TITLE:
Hurst’s Memory for Chaotic, Tree Ring, and SOI Series
AUTHORS:
Byung-Sik Kim, Hung-Soo Kim, Sun-Hong Min
KEYWORDS:
Hurst’s Memory; DFA; Chaotic Series; Tree-Ring Series; SOI; BDS Statistic
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.1,
January
14,
2014
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.