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fractional differencing parameter, (4) detrended fluctuation analysis (DFA)
(Peng et al., 1994; Kantelhardt et al., 2001) assuming local segmental trends
of linear type and (5) maximum likelihood based estimator (MLE) (Hipel and
McLeod, 1994; Hamed, 2008). In order to determine whether the estimated
H
value by using the first four selected techniques for a given low flow time
series significantly deviates from 0.5, a simulated distribution of
H
was
developed by using a large number of randomly generated samples. The
random samples with sample size being equal to observed sample were
generated from a white noise process (i.e. normally distributed values with
zero mean and unit variance). From the simulated distribution of
H
, 2.5
th
and
97.5
th
percentile values of
H
were obtained to define a 95% confidence interval
for each of the selected techniques. For the MLE method, 95% confidence
interval was obtained using the parametric approximation described in Hipel
and McLeod (1994) and Hamed (2008).
Observed values of
H
along with 95% confidence intervals for selected
10 stations (out of 49) with longer records (ranging from 75 to 93 years) are
shown in Fig. 10.8. According to the results obtained for the RARS method,
none of the estimated
H
values fall at or outside the upper limit at 95%
confidence interval. However, the results for the ASD method suggest the
possibility of LTP in annual and winter low flow time series at a single station
(05BB001) while the results of the FARIMA method suggest the possibility of
Fig. 10.8.
The Hurst exponent estimated using the (a) RARS, (b) ASD, (c) FARIMA
modelling and (d) DFA methods for time series of annual, winter and summer
low flows observed at 10 selected stations. The 95% confidence intervals for
each of the four methods and selected stations were obtained from the simulated
distribution of the Hurst exponent developed by generating 10,000 random
samples, of size equal to the observed samples, from Normal (0, 1) distribution.
Dotted horizontal line corresponds to an independent time series.
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