Analysis of Trends in Low-Flow Time Series of Canadian Rivers - Hydrologic Time Series Analysis: Theory and Practice - page 205

2000, 2004; Khaliq et al., 2008), can also be used to address the influence of
serial dependence on trend significance provided data over longer time periods
are available. The BBS approach is more flexible and hence it can address the
influence of autocorrelations of higher lags also and not just that of the lag-1
autocorrelation. The BBS approach is considered for the analysis of trends in
the case study presented in Section 10.3.
In a similar manner as presented above for the STP case, hundred thousand
samples were simulated from the FARIMA (0, d , 0) model with values of the
fractional differencing parameter d ranging from 0.001 to 0.498. In its one
parametric form, FARIMA (0, d , 0) is the most simplistic LTP model. Other
complicated structures of FARIMA model with autoregressive and moving
average components were not explored because of simplicity reasons. With
the increase in the value of the parameter d , the strength/intensity of LTP
increases. For an independent time series, the value of the parameter d is zero.
Thus, d = 0.001 would approximately result in an independent time series and
d = 0.498 would result in a highly long-term persistent time series. It is
important to note that the FARIMA is a stationary model and therefore it
would generate stationary time series meaning that the statistical characteristics
of such time series do not change over time. The results of this investigation
are shown in Fig. 10.3. The MK, with the independence assumption, and the
MMK1 and MMK2 tests, with the AR(1) assumption, were used to identify
trends in each simulated time series. It is clear from Fig. 10.3 that the rejection
rate of the null hypothesis increases as the sample size increases meaning that
the MK, MMK1 and MMK2 tests would result in significant trends more
often for the longer samples than for the smaller samples. The MK test
converges to the nominal significance level nearly for all sample sizes when
there is no/weak LTP suggesting that the MK test is unbiased. However, it
considerably deviates from the nominal significance level in the presence of
strong LTP. For example, for a sample size of 50 and d = 0.498, there are
nearly 50% chances that the MK test would suggest a significant trend, given
that there is no trend under the assumption of LTP. Such trends are not real
because they are merely due to fluctuations of the behaviour of the LTP
model. Figures 10.3(b, c) also suggest that STP based MMK1 and MMK2
tests are not adequate to address the influence of LTP on trend significance.
These observations as well as the results of the STP based investigation
presented above suggest that proper verification of the serial structure of the
time series being tested for temporal changes is very important before applying
the MK test for identifying trends.
10.2.4 Field Significance Analysis
As the presence of positive (negative) serial correlation in a hydrological time
series inflates (deflates) the rate of rejecting the null hypothesis of no trend,
the presence of positive cross correlation among a gauging network will
inflate the rate of rejecting the null hypothesis of no field significance of