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In this work, the scaling behavior of the sleep process is evaluated by using detrended fluctuation analysis based on wavelets. The analysis is carried out from arrivals of short and recurrent cortical events called A-phases, which in turn build up the Cyclic Alternating Pattern phenomenon, and are classified in three types: A1, A2 and A3. In this study, 61 sleep recordings corresponding to healthy, nocturnal frontal lobe epilepsy patients and sleep-state misperception subjects, were analyzed. From the A-phase annotations, the onsets were extracted and a binary sequence with one second resolution was generated. An item in the sequence has a value of one if an A-phase onset occurs in the corresponding window, and a value of zero otherwise. In addition, we consider other different temporal resolutions from 2[Formula: see text]s to 256[Formula: see text]s. Furthermore, the same analysis was carried out for sequences obtained from the different types of A-phases and their combinations. The results of the numerical analysis showed a relationship between the time resolutions and the scaling exponents; specifically, for higher time resolutions a white noise behavior is observed, whereas for lower time resolutions a behavior towards to [Formula: see text]-noise is exhibited. Statistical differences among groups were observed by applying various wavelet functions from the Daubechies family and choosing the appropriate sequence of A-phase onsets. This scaling analysis allows the characterization of the free-scale dynamic of the sleep process that is specific for each sleep condition. The scaling exponent could be useful as a diagnosis parameter in clinics when sleep macrostructure does not offer enough information.

In this paper, the complexity-entropy causality plane approach is applied to analyze traffic data. The R/S analysis and detrended fluctuation analysis (DFA) methods are also used to compare with this approach. Moreover, based on the concept of entropy, we propose to use permutation to calculate the probability distribution of the time series when applying the representation plane. The empirical results indicate that traffic dynamics exhibit different levels of traffic congestion and demonstrate that this statistical method can give a more refined classification of traffic states than the R/S analysis and DFA.

We introduce a new method, multifractal cross-correlation analysis based on statistical moments (MFSMXA), to investigate the long-term cross-correlations and cross-multifractality between time series generated from complex system. Efficiency of this method is shown on multifractal series, comparing with the well-known multifractal detrended cross-correlation analysis (MFXDFA) and multifractal detrending moving average cross-correlation analysis (MFXDMA). We further apply this method on volatility time series of DJIA and NASDAQ indices, and find some interesting results. The MFSMXA has comparative performance with MFXDMA and sometimes perform slightly better than MFXDFA. Multifractal nature exists in volatility series. In addition, we find that the cross-multifractality of volatility series is mainly due to their cross-correlations, via comparing the MFSMXA results for original series with those for shuffled series.

Rescaled range analysis (R/S analysis), detrended fluctuation analysis (DFA) and detrended moving average (DMA) are widely-used methods for detection of long-range correlations in time series. Detrended cross-correlation analysis (DCCA) is a recently developed method to quantify the cross-correlations of two non-stationary time series. Another method for studying auto-correlations and cross-correlations was presented by Sergio Arianos and Anna Carbone in 2009. Recent studies have reported the susceptibility of this methods to periodic trends, which can result in spurious crossovers. In this paper, we propose the modified methods base on Laplace transform to minimizing the effect of periodic trends. The effectiveness of our techniques are demonstrated on stock data corrupted with periodic trends.

In the present paper, the influence of the signal class (fBm/fGn) and the data length of time series on choosing the robust fractal analysis method have been studied. More than 1000 fBm/fGn generated time series in short, intermediate and long ranges have been analyzed using common fractal analysis methods. The chosen techniques were power spectral density, detrended fluctuation analysis, rescaled range analysis, box counting, average wavelet coefficients, and the variation method. Numerous graphs indicating the suitability of each method in terms of biases in calculating the fundamental fractal feature of time series, Hurst coefficient, were employed. The results strongly emphasized the crucial influence of the signal class as well as the data length when choosing the appropriate fractal analysis method. Furthermore, as a step forward, a study on the number of data points present in a classified class/length was performed. The effect of the number of data points could not be neglected either. Based on the results, a strategy flowchart for fractal analysis of time series has been proposed. Finally, as an empirical example, the monthly, weekly and daily scaled flow time series of Ghar-e-Aghaj River have been analyzed within the framework of the strategy flowchart.

The water pH series from Dongting Lake Inlet and Outlet in China are analyzed by detrended fluctuation analysis (DFA), spectral analysis and multifractal methods. The results show that these pH series are characterized by long-term memory, 1/f noise and multifractal scaling, and these characteristics have obvious difference between the Lake Inlet and Outlet. The comparison results show that monofractal (DFA exponent) and multifractal (Δα, Δf, B) parameters can be quantitative dynamical indexes reflecting the capability of anti-acidification of Dongting Lake. Furthermore, we investigate the frequency-size distribution of pH series from Dongting Lake Inlet and Outlet. Our findings suggest that water pH is an example of a self-organized criticality (SOC) process. Based on concept of self-organized ctiticality, we analysis the cause that different scale-free power-law behavior between pH series from Dongting Lake Inlet and Outlet. This work can be helpful to improvement of modeling of lake water quality.

Each fractal method of time series analysis gives origin to a parameter that characterizes them. For example, β is the resulting parameter of the application of spectral analysis, α for detrended fluctuation analysis (DFA), D for Higuchi's method and Ha for semivariogram, to mention only some of them. Diverse linear relations between the mentioned parameters have been reported; some of these relations have been obtained from theoretical approaches by assuming time series properties such as their self-similarity. The obtained relations are linear in all cases; nevertheless, these relations only have a limited validity range. In this work, we determine the relation between each one of the parameters ß, α and D for a suitable value range. This is done by means of numerical experiments, consisting of the generation of self-affine and fractal synthetic series. Each one of these sets has a minimum of 500 series with pre-established values of ß, D and Ha. In addition, these are constructed in a suitable value range for each generation method. Later, we apply fractal methods to each one of the three sets of series and the relations between the parameters are obtained. It is observed that all the relations have a linear part as it is expected, but there exist differences in the validity range depending on the generation method.

A two-phase phenomenon in three financial exchange prices is studied. To understand the underlying mechanism for the formation of market prices, we perform the multifractal analysis and the detrended fluctuation analysis in terms of time series of market prices. We also examine higher order temporal correlations for the market price. Although the multifractal properties of market prices are obtained, it cannot be reproduced the binomial multiplicative process through that was used to understand fully developed turbulence.

The time series data of the monthly rainfall records (for the time period 1871–2002) in All India and different regions of India are analyzed. It is found that the distributions of the rainfall intensity exhibit perfect power law behavior. The scaling analysis revealed two distinct scaling regions in the rainfall time series.

Sea clutter refers to the radar returns from a patch of ocean surface. Accurate modeling of sea clutter and robust detection of low observable targets within sea clutter are important problems in remote sensing and radar signal processing applications. Due to lack of fundamental understanding of the nature of sea clutter, however, no simple and effective methods for detecting targets within sea clutter have been proposed. To help solve this important problem, we apply three types of fractal scaling analyses, fluctuation analysis (FA), detrended fluctuation analysis (DFA), and the wavelet-based fractal scaling analysis to study sea clutter. Our analyses show that sea clutter data exhibit fractal behaviors in the time scale range of about 0.01 seconds to a few seconds. The physical significance of these time scales is discussed. We emphasize that time scales characterizing fractal scaling break are among the most important features for detecting patterns using fractal theory. By systematically studying 392 sea clutter time series measured under various sea and weather conditions, we find very effective methods for detecting targets within sea clutter. Based on the data available to us, the accuracy of these methods is close to 100%.

Monthly averages of the sunspot number visible on the sun, observed from 1749, Zurich Observatory, and from 1848 other observatories, have been analyzed. This time signal presents a frequency power spectra with a clear 1/fα behavior with α≃0.8±0.2. The well-known cycle of approximately 11 years, clearly present in the spectrum, does not produce a sensible distortion of that behavior. The eventual characterization of the sunspot time series as a fractal is analyzed by means of the detrended fluctuation analysis (DFA). The jump-size distribution of the signal is also studied.

Chromatin appearance in breast epithelial cells has been shown to have fractal properties, and detrended fluctuation analysis (DFA) is an effective method for characterizing the scaling in non-stationary fractal signals in terms of a scaling exponent. This study examines the use of DFA for the characterization of chromatin appearance in breast epithelial cells. Images of nuclei representative of fine-needle aspiration biopsies of the breast are characterized in terms of the scaling exponent for 19 patients with benign lesions and 22 patients with invasive ductal carcinoma. Characterizing patients by the standard deviations of the values of the scaling exponent for their representative nuclei, a statistically significant difference is noted between benign and malignant cases. This reflects that malignancies exhibit less variability in chromatin roughness than do benign cases. Previous logistic regression models for the diagnosis of breast epithelial cell lesions are improved upon by incorporating the standard deviation of the scaling exponent. Using leave-one-out cross-validation, the best logistic regression classifiers demonstrate a sensitivity of 95% and a specificity of 100%. A combination of DFA and lacunarity analysis is seen to provide the best approach to characterizing chromatin in breast epithelial cell nuclei.

Since the pioneer contributions due to Vandewalle and Ausloos, the Hurst exponent has been applied by econophysicists as a useful indicator to deal with investment strategies when such a value is above or below [Formula: see text], the Hurst exponent of a Brownian motion. In this paper, we hypothesize that the self-similarity exponent of financial time series provides a reliable indicator for herding behavior (HB) in the following sense: if there is HB, then the higher the price, the more the people will buy. This will generate persistence in the stocks which we shall measure by their self-similarity exponents. Along this work, we shall explore whether there is some connections between the self-similarity exponent of a stock (as a HB indicator) and the stock’s future performance under the assumption that the HB will last for some time. With this aim, three approaches to calculate the self-similarity exponent of a time series are compared in order to determine which performs best to identify the transition from random efficient market behavior to HB and hence, to detect the beginning of a bubble. Generalized Hurst Exponent, Detrended Fluctuation Analysis, and GM2 algorithms have been tested. Traditionally, researchers have focused on identifying the beginning of a crash. We study the beginning of the transition from efficient market behavior to a market bubble, instead. Our empirical results support that the higher (respectively the lower) the self-similarity index, the higher (respectively the lower) the mean of the price change, and hence, the better (respectively the worse) the performance of the corresponding stock. This would imply, as a consequence, that the transition process from random efficient market to HB has started. For experimentation purposes, S&P500 stock Index constituted our main data source.

Conscious activity is generally consistent with a deterministic mode of action, however some activities demand randomness. The goal of this study was to find whether mind can produce randomness. We asked various human subjects, aged between 21 and 57, to produce sequences of random integers. The non-repetition of the terms in the series was the "algorithm" generally used by all the subjects. The series were analyzed by using spectral analysis (SA), detrended fluctuation analysis (DFA), information (I2) and approximate entropy (AE) and correlation dimension analysis (CDA). SA of the series revealed 1/fβ(-0.42 of mental series revealed short-range correlation for pairs of successive numbers in the series. The correlation decreased gradually and it disappeared at a delay step of about 7 which is the characteristic number of short memory. AE had a minimum value at α=1 and a maximum value at α=0.5 for mental and random series, respectively. CDA showed no presence of attractors in the series, therefore none of the randomness produced by the subjects was of deterministic origin. A comparison of the scaling exponents revealed that the mental series have a certain long-range characteristic which is common to the human language as evident in the literature writings.

We compare three recently applied methods for analyzing heart rate variability: detrended fluctuation analysis (DFA), multiresolution wavelet analysis (WAV) and detrended time series analysis (DTS). In the comparison, both scale-dependent and scale-independent measures are considered. In agreement with recent results by Thurner et al.,9 we conclude that scale-dependent measures are well suited to separate healthy subjects from patients with heart disease. However, as regards the use in Kaplan-Meier cumulative survival curves, scale-independent measures (generally slope values) clearly outperform scale dependent measures (generally rms values). The comparison is mainly based on a database containing recordings from 428 patients with heart disease (myocardial infarct) and on a database containing 105 healthy subjects and 11 heart patients.

Multiresolution Wavelet Transform and Detrended Fluctuation Analysis have recently been proven to be excellent methods in the analysis of Heart Rate Variability and in distinguishing between healthy subjects and patients with various dysfunctions of the cardiac nervous system. We argue that it is possible to obtain a distinction between healthy subjects/patients of at least similar quality by, first, detrending the time-series of RR-intervals by subtracting a running average based on a local window with a length of around 32 data points, then calculating the standard deviation of the detrended time-series. The results presented here indicate that the analysis can be based on very short time-series of RR-data (7–8 minutes), which is a considerable improvement relative to 24-hour Holter recordings.

A method to estimate the persistent behavior from a chaotic time series is proposed. Persistency means that here each value depends to some extent on the previous values and not only on the recent ones. The data were analyzed by means of Hurst's rescaled range method, i.e., R/S analysis (which was introduced by Mandelbrot and Wallis). The relation of the Hurst exponent to the self-affine and self-simialr fractal dimension is discussed.

The dynamics seismic activity occurred in the Cocos Plate—Mexico is analyzed through the evolution of Hurst exponent and 3D fractal dimension, under the mathematical fractal structure based on seismic activity time series, taking into account the magnitude ([Formula: see text]) as the main parameter to be estimated. The seismic activity time series and, annual intervals are considered first for finding the Hurst exponent of each year since 1988 (the year in which the database is consistent) until 2012, and then the following years are accumulated describing the cumulative Hurst exponent. The seismic activity description is based on Cocos Plate data information; during a period conform from 1 January 1988 to 31 December 2012. Analyses were performed following methods, mainly considering that the Hurst exponent analysis provides the ability to find the seismicity behavior time–space, described by parameters obtained under the fractal dimension and complex systems.

The paper refers to a recently introduced fractal signal analysis method, which relies on scaling properties of signal parameters regarding the cutoff frequency used to smoothen the signal’s graph. The relations between the smoothing dimensions and other exponents (fractal dimension df, power spectrum exponent ß, Hurst exponent H) are determined theoretically and tested by numerical experiments.

Detrended fluctuation analysis (DFA) is used to examine long-range dependence in variations and volatilities of American treasury bills (TB) during periods of low and high movements in TB rates. Volatility series are estimated by generalized autoregressive conditional heteroskedasticity (GARCH) model under Gaussian, Student, and the generalized error distribution (GED) assumptions. The DFA-based Hurst exponents from 3-month, 6-month, and 1-year TB data indicates that in general the dynamics of the TB variations process is characterized by persistence during stable time period (before 2008 international financial crisis) and anti-persistence during unstable time period (post-2008 international financial crisis). For volatility series, it is found that; for stable period; 3-month volatility process is more likely random, 6-month volatility process is anti-persistent, and 1-year volatility process is persistent. For unstable period, estimation results show that the generating process is persistent for all maturities and for all distributional assumptions.

The First Return method has proven to be an efficient method for determining the Hurst exponent, H, of self-affine surfaces. We discuss its foundations and some corrections to scaling which must be taken into account for an adequate estimation of H. Using this method, we analyze a set of artificially generated curves with known Hurst exponent and compare the result with Fourier analysis. We also discuss the case where the surface to be analyzed has a curved bias with a maximum or minimum. In this case, the relation between the scaling exponents associated with the first-return histograms and the Hurst exponent H is different from the unbiased case.

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.

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.

Since the existence of market memory could implicate the rejection of the efficient market hypothesis, the aim of this paper is to find any evidence that selected emergent capital markets (eight European and BRIC markets, namely Hungary, Romania, Estonia, Czech Republic, Brazil, Russia, India and China) evince long-range dependence or the random walk hypothesis. In this paper, the Hurst exponent as calculated by R/S fractal analysis and Detrended Fluctuation Analysis is our measure of long-range dependence in the series. The results reinforce our previous findings and suggest that if stock returns present long-range dependence, the random walk hypothesis is not valid anymore and neither is the market efficiency hypothesis.

The time series of gold price in the Indian market and the global consumer price index for the period of January 1985 to June 2013 are analyzed in terms of the multifractal detrended fluctuation analysis (MF-DFA). Multifractal variables, such as the generalized Hurst exponent, the multifractal mass exponent, the singularity spectrum, are extracted for both the series. Special emphasis is given on the possible source(s) of correlations in these series. The multifractal results are fitted to the generalized binomial multifractal model consists of only two parameters. Our analysis show that the multifractal nature of the Indian gold market time series and the global consumer price index series is due to both the long-range temporal correlation and the fat-tailed probability density function of the values. Surprisingly, the series are well described by the two-parameter binomial multifractal model used.

According to current literature, the Mike-Farmer (MF) model1 is constructed empirically based on the continuous double auction mechanism in an order-driven market, which can successfully capture the diffusive behavior of stock prices at the transaction level. In our paper, we revisit the statistical properties of the generated series of prices based on the MF model to clarify whether it can reproduce the stylized facts in real world markets. However, the Detrended Fluctuation Analysis (DFA) scaling exponent of volatility Hv ≈ 0.6, which may be slightly lower than that in real markets; while a modified version of the MF model proposed by Gu and Zhou2 can improve the DFA scaling exponent of volatility Hv ≈ 0.75, which is closer to the empirical findings. Finally, we test the existence of another commonly found two stylized facts in the real world: the volatility clustering, and leverage effect.

The purpose of the present work is to study the fractal behaviour of prime Indian stock exchanges, namely Bombay Stock Exchange Sensitivity Index (BSE Sensex) and National Stock Exchange (NSE). To analyze the monofractality of these indices we have used Higuchi method and Katz method separately. By applying Mutifractal Detrended Fluctuation Analysis (MFDFA) technique we have calculated the generalized Hurst exponents, multifractal scaling exponents and generalized multifractal dimensions for the present indices. We have deduced Hölder exponents as well as singularity spectra for BSE and NSE. It has been observed that both the stock exchanges are possessing self-similarity at different small ranges separately and inhomogeneously. By comparing the multifractal behaviour of the BSE and NSE indices, we have found that the second one exhibits a richer multifractal feature than the first one.