## 2 Materials and Methods
### Methods
#### 2.2.3 Trend Analysis  
Trend analysis can be carried out using both non-parametric and parametric tests [54]. Normal distribution of the data set is not a prerequisite for conducting non-parametric tests. Moreover, such tests are highly insensitive to outliers; thus, they are widely used by researchers. Mann-Kendall is one of the most commonly used non-parametric tests for detecting trends in time series data [55; 56; 57].  
This test was initially used by [PERSON] in 1945 and subsequently developed by [PERSON] in 1970 [58; 59]. The null hypothesis in this test states that the data are randomly distributed and do not form any trend. Rejecting the null hypothesis confirms the presence of a trend in the data set. First, the differences between each observation and the other ones were calculated and then parameter S was obtained using Equation (5).  
\[\text{S}=\sum_{k=1}^{n-1}\sum_{j=k+1}^{n}sgn(x_{j}-x_{k}) \tag{5}\]where n is the number of observations, \(x_{i}\) and \(x_{k}\) show the \(j\)th and \(k\)th series, respectively. The \(sgn\) function was calculated as follows:  
\[sgn(x_{j}-x_{k})=\begin{cases}1\;if\left(x_{j}-x_{k}\right)>0\\ 1\;if\left(x_{j}-x_{k}\right)=0\\ 1\;if\left(x_{j}-x_{k}\right)<0\end{cases} \tag{6}\]  
The values of \(S\) and \(V(S)\) were used to compute the test statistic \(Z\) as follows:  
\[Z=\left\{\begin{array}{ll}x=\frac{S-1}{\sqrt{\text{var}(s)}}\\ x=\frac{S+1}{\sqrt{\text{var}(s)}}\end{array}\right.\qquad\qquad\left\{ \begin{array}{ll}if&S>0\\ if&S=0\\ if&S<0\end{array}\right. \tag{7}\]  
If \(\mid Z\mid\) is larger than \(Z\) critical value, then the null hypothesis is invalid, indicating that the trend is significant.  
All the abovementioned steps were coded in the R programming platform. All NPP images during 2000-2020 were recalled. Then, the data of each pixel were extracted as a matrix, matched with the ecosystem boundaries, and analyzed. The magnitude of a trend can be estimated using [PERSON]'s slope estimator, which is a non-parametric method [60,61]. Equation (8) was used to calculate [PERSON]'s estimate:  
\[\text{f(1)}=\text{Qt}+\text{C} \tag{8}\]  
where C is a constant and Q is the trend obtained through the following equation:  
\[Q_{\text{ij}}=\frac{X_{j}-X_{k}}{j-k} \tag{9}\]  
where \(Q\) indicates the trend magnitude, \(X_{k}\) and \(X_{j}\) are the data values in the time series, and \(Q_{i}\) is the [PERSON]'s slope estimate obtained from the medians of the A number of observations (N). The number of odd observations is obtained using Equation (10).  
\[Q\text{med }=Q\left(\frac{N+1}{2}\right) \tag{10}\]  
The number of even observations is obtained through Equation (11).  
\[Q\text{med }=\frac{1}{2}\Big{(}Q_{\lfloor\frac{N}{2}\rfloor}+Q_{\lfloor\frac{N+2 }{2}\rfloor}\Big{)} \tag{11}\]  
The confidence interval is also calculated from the specific probability of determining whether the median trend is statistically different from zero. It is calculated using Equation (12).  
\[\text{Ca}=Z_{1-\frac{\text{a}}{2}\sqrt{\text{var}(s)}} \tag{12}\]  
where \(Z_{\left(1-\text{a}/2\right)}\) is typically obtained from the standard normal distribution table [62].