2.3.4. Non-Parametric Mann–Kendall Test

The non-parametric Mann–Kendall (MK) test [25,26] is widely applied to detect the possible trends in many countries such as in China [47], in Serbia [48] in Brazil [49], in Canada [50]. In this study, the MK test statistic, S, is applied and briefly represented by:

$$S = \sum\_{\mathbf{k}}^{n-1} \sum\_{\mathbf{j} = \mathbf{k}+1}^{n} \text{sign}(\mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_{\mathbf{k}}) \tag{5}$$

where n is the number of data points; xj and xk are the data values in time series j and k respectively, and

$$\text{sign}(\mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_{\mathbf{k}}) = \begin{cases} +1 & \text{if } \mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_{\mathbf{k}} > 0 \\ 0 & \text{if } \mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_{\mathbf{k}} = 0 \\ -1 & \text{if } \mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_{\mathbf{k}} < 0 \end{cases} \tag{6}$$

In this test, the null hypothesis (Ho) assumes that there is no trend in meteorological droughts over time; the alternative hypothesis (H1) assumes that there is an upward or downward trend over time. The mathematical equations for calculating Var(S) and standardized test statistics Z are presented in previous studies [25,26,47,51,52]. An upward, downward, or no trend will be assessed at α significance level of 0.05. The computed probability is greater than the specific significance level α (Ho is rejected); the increasing trend responds to a positive value of Z and a negative value of Z indicates a decreasing trend. There is no trend if the computed probability is less than the level of significance (Ho is accepted). At the α significance level of 0.05, the null hypothesis of no trend is rejected if |ZMK| > 1.96.
