## 2 Materials and Methods
### Analysis of Ecosystem Service Trade-Offs and Synergies
#### 2.4.2 Quantification of Local Ecosystem Service Trade-Offs and Synergies  
1. Ecosystem Service Cold and Hot Spot Analysis Cold spots and hot spots refer to regions where the distribution of ecosystem service values is either significantly dispersed (cold spots) or highly concentrated (hot spots). A hot spot is identified when a given service exhibits high values that are spatially surrounded by similarly high values. In spatial statistics, the \(G_{i}\) statistic, introduced by [PERSON] [56], is widely utilized as a local spatial autocorrelation measure. This statistic, based on a \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ecosystem Service** & **Proportion of** & **Subclass** & **Proportion of** & **Type** & **Example** \\
**Relationships** & **Study Area** & **Subclass** & **Study Area** & **Type** & **Example** \\ \hline Trade-off & 68.67\% & Strong Trade-off & 35.50\% & High (1) Medium (0) Low (4) & 11,311 \\  & & & High (1) medium (1) Low (3) & 11,321 \\  & & & High (1) medium (1) Low (3) & 12,113 \\ \hline  & & & High (1) Medium (2) Low (2) & 12,312 \\  & & & High (1) Medium (3) Low (1) & 22,312 \\  & & & High (1) Medium (3) Low (1) & 22,312 \\ \hline  & Weak & & High (2) Medium (0) Low (3) & 11,133 \\  & Trade-off & 33.17\% & High (2) Medium (0) Low (3) & 11,313 \\ \hline  & & & High (2) Medium (1) Low (2) & 23,113 \\  & & & High (2) Medium (1) Low (2) & 31,123 \\ \hline  & & & High (2) Medium (2) Low (1) & 23,213 \\ \hline  & & & High (3) Medium (0) Low (2) & 33,113 \\  & & & High (3) Medium (1) Low (1) & 33,213 \\ \hline  & & & High (4) Medium (0) Low (1) & 33,313 \\ \hline Synergy & 31.33\% & High Synergy & 2.24\% & High (5) Medium (0) Low (0) & 33,333 \\ \hline  & & & High (4) Medium (1) Low (0) & 33,233 \\  & & & High (3) Medium (2) Low (0) & 33,233 \\  & & & High (2) Medium (3) Low (0) & 22,332 \\ \hline  & & & High (1) Medium (4) Low (0) & 22,322 \\ \hline  & & & High (0) Medium (5) Low (0) & 22,222 \\ \hline  & Low & & High (0) Medium (1) Low (4) & 11,211 \\ Synergy & & & High (0) Medium (2) Low (3) & 12,112 \\  & & & High (0) Medium (3) Low (2) & 12,212 \\ \hline  & & & High (0) Medium (4) Low (1) & 22,212 \\  & & & High (0) Medium (0) Low (5) & 11,111 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Classification criteria and statistics of trade-offs/synergies among ecosystem services. complete distance matrix, is employed to examine the degree of spatial clustering for high or low values of ecosystem services within ecological restoration zones. The calculation formula is as follows:  
\[G_{i}^{*}=\frac{\sum_{j}^{n}W_{ij}x_{j}}{\sum_{j}^{n}x_{j}} \tag{11}\]  
The \({G_{i}}^{*}\) test yields the following \(Z({G_{i}}^{*})\) values [56]:  
\[Z({G_{i}^{*}})=\frac{G_{i}^{*}-E({G_{i}^{*}})}{\sqrt{VAR\big{(}G_{i}^{*}\big{)} }}=\frac{\sum_{j}W_{ij}x_{j}}{\sqrt[3]{\frac{n\sum_{j}^{n}W_{ij}^{2}-\left(\sum _{j}^{n}W_{ij}\right)^{2}}{n-1}}} \tag{12}\]  
\[X=\frac{\sum_{j}^{n}x_{j}}{n},\ S=\sqrt{\frac{\sum_{j}^{n}x_{j}^{2}}{n}-X^{2}} \tag{13}\]  
In the equation: \(w_{ij}\) is the spatial weight matrix; \(x_{j}\) is the attribute value; \(x\) is the mean of all attribute values; and \(n\) is the number of patches. The classification based on \(Z({G_{i}}^{*})\) is as follows: \(Z({G_{i}}^{*})>2.58\): Extreme hot spot; \(1.96<Z({G_{i}}^{*})\leq 2.58\): Hot spot; \(1.65<Z({G_{i}}^{*})\leq 1.96\): Sub-hot spot; \(Z({G_{i}}^{*})<-2.58\): Extreme cold spot; \(-2.58\leq Z({G_{i}}^{*})<-1.96\): Cold spot; \(-1.95\leq Z({G_{i}}^{*})\leq-1.65\): Sub-cold spot; \(-1.65\leq Z({G_{i}}^{*})\leq 1.65\): Non-significant area. 2 Pearson Correlation Analysis  
This study utilizes the Pearson correlation coefficient method to analyze the trade-offs and synergies between local ecosystem services. Based on the significance test, a positive correlation coefficient indicates synergy, while a negative correlation coefficient reflects a trade-off relationship.