification of the conventional detrended fluctuation analysis (DFA) technique. We start from analysing both methods from the physical point of view and demonstrate the difference between them which results in a higher accuracy of the new method compared to the conventional DFA. Then, applying the developed method to estimate the measure of chaoticity of a real dynamical system — the Earth’s crust, we reveal that the latter exhibits two distinct mechanisms of transition to a critical state: while the first mechanism has already been known due to numerous studies of other dynamical systems, the second one is new and has not previously been described. Using GPS time series, we demonstrate efficiency of the developed method in identification of critical states of the Earth’s crust. Finally we employ the method to solve a practically important task: we show how the developed measure of chaoticity can be used for early detection of disastrous seismic events and provide a detailed discussion of the numerical results, which are shown to be consistent with outcomes of other researches on the topic.

The primary purpose of this study was to find Bayesian estimates for the Hurst dimension of a Fractional Brownian motion with a Beta prior when the process is observed at discrete times. Overestimation is observed though the overestimation is less severe as real H goes up. In addition, the estimated H decreases as Beta parameters go up given an Alpha value. In contrast, the estimated H increases as Alpha parameters go up given a Beta value. For the real-world data, the 2011 daily Taiwan stock index was used and the estimated Hurst index was 0.21.

Many complex systems generate multifractal time series which are long-range cross-correlated. This paper introduces three multifractal cross-correlation analysis methods, such as multifractal cross-correlation analysis based on the partition function approach (MFXPF), multifractal detrended cross-correlation analysis (MFDCCA) methods based on detrended fluctuation analysis (MFXDFA) and detrended moving average analysis (MFXDMA), which only consider one moment order. We do comparative analysis of the artificial time series (binomial multiplicative cascades and Cantor sets with different probabilities) by these methods. Then we do a feasibility test of the fixed threshold target detection within sea clutter by applying the multifractal cross-correlation analysis methods to the IPIX radar sea clutter data. The results show that it is feasible to use the method of the fixed threshold based on the multifractal feature parameter Δ f ( α ) by the MFXPF and MFXDFA-1 methods. At last, we give the main conclusions and provide a valuable reference on how to choose the multifractal algorithms, the detection parameters and the target detection methods within sea clutter in practice.

Fractal investigation of a signal often involves estimating its fractal dimension or Hurst exponent H when considered as a sample of a fractional process. Fractional Gaussian noise (fGn) belongs to the family of self-similar fractional processes and it is dependent on parameter H. There are variety of traditional methods for Hurst exponent estimation. Our novel approach is based on zero-crossing principle and signal segmentation. Thanks to the Bayesian analysis, we present a new axiomatically based procedure of determining the expected value of Hurst exponent together with its standard deviation and credible intervals. The statistical characteristics are calculated at the interval level at first and then they are used for the deduction of the aggregate estimate. The methodology is subsequently used for the EEG signal analysis of patients suffering from Alzheimer disease.

Interaction patterns among different warehouses could make the warehouse-out behavioral sequences less predictable. We firstly take a coupling detrended fluctuation analysis on the warehouse-out quantity, and find that the multivariate sequences exhibit significant coupling multifractal characteristics regardless of the types of steel products. Secondly, we track the sources of multifractal warehouse-out sequences by shuffling and surrogating original ones, and we find that fat-tail distribution contributes more to multifractal features than the long-term memory, regardless of types of steel products. From perspective of warehouse contribution, some warehouses steadily contribute more to multifractal than other warehouses. Finally, based on multiscale multifractal analysis, we propose Hurst surface structure to investigate coupling multifractal, and show that multiple behavioral sequences exhibit significant coupling multifractal features that emerge and usually be restricted within relatively greater time scale interval.

In this work, we develop the asymptotic theory of the Detrended Fluctuation Analysis (DFA) and Detrended Cross-Correlation Analysis (DCCA) for trend-stationary stochastic processes without any assumption on the specific form of the underlying distribution. All results are presented and derived under the general framework of potentially overlapping boxes for the polynomial fit. We prove the stationarity of the DFA and DCCA, viewed as stochastic processes, obtain closed forms for moments up to second order, including the covariance structure for DFA and DCCA and a miscellany of law of large number related results. Our results generalize and improve several results presented in the literature. To verify the behavior of our theoretical results in small samples, we present a Monte Carlo simulation study and an empirical application to econometric time series.

Numerous investigations have been performed to establish a correlation between earthquakes and the geological parameters. Radon is one of the general criterions that is referred for the examination of this correlation. Chaotic signal processing methods has a very important explain of complex dynamical system for example earthquakes. Scale analyses technique, Detrended Fluctuation Analysis (DFA), has similar exponents with RMS (Root Mean Square). Hurst exponent is an enormous method to prediction long-range power law. Multifractal time series have both excessively great and little ripple. Made in this study, Detrended Fluctuation Analysis was performed for 222Rn data that was gathered from Lesvos Island in Greece. The results of Detrended Fluctuation Analysis have a robust correlation for each 222Rn time series.

Detrended fluctuation analysis (DFA) is a popular tool in physiological and medical studies for estimating the self-similarity coefficient, α, of time series. Recent researches extended its use for evaluating multifractality (where α is a function of the multifractal parameter q) at different scales n. In this way, the multifractal-multiscale DFA provides a bidimensional surface α(q,n) to quantify the level of multifractality at each scale separately. We recently showed that scale resolution and estimation variability of α(q,n) can be improved at each scale n by splitting the series into maximally overlapped blocks. This, however, increases the computational load making DFA estimations unfeasible in most applications. Our aim is to provide a DFA algorithm sufficiently fast to evaluate the multifractal DFA with maximally overlapped blocks even on long time series, as usually recorded in physiological or clinical settings, therefore improving the quality of the α(q,n) estimate. For this aim, we revise the analytic formulas for multifractal DFA with first- and second-order detrending polynomials (i.e., DFA1 and DFA2) and propose a faster algorithm than the currently available codes. Applying it on synthesized fractal/multifractal series we demonstrate its numerical stability and a computational time about 1% that required by traditional codes. Analyzing long physiological signals (heart-rate tachograms from a 24-h Holter recording and electroencephalographic traces from a sleep study), we illustrate its capability to provide high-resolution α(q,n) surfaces that better describe the multifractal/multiscale properties of time series in physiology. The proposed fast algorithm might, therefore, make it easier deriving richer information on the complex dynamics of clinical signals, possibly improving risk stratification or the assessment of medical interventions and rehabilitation protocols.