3.2 Trend analysis
Trend analysis has been performed using the modified Mann-Kendall (MK)
test (Mann, 1945, Hamed and Rao, 1998) on the annual and seasonal
metrics of duration and occurrence and the mean date of occurrence .
The MK rank correlation test for two sets of observations X =
x1,x2, …,xn andY=y1,y2,…,ynis formulated as follows, with the S statistic calculated as:
\(S=\sum_{i<j}{a_{\text{ij}}b_{\text{ij}}}\) (6)
where
\(a_{\text{ij}}=sgn\left(x_{j}-x_{i}\right)=\left\{\par
\begin{matrix}1,\ if\ x_{i}<x_{y}\\
0,\ if\ x_{i}=x_{y}\\
-1,\ if\ x_{i}>x_{y}\\
\end{matrix}\right.\ \) (7)
and bij is similarly defined for the observations
in Y . Under the null hypothesis that X and Y are
independent and randomly ordered, the statistic S tends to
normality for large n . In the current work, the modified MK test
proposed by Hamed and Rao (1998) is considered, that is robust in the
presence of autocorrelation in the time series tested by modifying the
variance of the S statistic.
In addition, to consider the issue of false positives due to repeated
statistical tests (Wilks, 2016), the False Discovery Rate (FDR)
procedure introduced by Benjamini and Hochberg (1995) has been
implemented to identify field-significant test results. With this
method, the results are considered field significant (or regionally
significant) if at least one local p-value of the test is below the
global significance level. Only 254 of the 452 selected stations, those
with at least 10 years with more than five consecutive zero-flow days,
have been considered for this analysis to avoid testing trends on very
small sample size.