A new combined approach on Hurst exponent estimate and its applications in realized volatility
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Hurst exponent
Rescaled range
Exponent
In order to investigate self-similarity and long-rang dependence of wind speed time series, the rescaled range analysis and the detrended fluctuation analysis (DFA) were carried out to calculate the Hurst exponent of the wind speed time series. The power spectral density was analyzed and the spectral indexes were calculated. The results show that, the Hurst exponents calculated with the both methods are close to 1, which means that the wind speed time series has strong self-similarity and long-range positive dependence. However, the Hurst indexes from the 2 methods are different, which means that the DFA method could reflect the power-law feature of the non-stationary wind speed time series. Furthermore, the results of the Hurst exponents and the spectral indexes show that the wind speed fluctuation belongs to the “1/f fluctuation”. The conclusions offer some references for further study on fractal chaos and short-time wind speed prediction.
Hurst exponent
Rescaled range
Self-similarity
Similarity (geometry)
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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.
Hurst exponent
Rescaled range
Statistic
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Detrended Fluctuation Analysis (DFA) is the most popular fractal analytical technique used to evaluate the strength of long-range correlations in empirical time series in terms of the Hurst exponent, $H$. Specifically, DFA quantifies the linear regression slope in log-log coordinates representing the relationship between the time series' variability and the number of timescales over which this variability is computed. We compared the performance of two methods of fractal analysis -- the current gold standard, DFA, and a Bayesian method that is not currently well-known in behavioral sciences: the Hurst-Kolmogorov (HK) method -- in estimating the Hurst exponent of synthetic and empirical time series. Simulations demonstrate that the HK method consistently outperforms DFA in three important ways. The HK method: (i) accurately assesses long-range correlations when the measurement time series is short, (ii) shows minimal dispersion about the central tendency, and (iii) yields a point estimate that does not depend on the length of the measurement time series or its underlying Hurst exponent. Comparing the two methods using empirical time series from multiple settings further supports these findings. We conclude that applying DFA to synthetic time series and empirical time series during brief trials is unreliable and encourage the systematic application of the HK method to assess the Hurst exponent of empirical time series in behavioral sciences.
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Rescaled range
Exponent
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Monthly rainfall data of twenty-one years (1980 – 2000) were analyzed for the six regions of Nigeria using the rescaled range (R/S) statistic, the standard fluctuation analysis (FA) and the detrended fluctuation analysis (DFA). The results indicated that the distribution of monthly rainfall has a Hurst exponent of 1 in the short term which is an evidence of self-organized criticality,and a Hurst exponent of about 0.5 in the long run which is an evidence of a random walk. The Hurst exponents decreased from around 1 for small window sizes and reached 0.5 after 36 months and then remained fairly constant. The R/S statistic was found to perform better than the FA and the linear DFA used in the analysis.
Hurst exponent
Rescaled range
Statistic
Self-organized Criticality
Exponent
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This paper gives a basic overview of the various attempts at modelling stochastic processes for stock markets with a specific application to the Portuguese stock market data. Long-memory dependence in the stock prices would completely alter the data generation process and econometric models not considering the long-range dependence would exhibit poor forecasting abilities. The Hurst exponent is used to identify the presence of long-memory or fractal behaviour of the data generation process for the daily returns to ascertain if the process follows a fractional brownian motion. Detrended fluctuation analysis (DFA) using linear and quadratic trends and the Geweke Porter-Hudak methods are applied to detect the presence of long-memory or persistence. We find that the daily returns exhibit a small amount of long memory and that the quadratic trend used in the DFA overestimates the value of the Hurst exponent. These findings are corroborated by the use of the Geweke Porter-Hudak method wherein the Hurst exponent is close to the DFA using the linear trend.
Hurst exponent
Rescaled range
Fractional Brownian motion
Stock (firearms)
Exponent
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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.
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Rescaled range
Random walk hypothesis
Efficient-market hypothesis
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Air temperature records are commonly subjected to inhomogeneities, e.g., sudden jumps caused by a relocation of the measurement station or by installing a new type of shelter. We study the effect of these inhomogeneities on the estimation of the Hurst exponent and show that they bias the estimates toward larger values. The Hurst exponent is a parameter to measure long‐range dependence (LRD), which is a characteristic frequently used to describe the natural variability of temperature records. Analyzing a set of temperature time series before and after homogenization with respect to LRD, we find that the average Hurst exponent is clearly reduced for the homogenized series. To test whether (1) jumps cause this positive bias and (2) the homogenization does not artificially reduce the Hurst exponent estimates, we perform a simulation study. This test shows that inhomogeneities in the form of jumps bias the Hurst exponent estimation and that the homogenization procedure is able to remove this bias, leaving the Hurst exponent unchanged. This result holds for fractional autoregressive integrated moving average (FARIMA)‐based as well as for detrended fluctuation analysis‐based estimation. We conclude that the use of homogenized series is necessary to prevent misleading conclusions about the dependence structure and thus about subsequent analysis such as trend tests.
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Homogenization
Exponent
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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.
Hurst exponent
Rescaled range
Exponent
Periodogram
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Hurst exponent
Rescaled range
Exponent
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A fractal dimension may be thought of as a measure of randomness. Fractal dimensions based on semivariograms have been used to determine degree of randomness in yearly crop yields. Through rescaled range analysis Hurst exponents also define fractal dimensions. This method of obtaining fractal dimensions gives more reasonable and sensitive measures than the semivariogram method. To address the inherent randomness due to yearly variations, global trends in yield must be removed before either method is applied. After detrending, a fractal dimension obtained from semivariogram is usually that of a random process. The Hurst method yields an exponent H, which results in a fractal dimension D = 2-H. The Hurst exponent H = 0.5 corresponds to a completely random system, H = 1 to a completely deterministic system. Natural processes such as river discharges, temperatures, precipitation, and tree rings have Hurst exponents about 0.72. The Hurst phenomenon is the occurrence of H greater than 0.5 corresponding to persistent Brownian motions, rather than equal to 0.5 corresponding to a random process. There is not much auto-correlation in the detrended crop yields, but the Hurst exponents from detrended yearly crop yields of Illinois soybean and wheat and of US soybean, wheat, and cotton are mainly between 0.5 and 1 suggesting long-term dependence similar to that of other natural processes. Illinois and US detrended yearly com yield have exponents less than 0.5, corresponding to anti-persistent Brownian motions. Com data from the Morrow Plots also have Hurst exponents less than 0.5 for two plots that were either over-fertilized or previously not treated, while an optimally treated (properly fertilized and previously manured) plot had an exponent greater than 0.5.
Hurst exponent
Fractional Brownian motion
Rescaled range
Variogram
Decorrelation
Multifractal system
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