Abstract
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.
Highlights
Hydrologic or geophysical time series may have a certain dependent structure in itself
(5) Detrended Fluctuations Analysis The method of detrended fluctuations analysis (DFA) has proven as a useful tool in revealing the extent of long-range correlation in time series
(7) Maximum Likehood Estimation(MLE) Here the d is estimated by the S-Maximum Likelihood Estimation (MLE) function of S-Plus [18], and the d is related to the Hurst exponent as follows; d= H − 0.5 (28)
Summary
Hydrologic or geophysical time series may have a certain dependent structure in itself. If this month has a monthly streamflow with high level at a station, month could have a high streamflow. This describes that the consecutive values of hydrologic or geophysical time series show self-dependence between the values which has been known as short-range (or short-term) dependence, persistence, or memory [1]. [3] suggested a method called the R/S analysis for detecting long memory and the method allowed the calculation of Hurst’s exponent or self-similarity parameter. We select an appropriate technique by the examination and apply it to the SOI (southern Oscillation Index) and 6 tree-ring series
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