*2.4. Data Analysis*

The Mann–Kendall test, as a nonparametric method for testing trends, and is also satisfactory for examining the significance of trends in a time series [49]. The statistics of variance can be described as follows:

$$S = \sum\_{n=1}^{i-1} \sum\_{m=n+1}^{i} \text{sgn}(\mathbf{x}\_m - \mathbf{x}\_n),\tag{1}$$

$$\text{sgn}(\mathbf{x}\_m - \mathbf{x}\_n) = \begin{cases} 1 & \mathbf{x}\_m - \mathbf{x}\_n > 0 \\ 0 & \mathbf{x}\_m - \mathbf{x}\_n = 0 \\ -0 & \mathbf{x}\_m - \mathbf{x}\_n < 0 \end{cases},\tag{2}$$

$$Z = \begin{cases} \begin{array}{cc} \frac{s-1}{\sqrt{Var(S)}} & S > 0\\ 0 & S = 0\\ \frac{s+1}{\sqrt{Var(S)}} & S < 0 \end{array} \end{cases} \tag{3}$$

$$Var(S) = \frac{1}{18} [i(i+1)(2i+5) - \sum\_{i=1}^{n} t\_i(t\_i - 1)(2t\_i + 5)],\tag{4}$$

where *i* is the number of data points in the sequence, and *ti* is the number of data values. Statistic *Z*, as a standard normal variable, was used to evaluate the statistical significance. The Mann–Kendall test is applied on a time series for all biophysical variables, and if the *Z* value is less than or equal to the significance level (α = 0.05), a significant trend of the variable will be detected. In this study, the Mann–Kendall test for trends and linear regression analysis was used to detect and estimate the annual and seasonal trend of biophysical variables, with significance defined as *p* < 0.05.

Pearson's Correlation Coe fficient was used to evaluate the correlation between the climate variables and vegetation index as well as between the climate variables and ET to determine the response of vegetation and ET to climate change.

$$r = \frac{\sum\_{i=1}^{n} \left(X\_i - \overline{X}\right) \left(Y\_i - \overline{Y}\right)}{\sqrt{\sum\_{i=1}^{n} \left(X\_i - \overline{X}\right)^2} \sqrt{\sum\_{i=1}^{n} \left(Y\_i - \overline{Y}\right)^2}},\tag{5}$$

where *X* represents the climate variables, *Y* represents the vegetation index or ET, and *n* is the number of samples.
