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The recognition of denatured biological tissue is an indispensable part in the process of high intensity focused ultrasound treatment. As a nonlinear method, multi-scale permutation entropy (MPE) is widely used in the recognition of denatured biological tissue. However, the traditional MPE method neglects the amplitude information when calculating the time series complexity. The disadvantage will affect the recognition effect of denatured tissues. In order to solve the above problems, the method of multi-scale rescaled range permutation entropy (MRRPE) is proposed in this paper. The simulation results show that the MRRPE not only includes the amplitude information of the signal when calculating the signal complexity, but also extracts the extreme volatility characteristics of the signal effectively. The proposed method is applied to the HIFU echo signals during HIFU treatment, and the support vector machine (SVM) is used for recognition. The results show that compared with MPE and the multi-scale weighted permutation entropy (MWPE), the recognition rate of denatured biological tissue based on the MRRPE is higher, up to 96.57%, which can better recognize the non-denatured biological tissues and the denatured biological tissues.

Uncovering dynamic nonlinearity of slug flow with structural mutation constitutes a challenging problem of significant importance. Rescaled range permutation entropy has been recently introduced as a novel measure for the complexity of chaotic time series with extreme volatility by assigning the ratio of range and standard deviation as the weight for each extracted vector. However, previous work is only limited to the analysis of typical chaotic systems. The intrinsic advancement and the potentiality in the practical application of the algorithm still need to be investigated. Therefore, we firstly conduct simulations on characterizing impulsive signal to demonstrate that rescaled range permutation entropy is more appropriate to characterize the extreme volatility of system. In addition, a practical application in experimental signals analysis of gas–liquid two-phase slug flow is discussed in detail. The result shows that the rescaled range permutation entropy has a significant discernibility for the flow structure sudden change and paves a new way on investigating the dynamic complexity of slug flow.

This paper investigates the existence of chaos in concentration dynamics of particulate matter with an aerodynamic diameter less than or equal to $$10\,\upmu \hbox {m}$$ ( $$\hbox {PM}_{10}$$ ) in the greater Athens area (GAA), Greece. It reports findings on three 16-year $$\hbox {PM}_{10}$$ time series recorded by three different air pollution monitoring stations located in GAA and examines if critical fractal epochs with long memory exist. Detrended Fluctuation Analysis (DFA) and Rescaled Range (R/S) Analysis were used via sliding windows of approximately 1-month duration. In all $$\hbox {PM}_{10}$$ time series, several segments were found with critical fractal behaviour and hidden long-memory patterns. All these segments exhibited Hurst exponents above 0.75 and DFA exponents above 1.75. Twelve $$\hbox {PM}_{10}$$ segments with fractality and long memory were commonly identified by both techniques. In one case, long memory was identified concurrently across all three air pollution monitoring stations and in another case, across two stations. The importance of the agreement between two different and independent chaos-analysis techniques is discussed in association with the proper selection of threshold values. This is the second time to address chaos in $$\hbox {PM}_{10}$$ data series in GAA, and the first time to combine two widely accepted techniques, DFA and R/S analysis.

When conducting an analysis of nature’s time series, such as meteorological ones, an important matter is a long-range dependence to quantify the global behavior of the series and connect it with other physical characteristics of the region of study. In this paper, we applied the Higuchi fractal dimension and the Hurst exponent (rescaled range) to quantify the relative trend underlying the time series of historical data from 17 of the 34 weather stations located in the Río Bravo-San Juan Basin, Mexico; these data were provided by the National Water Commission CONAGUA) in Mexico. In this way, this work aims to perform a comparative study about the level of persistency obtained by using the Higuchi fractal dimension and Hurst exponent for each station of the basin. The comparison is supported by a climate clustering of the stations, according to the Köppen classification. Results showed a better fitting between the climate of each station and its Higuchi fractal dimension obtained than when using the Hurst exponent. In fact, we found that the more the aridity of the zone the more the persistency of rainfall, according to Higuchi’s values. In turn, we found more relation between the Hurst exponent and the accumulated amount of rainfall. These are relations between the climate and the long-term persistency of rainfall in the basin that could help to better understand and complete the climatological models of the study region. Trends between the fractal exponents used and the accumulated annual rainfall were also analyzed.

Control Performance Assessment (CPA) has great practical importance. Control quality significantly affects final production throughput, efficiency and environmental impact. There are many approaches starting from time-domain methods, through Minimum Variance, Gaussian and non-Gaussian statistics up to alternative wavelet, fractal or entropy measures. Analysis of production data from process industry shows that signals are often described by non-Gaussian distributions, mostly fat-tail. On the other hand, simulations show that strong disturbances may significantly screen ability of proper detection. This work tests different approaches, i.e. Gaussian standard deviation and fat-tail distribution factors, integral indexes and focuses on persistence measures of rescaled range R/S plot. Robustness of above measures against disturbances with varying statistical properties is investigated. Results confirm that fractal measures may be applied as robust alternative to standard statistics.

The contribution of this article is mainly to develop a new stochastic sequence forecasting model, which is also called the difference iterative forecasting model based on the Generalized Cauchy (GC) process. The GC process is a Long-Range Dependent (LRD) process described by two independent parameters: Hurst parameter H and fractal dimension D. Compared with the fractional Brownian motion (fBm) with a linear relationship between H and D, the GC process can more flexibly describe various LRD processes. Before building the forecasting model, this article demonstrates the GC process using H and D to describe the LRD and fractal properties of stochastic sequences, respectively. The GC process is taken as the diffusion term to establish a differential iterative forecasting model, where the incremental distribution of the GC process is obtained by statistics. The parameters of the forecasting model are estimated by the box dimension, the rescaled range, and the maximum likelihood methods. Finally, a real wind speed data set is used to verify the performance of the GC difference iterative forecasting model.

The characterization of time series requires knowledge of certain para- meters. One of those parameters is the Hurst exponent, which is an indicator of long-range dependence characteristics. Rescaled range (R/S), proposed by E. Hurst, is the most commonly used method to compute this exponent. On the other hand, wavelet analysis is known to reflect better the nonlinear dynamics of the biological, climatic, or economic series than the statistical tools often used for this analysis. The average wavelet coefficient (AWC) is a wavelet method that has been used for the last years to compute the Hurst exponent. In this paper, we present a modification to the AWC method, and we compare its performance with the original version of AWC and with R/S methods. The results obtained for the synthetic series were so promising that we decided to apply our proposal in rainfall series. Therefore, these results were compared with the ones reported from La Pampa. After that, series from different climatic regions of the Argentine Republic were analyzed.

The long-range correlations within the current signal time series of the Beta vulgaris vacuolar membrane under the influence of organolead compound (Met3PbCl) are investigated. The current time series is transformed into a dwell time series. Then the rescaled range and detrended fluctuations analyses are used. It is shown that the presence of Met3PbCl in the solution decreases the mean value of the Hurst exponent and therefore influences the long-range correlations in ionic channel current. This observation is statistically significant. An ion channel model is built and the experimental results reconstructed and analysed.

This paper investigates the complexity of soil radon time series (RTS) recorded from March 2017 to April 2018 in Pakistan via RTM 1688-2 monitor, in direct linkage to parallel series of temperature, relative humidity, and pressure. Several complexity measures are employed. The normalized Lempel Ziv Complexity (LZC) of the complete RTS is 0.30 and 0.14, 0.22, and 0.14 for the complete time series of temperature, relative humidity and pressure respectively. LZC ranges between 0.3 and 0.5 for RTS and between 0.1 and 0.3 for temperature (T), relative humidity (RH) and pressure (P). Hurst exponent (HE) values via rescaled range (R/S) analysis show an inverse relationship with LZC. HE ranges between 0.60 ≤ HE ≤ 0.98 for RTS, 0.60 ≤ HE ≤ 0.92 for T series, 0.61 ≤ HE ≤ 0.93 for RH series, and 0.87 ≤ HE ≤ 0.98 for P series. Permutation Entropy (PE) is 1.7809 for complete RTS with a minimum of 1.7668 in August 2018. Highest value of Sample Entropy (SE) is 1.3716 for February 2018. The highest symbolic Normalized Corrected Shannon Entropy (NCSE) value is found in January 2018 with a value of 0.7864. Finally, the Fractal Dimensions varied significantly for calculation with the Sevcik's, Higuchi's and Katz's methods. Common dates of below threshold areas are reported. The different aspects of the derived results are discussed.

Long range memory in share indices show temporal dependence between observations spaced by long intervals of time and has distinct non-periodic cycles. This paper examines the presence of long memory of various indices of National Stock Exchange (NSE). The data consists of closing values of indices over different periods of time. The tests applied to examine long memory are Hurst exponent, Manderbolt-Hurst exponent, Lo’s rescaled-range analysis and Geweke and Porter-Hudak (GPH) test. The results of the estimated Hurst exponent, Manderbolt-Hurst exponent and GPH test show that invariably all NSE indices series have long memory. However, the results of Lo’s rescaled-range analysis indicate the absence of long memory for all indices. Key words: Long memory, rescaled range analysis, fractional dimension, Hurst exponent, GPH test, NSE indices

The Hurst phenomenon is a well-known feature of long-range persistence first observed in hydrological and geophysical time series by E. Hurst in the 1950s. It has also been found in several cases in turbulence time series measured in the wind tunnel, the atmosphere, and in rivers. Here, we conduct a systematic investigation of the value of the Hurst coefficient H in atmospheric surface-layer data, and its impact on the estimation of random errors. We show that usually $$H > 0.5$$ , which implies the non-existence (in the statistical sense) of the integral time scale. Since the integral time scale is present in the Lumley–Panofsky equation for the estimation of random errors, this has important practical consequences. We estimated H in two principal ways: (1) with an extension of the recently proposed filtering method to estimate the random error ( $$H_p$$ ), and (2) with the classical rescaled range introduced by Hurst ( $$H_R$$ ). Other estimators were tried but were found less able to capture the statistical behaviour of the large scales of turbulence. Using data from three micrometeorological campaigns we found that both first- and second-order turbulence statistics display the Hurst phenomenon. Usually, $$H_R$$ is larger than $$H_p$$ for the same dataset, raising the question that one, or even both, of these estimators, may be biased. For the relative error, we found that the errors estimated with the approach adopted by us, that we call the relaxed filtering method, and that takes into account the occurrence of the Hurst phenomenon, are larger than both the filtering method and the classical Lumley–Panofsky estimates. Finally, we found that there is no apparent relationship between H and the Obukhov stability parameter. The relative errors, however, do show stability dependence, particularly in the case of the error of the kinematic momentum flux in unstable conditions, and that of the kinematic sensible heat flux in stable conditions.

Global Navigation Satellite System (GNSS) is an efficient space geodetic method for precise spatial coordinates determination and time series generation. Time series analysis is very important in in order to find tendencies and make better predictions. In this study we, analyze time series of coordinates ENU (East, North, Up) to determine its performance of GNSS network in the Ecuador using Hurst coefficient estimation. This network is composed by 30 stations. Hurst analysis was done using two approximations. One is Rescaled Range (R/S) technic, and the second is Wavelet transformation. Times series of ENU coordinates were attained from the GNSS stations diary solutions to calculate of Hurst coefficient. Hurst coefficient estimated using both methods were higher than 0.5 which is an indication that the time series indicate persistent properties, that is, a positive correlation.

A monthly average solar green coronal index time series for the period from January 1939 to December 2008 collected from NOAA (The National Oceanic and Atmospheric Administration) has been analysed in this paper in perspective of scaling analysis and modelling. Smoothed and de-noising have been done using suitable mother wavelet as a pre-requisite. The Finite Variance Scaling Method (FVSM), Higuchi method, rescaled range (R/S) and a generalized method have been applied to calculate the scaling exponents and fractal dimensions of the time series. Autocorrelation function (ACF) is used to find autoregressive (AR) process and Partial autocorrelation function (PACF) has been used to get the order of AR model. Finally a best fit model has been proposed using Yule-Walker Method with supporting results of goodness of fit and wavelet spectrum. The results reveal an anti-persistent, Short Range Dependent (SRD), self-similar property with signatures of non-causality, non-stationarity and nonlinearity in the data series. The model shows the best fit to the data under observation.

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.

In this article, we propose a test of the dynamics of stock market indexes typical of the US and EU capital markets in order to determine which of the two fundamental hypotheses, efficient market hypothesis (EMH) or fractal market hypothesis (FMH), best describes market behavior. The article’s major goal is to show how to appropriately model return distributions for financial market indexes, specifically which geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM) dynamic equations best define the evolution of the S&P 500 and Stoxx Europe 600 stock indexes. Daily stock index data were acquired from the Thomson Reuters Eikon database during a ten-year period, from January 2011 to December 2020. The main contribution of this work is determining whether these markets are efficient (as defined by the EMH), in which case the appropriate stock indexes dynamic equation is the GBM, or fractal (as described by the FMH), in which case the appropriate stock indexes dynamic equation is the GFBM. In this paper, we consider two methods for calculating the Hurst exponent: the rescaled range method (RS) and the periodogram method (PE). To determine which of the dynamics (GBM, GFBM) is more appropriate, we employed the mean absolute percentage error (MAPE) method. The simulation results demonstrate that the GFBM is better suited for forecasting stock market indexes than the GBM when the analyzed markets display fractality. However, while these findings cannot be generalized, they are verisimilar.

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.

This work examines if chaos and long memory exist in PM10 concentrations recorded in Athens, Greece. The algorithms of Katz, Higuchi, and Sevcik were employed for the calculation of fractal dimensions and Rescaled Range (R/S) analysis for the calculation of the Hurst exponent. Windows of approximately two months&rsquo, duration were employed, sliding one sample forward until the end of each utilized signal. Analysis was applied to three long PM10 time series recorded by three different stations located around Athens. Analysis identified numerous dynamical complex fractal time-series segments with patterns of long memory. All these windows exhibited Hurst exponents above 0.8 and fractal dimensions below 1.5 for the Katz and Higuchi algorithms, and 1.2 for the Sevcik algorithm. The paper discusses the importance of threshold values for the postanalysis of the discrimination of fractal and long-memory windows. After setting thresholds, computational calculations were performed on all possible combinations of two or more techniques for the data of all or two stations under study. When all techniques were combined, several common dates were found for the data of the two combinations of two stations. When the three techniques were combined, more common dates were found if the Katz algorithm was not included in the meta-analysis. Excluding Katz&rsquo, s algorithm, 12 common dates were found for the data from all stations. This is the first time that the results from sliding-window chaos and long-memory techniques in PM10 time series were combined in this manner.

This paper examines the existence of long memory in the daily stock market EGX 30 index. The long memory of the series was investigated and a fractionally integrated autoregressive moving-average (ARFIMA) model was fitted using 3152 daily data (from June 20, 2004 to July 18, 2017). This paper shows the efficiency of several methods used to estimate fractional parameter in the ARFIMA model, long memory was investigated by Rescaled Range statistic (R/S), Aggregated variance, Higuchi and absolute moment and estimates has obtained by semi parametric methods such as GPH and modified GPH. The results indicate that the modified GPH outperformed the GPH.This paper examines the existence of long memory in the daily stock market EGX 30 index. The long memory of the series was investigated and a fractionally integrated autoregressive moving-average (ARFIMA) model was fitted using 3152 daily data (from June 20, 2004 to July 18, 2017). This paper shows the efficiency of several methods used to estimate fractional parameter in the ARFIMA model, long memory was investigated by Rescaled Range statistic (R/S), Aggregated variance, Higuchi and absolute moment and estimates has obtained by semi parametric methods such as GPH and modified GPH. The results indicate that the modified GPH outperformed the GPH.

Radio refractivity index is used to quantify the effect of atmospheric parameters in communication systems. Scaling and dynamical complexities of radio refractivity across different climatic zones of Nigeria have been studied. Scaling property of the radio refractivity across Nigeria was estimated from the Hurst Exponent obtained using two different scaling methods namely: The Rescaled Range (R/S) and the detrended fluctuation analysis(DFA). The delay vector variance (DVV), Largest Lyapunov Exponent (λ1) and Correlation Dimension (D2) methods were used to investigate nonlinearity and the results confirm the presence of deterministic nonlinear profile in the radio refractivity time series. The recurrence quantification analysis (RQA) was used to quantify the degree of chaoticity in the radio refractivity across the different climatic zones. RQA was found to be a good measure for identifying unique fingerprint and signature of chaotic time series data. Microwave radio refractivity was found to be persistent and chaotic in all the study locations. The dynamics of radio refractivity increases in complexity and chaoticity from the Coastal region towards the Sahelian climate. The design, development and deployment of robust and reliable microwave communication link in the region will be greatly affected by the chaotic nature of radio refractivity in the region.

In his 1951 study of Nile River data, H.E. Hurst introduced the rescaled range statistic-the R / S statistic. He argued via a small simulation study that if X i , i = 1 , … , n , are i.i.d. normal then the R / S statistic should grow in the order of n . However, Hurst found that for the Nile River data, the R / S statistic increased not in the order of n , but in the order n H , where H ranged between 0.75 and 0.80 . He discovered that the effect also appeared in other sets of data. This is now called the Hurst phenomenon. We shall establish some unexpected universal asymptotic properties of the R / S statistic, which show conclusively that the Hurst phenomenon can never appear for i.i.d. data.

Made available in DSpace on 2018-11-26T16:18:59Z (GMT). No. of bitstreams: 0 Previous issue date: 2016-02-15 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) In the context of dynamical systems, time series analysis is frequently used to identify the underlying nature of a phenomenon of interest from a sequence of observations. For signals with a self-affine structure, like fractional Brownian motions (fBm), the Hurst exponent H is one of the key parameters. Here, the use of quantile graphs (QGs) for the estimation of H is proposed. A QG is generated by mapping the quantiles of a time series into nodes of a graph. H is then computed directly as the power-law scaling exponent of the mean jump length performed by a random walker on the QG, for different time differences between the time series data points. The QG method for estimating the Hurst exponent was applied to fBm with different H values. Comparison with the exact H values used to generate the motions showed an excellent agreement. For a given time series length, estimation error depends basically on the statistical framework used for determining the exponent of the power-law model. The QG method is numerically simple and has only one free parameter, Q, the number of quantiles/nodes. With a simple modification, it can be extended to the analysis of fractional Gaussian noises. (C) 2015 Elsevier B.V. All rights reserved. Univ Estadual Paulista, Inst Biociencias, Dept Bioestat, Botucatu, SP, Brazil Inst Nacl Pesquisas Espaciais, Lab Comp & Matemat Aplicada, BR-12201 Sao Jose Dos Campos, SP, Brazil Univ Estadual Paulista, Inst Biociencias, Dept Bioestat, Botucatu, SP, Brazil FAPESP: 2014/05145-0 FAPESP: 2013/19905-3 CNPq: 501221/2012-3 CNPq: 303437/2012-0

Information entropy, as a quantitative measure of complexity in nonlinear systems, has been widely researched in a variety of contexts. With the development of a nonlinear dynamic, the entropy is faced with severe challenges in dealing with those signals exhibiting extreme volatility. In order to address this problem of weighted permutation entropy, which may result in the inaccurate estimation of extreme volatility, we propose a rescaled range permutation entropy, which selects the ratio of range and standard deviation as the weight of different fragments in the time series, thereby effectively extracting the maximum volatility. By analyzing typical nonlinear systems, we investigate the sensitivities of four methods in chaotic time series where extreme volatility occurs. Compared with sample entropy, fuzzy entropy, and weighted permutation entropy, this rescaled range permutation entropy leads to a significant discernibility, which provides a new method for distinguishing the complexity of nonlinear systems with extreme volatility.

This paper examines the volatility of cryptocurrencies, with particular attention to their potential long memory properties. Using daily data for the three major cryptocurrencies, namely Ripple, Ethereum, and Bitcoin, we test for the long memory property using, Rescaled Range Statistics (R/S), Gaussian Semi Parametric (GSP) and the Geweke and Porter-Hudak (GPH) Model Method. Our findings show that squared returns of three cryptocurrencies have a significant long memory, supporting the use of fractional Generalized Auto Regressive Conditional Heteroscedasticity (GARCH) extensions as suitable modelling technique. Our findings indicate that the Hyperbolic GARCH (HYGARCH) model appears to be the best fitted model for Bitcoin. On the other hand, the Fractional Integrated GARCH (FIGARCH) model with skewed student distribution produces better estimations for Ethereum. Finally, FIGARCH model with student distribution appears to give a good fit for Ripple return. Based on Kupieck&rsquo, s tests for Value at Risk (VaR) back-testing and expected shortfalls we can conclude that our models perform correctly in most of the cases for both the negative and positive returns.

El presente trabajo corrobora la dependencia de largo plazo en los retornos de índices bursátiles de Chile, Argentina, Brasil y México. Utilizando el estadístico rango reescalado (R/S) propuesto por Hurst (1951), y posteriormente afinados por Mandelbrot y Wallis (1969) y Mandelbrot (1972, 1975), y el estadístico rango reescalado modificado (Lo, 1991), en general, se confirma la existencia de memoria de largo plazo. Sin embargo, cuando la muestra es separada en dos períodos (pre y poscrisis asiática) se encuentra evidencia de un efecto reset, el cual consiste en que las series en estudio mantienen las propiedades estadísticas pero pierden memoria.

The statistical distribution of values in the signal and the autocorrelations (interpreted as the memory or persistence) between values are attributes of a time series. The autocorrelation function values are positive in a time series with persistence, while they are negative in a time series with anti-persistence. The persistence of values with respect to each other can be strong, weak, or nonexistent. A strong correlation implies a "memory" of previous values in the time series. The long-range persistence in time series could be studied using semivariograms, rescaled range, detrended fluctuation analysis and Fourier spectral analysis, respectively. In this work, persistence analysis is to study interplanetary magnetic field (IMF) time series. We use data from the IMF components with a time resolution of 16 s. Time intervals corresponding to distinct processes around 41 magnetic clouds (MCs) in the period between March 1998 and December 2003 were selected. In this exploratory study, the purpose of this selection is to deal with the cases presenting the three periods: plasma sheath, MC, and post-MC. We calculated one exponent of persistence (e.g., α, β, Hu, Ha) over the previous three time intervals. The persistence exponent values increased inside cloud regions, and it was possible to select the following threshold values: α(j) = 1.392, Ha(j) = 0.327, and Hu(j) = 0.875. These values are useful as another test to evaluate the quality of the identification. If the cloud is well structured, then the persistence exponent values exceed thresholds. In 80.5% of the cases studied, these tools were able to separate the region of the cloud from neighboring regions. The Hausdorff exponent (Ha) provides the best results.

The aim of this paper is to estimate the Hurst parameter of Fractional Gaussian Noise (FGN) using Bayesian inference. We propose an estimation technique that takes into account the full correlation structure of this process. Instead of using the integrated time series and then applying an estimator for its Hurst exponent, we propose to use the noise signal directly. As an application we analyze the time series of the Nile River, where we find a posterior distribution which is compatible with previous findings. In addition, our technique provides natural error bars for the Hurst exponent.

Some special sampling of discrete scale invariant (DSI) processes are presented to provide a multi-dimensional self-similar process in correspondence. By imposing Markov property we show that the covariance functions of such Markov DSI sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion (sBm) with drift and scale invariant autoregressive model are presented and these properties are investigated. We present two new method to estimate Hurst parameter of DSI processes and apply them to some sBm and also to the SP500 indices for some period which has DSI property. We compare our estimates with the maximum-likelihood and rescaled range (R/S) method which are applied to the corresponding multi-dimensional self-similar processes.

This paper considers the problem of estimation of the Hurst parameter H (1/2, 1) from longitudinal data with the error term of a fractional Brownian motion with Hurst parameter H that gives the amount of the long memory of its increment. We provide a new estimator of Hurst parameter H using a two scale sampling method based on -Sahalia and Jacod (2009). Asymptotic behaviors (consistent and central limit theorem) of the proposed estimator will be investigated. For the proof of a central limit theorem, we use recent results on necessary and sufficient conditions for multi-dimensional vectors of multiple stochastic integrals to converges in distribution to multivariate normal distribution studied by Nourdin et al. (2010), Nualart and Ortiz-Latorre (2008), and Peccati and Tudor (2005).

We study long-range correlations and trends in time series extracted from the data of seismic events occurred from 1973 to 2011 in a rectangular region that contains mainly all the continental part of Colombia. The long-range correlations are detected by the calculation of the Hurst exponents for the time series of interevent intervals, separation distances, depth differences and magnitude differences. By using a modification of the classical R / S method that has been developed to detect short-range correlations in time series, we find the existence of persistence for all the time series considered except for magnitude differences. We find also, by using the D F A until the third order, that the studied time series are not influenced by trends. Additionally, an analysis of the Hurst exponent as a function of the number of events in the time and the maximum window size is presented.

We present a method for estimating a time-scale local Hurst exponent on time series.The method has proven to be sensitive to sudden behavior changes on time series.Lower scales evaluate short-range correlations.Larger scales evaluate long-range correlations.The analysis evaluates pattern changes regardless the amplitude and scale. This paper introduces a method to perform a Time-Scale Local Hurst Exponent (TS-LHE) analysis for time series. The traditional Hurst exponent methods usually analyze time series as a whole, providing a single value that characterizes their global behavior. In contrast, the methods based on the Local Hurst Exponent allow the evaluation of the fractal structure of a time series on local events. However, a critical parameter in these methods is the selection of scale. Here, a TS-LHE method is presented, based on a systematic implementation of the rescaled-range (R/S) method, in a set of sliding windows of different sizes. This method allows calculating instantaneous values of Local Hurst Exponents at different scales, associating them with individual samples of a time series. This paper is organized as follows: first, an overview of the TS-LHE is provided; then, a proof-of-concept of this analysis is presented, considering (a) different fractional Brownian motion series, (b) a synthetic seismic signal under different noise conditions, and (c) a group of real seismic traces. Finally, the obtained results show that the TS-LHE analysis is particularly sensitive to sudden behavior changes of the time series, such as frequency or phase variations. This sensitivity is independent of the amplitude of the data, and thus, it can be used to identify pattern changes as well as long- and short-range correlations within a time series.

The report considers the modified Peters algorithm to calculate the Hurst exponent (HE), the essential properties of HE are proved, an example, analysis of a random time series is given.

International audience; This paper extends the fractional Brownian motion to the complex-valued case. The model is defined as the centered, zero at zero, self-similar complex-valued stochastic process with stationary increments. We present a few properties of this new model and propose an estimation of its main index, the Hurst exponent characterizing the self-similarity property.

In this paper, we study long-range dependence of hydrological records with high frequent and massive data set. For detecting breakpoints, we apply the Evolutionary Wavelet Spectrum (EWS) to provide a segmentation of the original time series. And rescaled range analysis (R/S) for estimating the Hurst exponent that describe the long-range dependence phenomenon are used. The results affirm that the hydrological records have long-range dependent (LRD) behaviors.

Hurst’s memory that roots in early work of the British hydrologist H.E. Hurst remains an open problem in stochastic hydrology. Today, the Hurst analysis is widely used for the hydrological studies for the memory and characteristics of time series and many methodologies have been developed for the analysis. So, there are many different techniques for the estimation of the Hurst exponent (H). 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, rescaled range (RR) analysis, modified rescaled 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. We found that DFA is the most appropriate technique for the Hurst exponent estimation for both the short term memory and long term memory. We analyze the SOI (Southern Oscillations Index) and 6 tree-ring series for USA sites by means of DFA and the BDS statistic is used for nonlinearity test of the series. From the results, we found that SOI series is nonlinear time series which has a long term memory of H = 0.92. Contrary to earlier work, all the tree ring series are not random from our analysis. A certain tree ring series show a long term memory of H = 0.97 and nonlinear property. Therefore, we can say that the SOI series has the properties of long memory and nonlinearity and tree ring series could also show long memory and non-linearity.

This paper addresses issues of self-affinity, long-memory and self-organisation in variations of radon in soil recorded in Lesvos Island, Greece. Several techniques were employed, namely (a) power-law wavelet spectral fractal analysis, (b) estimation of Hurst exponents through (b1) rescaled-range, (b2) roughness-length, (b3) variogram and (a), (c) detrended fluctuation analysis, (d) investigation of fractal dimensions and (e) analysis of five block entropies: (e1) Shannon entropy, (e2) Shannon entropy per letter, (e3) conditional entropy, (e4) Tsallis entropy, and (e5) normalised Tsallis entropy. Long-lasting antipersistency was identified during a period of anomalous radon variations following fractional Brownian modelling. Remaining variations did not exhibit analogous behaviour and followed fractional Gaussian modelling. Antipersistent power-law-beta-exponent-values between 1.5 and 2.0 were detected during anomalies. Persistent values were also found. Hurst exponents were mainly within 0

For a sequence of i.i.d. zero mean random variables belonging to the domain of attraction of the normal law, two results concerning the rescaled range statistic are investigated in this paper. More specifically, we obtain precise asymptotics in the law of the logarithm related to complete convergence and a.s. convergence under some mild conditions.

In this study, we give a rescaled range analysis method employed to demonstrate long-range correlations and the presence of periodical features for a time series. An extension of the rescaled range method to estimate the Hurst exponent of high-dimensional fractals is proposed in this study. The two-dimensional rescaled range analysis is used to analyze traffic data, reveal interesting scaling behavior with physical grounds. The relation between the one-dimensional rescaled range method and two-dimensional rescaled range method is interpreted carefully.

This paper presents Hurst exponent footprints from pseudo-dynamic measurements of significantly varied activities on a damaged bridge structure during rehabilitation through continuous monitoring. The system is interesting due to associated uncertainty in large-scale structures and significant presence of human intervention arising from fundamentally different processes. Investigations into the variation of computed Hurst exponents on time series of limited lengths are carried out in this regard. The Hurst exponents are compared with respect to specific events during the rehabilitation, as well as with the data collection locations. The variations of local Hurst exponents about the values computed for each activity are presented. The scaling of Hurst exponents for different activities is also investigated; these are representative of the extent of multifractality for each event. The extent of multifractality is assessed along with its source and time dependency. The National Roads Authority, Maudlins, Naas, Kildare, Ireland South Dublin County Council, Tallaght, Dublin, Ireland Coastway, Naas, Kildare, Ireland Datum Monitoring Services Limited, Moira, Armagh, Northern Ireland Structural Concrete Bonding Services Limited, Newbridge, Kildare, Ireland Complete Highway Maintenance Group, Walkinstown, Dublin, Ireland Testconsult Ireland Limited, Portlaoise, Laoise, Ireland

We propose to estimate the Hurst parameter involved in fractional processes via a method based on the Karhunen–Loeve expansion of a Gaussian process. We specifically investigate the cases of the fractional Brownian motion, the fractional Ornstein–Uhlenbeck family and the fractional Brownian bridge. We numerically compare our results with the ones obtained by the maximum-likelihood method, which show the validity of our proposal.

Investigating long range dependence of river flows, especially in connection with various climate and storage related factors, is important in order to improve stochastic models for long range dependence and in order to understand deterministic and stochastic variability in long-term behaviour of streamflow. Long range dependence expressed by the Hurst coefficient H is estimated for 39 (deseasonalized) mean daily runoff time series in Europe of at least 59 years using five estimators (rescaled range, regression on periodogram, Whittle, aggregated variances, and least squares based on variance). All methods yield estimates of H > 0.5 for all data sets. The results from the different estimators are significantly positively correlated for all pairs of methods indicating consistency of the methods used. Correlations between H and various catchment attributes are also analysed. H is strongly positively correlated with catchment area. Apparently, increasing storage with catchment area translates into increasing long range dependence. H is also positively correlated with mean discharge and air temperature and negatively correlated with the mean specific discharge and the seasonality index (maximum Parde coefficient). No significant correlation is found between the Hurst coefficient and the length of the analyzed time series. The correlations are interpreted in terms of snow processes and catchment wetness. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, we evaluate the differences between Healthy subjects (HS), Parkinson Disease (PDS) subjects, and Epileptic subjects (EBS — Epileptic subject between seizures and ES — epileptic subjects in seizure) by computing some linear, statistic and non-linear features, such as: correlation dimension, maximum Lyapunov exponent and Hurst coefficient. A comparison is made between PDS, EBS, ES and HS groups using non-linear parameters. The EEG data are wavelet multiresolution decomposed into subbands of interest (alpha 8–12 Hz, beta 13–30 Hz, delta 0–4 Hz, theta 4–8 Hz, gamma 30–60 Hz). We applied rescaled range method to estimate the Hurst coefficient of the decomposed signals, with different types of wavelet transforms. The results obtained using the maximum Lyapunov exponent do not highlight very well differences between the 4 groups analyzed (HS, PDS, EBS and ES) but using Hurst coefficient we can make a very good differentiation of EEG signals. The results show that the Hurst coefficient is significant different for delta and theta rhythms extracted from the EEG signal in case of the healthy subjects compared with PDS or EBS subjects. The Hurst coefficient for healthy subjects has a value higher than 0.5 and for PD subjects or epileptic subject between seizures has a value lower than 0.5. Hence, the non-linear parameters such as Hurst coefficient can help in EEG interpretation and in neurological disorders diagnosis.

In this paper, we have utilized ANN (artificial neural network) modeling for the prediction of monthly rainfall in Mashhad synoptic station which is located in Iran. To achieve this black-box model, we have used monthly rainfall data from 1953 to 2003 for this synoptic station. First, the Hurst rescaled range statistical (R/S) analysis is used to evaluate the predictability of the collected data. Then, to extract the rainfall dynamic of this station using ANN modeling, a three-layer feed-forward perceptron network with back propagation algorithm is utilized. Using this ANN structure as a black-box model, we have realized the complex dynamics of rainfall through the past information of the system. The approach employs the gradient decent algorithm to train the network. Trying different parameters, two structures, M531 and M741, have been selected which give the best estimation performance. The performance statistical analysis of the obtained models shows with the best tuning of the developed monthly prediction model the correlation coefficient (R), root mean square error (RMSE), and mean absolute error (MAE) are 0.93, 0.99, and 6.02 mm, respectively, which confirms the effectiveness of the developed models.

This paper reflects the studies undertaken to understand the complexity of the Electromyography (EMG) signal during biceps muscle contraction (flexion), by estimation of Fractal Dimension using Hurst's Rescaled Range (R/S) analysis. The measurements were done using surface electrodes with necessary signal acquisition circuitry and signal data is collected using sound card of PC through Matlab for data analysis purpose. The signal is also subjected to Frequency Domain techniques to estimate the power generated by muscle under different loading conditions. The Fractal Dimension, as obtained here, is found to vary under different loading conditions. The study of non-linear behavior of the EMG signal under varying loading conditions can lead to significant findings, pertaining to time based monitoring of muscle rehabilitation and endurance in a quantitative way.

We propose some class of statistics suitable for estimation of the Hurst index of the fractional Brownian motion based on the second order increments of an observed discrete trajectory., Comment: 4 pages

In this paper, we present the results of Monte Carlo simulations for two popular techniques of long-range correlation detection — classical and modified rescaled range analyses. A focus is put on an effect of different distributional properties on an ability of the methods to efficiently distinguish between short-term memory and long-term memory. To do so, we analyze the behavior of the estimators for independent, short-range dependent, and long-range dependent processes with innovations from eight different distributions. We find that apart from a combination of very high levels of kurtosis and skewness, both estimators are quite robust to distributional properties. Importantly, we show that R / S is biased upwards (yet not strongly) for short-range dependent processes, while M - R / S is strongly biased downwards for long-range dependent processes regardless of the distribution of innovations.

In this article, we consider the robustness to fat tails of four stationarity tests. We also consider their sensitivity to the number of lags used in long-run variance estimation, and the power of the tests. Lo's modified rescaled range (MR/S) test is not very robust. Choi's Lagrange multiplier (LM) test has excellent robustness properties but is not generally as powerful as the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test. As an analytical framework for fat tails, we suggest local-to-finite variance asymptotics, based on a representation of the process as a weighted sum of a finite variance process and an infinite variance process, where the weights depend on the sample size and a constant. The sensitivity of the asymptotic distribution of a test to the weighting constant is a good indicator of its robustness to fat tails. This article has supplementary material online.

Sharp fluctuations (in particular, extreme fluctuations) of asset prices have a great impact on financial markets and risk management. Therefore, investigating the time dynamics of sharp fluctuation is a challenge in the financial fields. Using two different representations of the sharp fluctuations (inter-event times and series of counts), the time clustering behavior in the sharp fluctuation sequences of stock markets in China is studied with several statistical tools, including coefficient of variation, Allan Factor, Fano Factor as well as R/S (rescaled range) analysis. All of the empirical results indicate that the time dynamics of the sharp fluctuation sequences can be considered as a fractal process with a high degree of time-clusterization of the events. It can help us to get a better understanding of the nature and dynamics of sharp fluctuation of stock price in stock markets.