## 2 Materials and Methods
### Methods
#### 2.3.2 Trends Analysis  
This research utilizes the Theil-Sen slope estimator and Mann-Kendall test to analyze the trends in vegetation CUE and WUE within the NMR over the past 23 years. The Theil-Sen slope estimator minimizes the impact of missing data and outliers on the time series, providing more reliable outcomes compared to simple linear regression or conventional least squares methods [36]. The Mann-Kendall (MK) test, known for its robustness against measurement errors and its ability to handle outliers, is extensively applied in trend analysis for time series data [37]. Additionally, the MK test is applied to evaluate the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data** & **Data Source** & **Spatial Resolution** & **Temporal Resolution** \\ \hline \multirow{4}{*}{Temperature} & National Earth System Science Data & \multirow{4}{*}{1 km} & \multirow{2}{*}{1 mon} \\  & Center ([[https://loess.geodata.cn](https://loess.geodata.cn)]([https://loess.geodata.cn](https://loess.geodata.cn)), & \\  & accessed on 1 September 2024) & & \\ \cline{1-1}  & National Earth System Science Data & & \\ \cline{1-1}  & Center ([[https://loess.geodata.cn](https://loess.geodata.cn)]([https://loess.geodata.cn](https://loess.geodata.cn)), & \\ \cline{1-1}  & accessed on 1 September 2024) & & \\ \cline{1-1}  & MCD12Q1v061 ([[https://lpdaac.usgs.gov/](https://lpdaac.usgs.gov/)]([https://lpdaac.usgs.gov/](https://lpdaac.usgs.gov/)) & \\ \cline{1-1}  & products/mcd12q1v061/, & 500 m & 1a \\ \cline{1-1}  & accessed on 1 September 2024) & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._statistical significance of the observed trends in _CUE_ and _WUE_ in the study region. The respective calculation formulas are as follows:  
\[\theta_{\text{slope}}=\text{median}\bigg{(}\frac{x_{j}-x_{i}}{j-i}\bigg{)},\ 2001 \leq i<j\leq 2023 \tag{3}\]  
where \(\theta_{\text{slope}}\) represents the trend in vegetation NEP, a value of \(\theta_{\text{slope}}>0\) signifies an upward trend, \(\theta_{\text{slope}}=0\) signifies stability, and \(\theta_{\text{slope}}<0\) signifies a downward trend.  
\[Z=\begin{cases}\frac{S-1}{\sqrt{s(S)}},\ S>0\\ \ \ 0,S=0\\ \ \ \frac{S-1}{\sqrt{s(S)}},\ S<0\end{cases} \tag{4}\]  
\[\text{S}=\sum_{j=1}^{n-1}\sum_{i=j+1}^{n}sgn\big{[}x_{j}-x_{i}\big{]} \tag{5}\]  
\[\text{sgn}\big{[}x_{j}-x_{i}\big{]}=\begin{cases}\ 1,\ x_{j}-x_{i}>0\\ \ 0,\ x_{j}-x_{i}=0\\ \ -1,\ x_{j}-x_{i}<0\end{cases} \tag{6}\]  
\[\text{s}(\text{S})=\frac{n(n-1)(2n+5)}{18} \tag{7}\]  
where NEP\({}_{i}\) and NEP\({}_{j}\) represent the NEP values for the years \(i\) and \(j\), respectively, measured in grams per square meter (g\(\cdot\)m\({}^{-2}\)), \(n\) denotes the duration of the study period, sgn refers to the sign function, and \(Z\) is the standardized test statistic. For a specified significance level \(\alpha\), a trend is considered statistically significant if \(\mid\!Z\!\mid\!>\!Z_{1}\)-\(\alpha/2\), indicating a meaningful trend in the time series at the \(\alpha\) level. Conversely, if \(\mid\!Z\!\mid\!<\!Z_{1}\)-\(\alpha/2\), the trend is not deemed significant. In this study, a change is defined as minor when the trend significance level is below \(\alpha=0.05\) and significant when it is below \(\alpha=0.01\).