A modified Multifractal Detrended Fluctuation Analysis (MFDFA) approach for multifractal analysis of precipitation
Jorge Luis Morales MartínezIgnacio Segovia-DomínguezIsrael QuirósFrancisco Antonio Horta-RangelGuillermo Sosa-Gómez
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Multifractal system
Hurst exponent
Fractional Brownian motion
A relevant issue in time series analysis is the estimation of long-range dependence, that is, how much future values of a time series depend on current values. One of the ways to verify this dependence is by estimating the Hurst exponent using methods such as detrended fluctuation analysis. Here, we propose a new methodology to estimate the Hurst exponent, named leave one out detrended fluctuation analysis. Furthermore, based on this new estimator for the Hurst exponent, we propose the noise reduction by the leave one out detrended fluctuation analysis method. We apply this new denoising method to electrocardiogram noise reduction. The results presented in this work show that this new methodology outperforms the SureShrink and universal noise reduction methods.
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Abstract In this paper, we investigate the statistical and scaling properties of the California earthquakes’ inter-events over a period of the recent 40 years. To detect long-term correlations behavior, we apply detrended fluctuation analysis (DFA), which can systematically detect and overcome nonstationarities in the data set at all time scales. We calculate for various earthquakes with magnitudes larger than a given M. The results indicate that the Hurst exponent decreases with increasing M; characterized by a Hurst exponent, which is given by, H = 0:34 + 1:53/M, indicating that for events with very large magnitudes M, the Hurst exponent decreases to 0:50, which is for independent events.
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In the paper consistent estimates of the Hurst parameter of fractional Brownian motion are obtained and confidence intervals of the obtained estimates are constructed. In many applications related to data processing, it is necessary to estimate the Hurst parameter. Among such tasks is the task of signal processing and analysis, when the signal can be considered as the imposition of a useful signal and background noise. Background noise is usually a combination of stochastic and fractal components. Numerical indicators of these properties are, respectively, the Hurst index, the stability index, the coefficients of the relationship of increments, which generalize the autocorrelation function. Obviously, the estimation of the Hurst index is a priority in the analysis of self-similar processes. Currently, there are many methods for estimating the Hurst parameter, but they are all focused on individual cases of processes where the property of self-similarity is combined with either long-term dependence (fractional Brownian motion), or with heavy tails. RS-analysis, disperse-time analysis and deviation analysis are most often used in estimating the Hurst parameter. A common feature of these methods is that they are all based on the use of statistical properties of second-order samples (variance, standard deviation, correlation coefficients).
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Piecewise Fractional Brownian motion (p-fBm) is a continuous non-stationary Gaussian process having stationary Gaussian increments, named piecewise fractional Gaussian noise (p-fGn). Unlike fractional Brownian motion (fBm) governed by a unique parameter (Hurst exponent), p-fBm is defined by three parameters: the Hurst exponent in low frequencies, the Hurst exponent in high frequencies and the threshold frequency, which separates the two regimes. In this paper, we present a synthesis method that generates a finite approximation of p-fBm series. Moreover, we analyze the synthesized p-fBm series and test the Gaussianity of both p-fBm and p-fGn. We test the stationarity of the first order increments (p-fGn) and explore an approach to estimation of the process Hurst parameters. Our contribution is relevant to modeling and analysis of certain textures that are characteristic of certain medical and other natural images.
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Used of the fractional Brownian motion and fractional Gaussian noise sequence, the detrended fluctuation analysis (DFA) applied to estimate the Hurst exponent to verify the stability and dependability of the method by changing the data length and regression trend order. The result shows that the Hurst exponent estimate is stable and efficient with the length of data for fractional Brownian motion and fractional Gaussian noise sequence. The influence on the Hurst exponent is not obvious when the regression trend order was changed, and the estimate accuracy is improved with the increasing of Hurst exponent value.
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허스트 지수를 산정하기 위하여 기존에 여러 방법론들이 제안되어 왔다. 그러나, 이들 방법론들은 시계열들의 지속성에 대하여 각기 다른 특성들을 보이고 있음을 기존의 연구에서 알 수 있다 따라서 본 연구에서는 수문학에서 주로 이용하고 있는 보정용량, 조정용량, 수정조정용량 방법 이외에 생리학 분야와 전자 분야 등에서 이용되고 있는 1/f 파워 스펙트럼 밀도 분석, DFA, AVT 방법, 최우도법 등을 이용하여 허스트 지수를 산정하여 보았다. 즉, 단기간과 장기간 기억을 가진 카오스와 추계학적 시계열들에 대하여 각각의 방법들을 적용하여 비교 분석하고자 하였으며, 각 방법론들에 대한 장점 및 단점 그리고 한계에 대하여 논의하였다. There are many different techniques for the estimation of the Hurst exponent. However, the techniques can produce different characteristics for the persistence of a time series each other. This study uses several techniques such as adjusted range, resealed range(RR) analysis, modified restated range(MRR) analysis, 1/f power spectral density analysis, Maximum Likelihood Estimation(MLE), detrended fluctuations analysis(DFA), and aggregated variance time(AVT)method for the Hurst exponent estimation. The generated time series from chaos and stochastic systems are analyzed for the comparative study of the techniques. Then this study discusses the advantages and disadvantages of the techniques and also the limitations of them.
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This work analyze the dynamics of the electroencephalographic (EEG) signals of normal and epileptic patients. The Detrended Fluctuation Anal- ysis (DFA) and the Hurst exponent methods are used for estimation of the presence of long term correlations in physiological time series observed in healthy and unhealthy brains. The presence of long-range correlation in a biological time series is an usually response observed in healthy organisms. The complexity of the recorded signal can guarantee some adaptability of the organism to the situations of disturbances. By other hand, the absence of this correlation indicates the loss of this complexity. Non-parametric Wilcoxon test was used for both, namely healthy and unhealthy groups, in order to compare the mean values of the Hurst exponents. Comparison of the means values of the Hurst exponents of normal and epileptic patients, using the Wilcoxon test results, point to some signicant dierence between two groups [1].
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The fractal scaling properties of the heartbeat time series are studied in different controlled ergometric regimes using both the improved Hurst rescaled range $(\mathrm{R}∕\mathrm{S})$ analysis and the detrended fluctuation analysis (DFA). The long-time ``memory effect'' quantified by the value of the Hurst exponent $H>0.5$ is found to increase during progressive physical activity in healthy subjects, in contrast to those having stable angina pectoris, where it decreases. The results are also supported by the detrended fluctuation analysis. We argue that this finding may be used as a useful new diagnostic parameter for short heartbeat time series.
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