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.