## 2 Materials and Methods
### [PERSON]'s Trend Analysis and Mann-Kendall Test  
[PERSON]'s trend analysis, a robust non-parametric statistical method, is well suited for calculating trends in long-term time series data [26]. It is insensitive to measurement errors and outliers and does not require data to follow a normal distribution, making it advantageous for analyzing long-term vegetation trends [27]. Compared to linear regression, [PERSON]'s method estimates trends by calculating the median of the series, providing a more accurate reflection of long-term vegetation changes. It is often combined with the Mann-Kendall test to assess the statistical significance of trends. The Mann-Kendall test is a non-parametric trend analysis method that is widely used to detect long-term trends in environmental data (e.g., climate, hydrology, and SPEI drought index trend analysis). We chose to use the Mann-Kendall Test for the following reasons [28]: (1) it is directly based on data ranks, avoiding issues caused by skewed distributions common in environmental data (e.g., precipitation, SPEI), it is robust to outliers, and it does not assume normality; (2) it can mitigate the impact of serial autocorrelation on significance testing using the Hamed-Rao pre-whitening method (or similar corrections); and (3) the Mann-Kendall test determines the statistical significance of a trend. [PERSON]'s slope estimates the magnitude and direction of the trend. It provides a comprehensive quantification of trends (significance, direction, and rate). The method was calculated in the pymannkendall module of Python 3.12.x. A statistical value of \(\mid\!\!\!\mid\!\!\mid\!\!\mid\!\! \[\beta=Median\left(\frac{x_{j}-x_{i}}{j-i}\right),1<i<j<n \tag{1}\]  
In the formula, \(\beta\) represents the slope of NDVI changes over the years. A value of \(\beta>0\) indicates an upward trend in vegetation coverage, while \(\beta<0\) indicates a downward trend. Here, \(x_{j}\) and \(x_{i}\) denote the time series data, representing the NDVI values for the \(j\)-th and \(i\)-th years, respectively, and \(n\) represents the length of the time series. \(p\)-value was used to assess the significance level [29; 30]:  
Hypotheses:  
Null hypothesis (H\({}_{0}\)): Slope = 0 (no trend). Alternative hypothesis (H\({}_{1}\)): Slope \(\
eq 0\) (trend exists). Decision Rule:  
If \(p<\) significance level (e.g., 0.05), reject H\({}_{0}\) and conclude the trend is statistically significant. If \(p\geq\) significance level (e.g., 0.05), fail to reject H\({}_{0}\) and conclude no significant trend. The formula for calculating the statistical measure \(S\) in the Mann-Kendall trend test is as follows:  
\[S=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}sgn(x_{j}-x_{i}) \tag{2}\]  
where \(sgn\left(x_{j}-x_{i}\right)\) is the sign function, defined as  
\[sgn\left(x_{j}-x_{i}\right)=\left\{ \begin{array}{l}1,x_{j}-x_{i}>0\\ 0,x_{j}-x_{i}=0\\ -1,x_{j}-x_{i}<0\end{array} \right. \tag{3}\]  
The test statistic \(S\) is calculated using Formula (2). The Trend Determination Criteria are as follows [31]:  
When \(n<10\):  
If \(S>0\), the sequence is considered to exhibit an upward trend. If \(S=0\), there is no significant trend. If \(S<0\), the sequence is considered to exhibit a downward trend. When \(n\geq 10\):  
The statistic \(S\) approximately follows a normal distribution. \(Var(S)\) is the variance of the statistic \(S\), calculated using Formula (4). The test statistic \(Z\) is calculated using Formula (5). In this study, the time series length is 19 years (2000-2018), and the test statistic \(Z\) is used for trend testing. \[Var(S)=\frac{n(n-1)(2n+5)-\sum_{i=1}^{n}t(i-1)(2i+5)}{18} \tag{4}\]  
The formula for calculating the standardized statistic \(Z\) is as follows:  
\[Z=\left\{ \begin{array}{l}\frac{S-1}{\sqrt{Var(S)}},S>0\\ 0\qquad,S=0\\ \frac{S+1}{\sqrt{Var(S)}},S<0\end{array} \right. \tag{5}\]  
At a given significance level \(\alpha\), if \(|Z|>z(1-\alpha/2)\), the null hypothesis (no trend) is rejected, indicating a statistically significant trend.