## 2 Materials and Methods
### Variable Selection and Data Sources
#### 2.2.3 Statistical Methods  
We explored the spatial attributes of carbon sinks across ecological spaces in the B-T-H agglomeration using the most common spatial analyses, the Global Moran's I, Local Moran's I, and the spatial econometric model. 1. Moran's I (1) Global Moran's I  
Empirical evidence suggests that climate factors exhibit both spatial dependence and heterogeneity. Therefore, global Moran's I was employed to examine the spatial attributes of carbon sinks across ecological spaces. Global Moran's I is the most widely used measure of spatial autocorrelation in ecological and environmental studies, which assesses the degree of clustering among geographic features. Moran's I calculates spatial autocorrelation by comparing the correlation between the attribute values of each geographic unit and those of its neighboring units. Moran's I was calculated as follows:  
\[\text{Moran's I }=\frac{N}{s_{0}}\frac{\sum_{i=1}^{N}\sum_{j=1}^{N}\omega_{ij} (y_{i}-\overline{y})(y_{i}-\overline{y})}{\sum_{i}^{N}\left(y_{i}-\overline{y }\right)^{2}}\]  
In Moran's I, \(s_{0}\) was calculated as follows:  
\[s_{0}=\sum_{i=1}^{N}\sum_{j=1}^{N}\omega_{ij}\]  
Moran's I statistics range from \(-1\) to \(1\), where positive values indicate positive spatial autocorrelation or clustering (i.e., neighboring units are similar), suggesting spatial dependency. In contrast, negative values indicate negative spatial autocorrelation or spatial dispersion (i.e., neighboring values are dissimilar), suggesting inverse spatial dependency. An autocorrelation value of \(0\) implies a random distribution in space. 1. Local Moran's I  
The local Moran's I is a local autocorrelation statistic based on global Moran's I and provides the contribution of each spatial unit's observation. The local Moran's I statistic was calculated as follows:  
\[\text{Local Moran's I }=\frac{y_{i}-\overline{y}}{S_{i}^{2}}\sum_{j=1,j\
eq 1}^{N} \omega_{ij}(y_{i}-\overline{y})\]where \(S_{i}^{2}\) is defined as follows:  
\[s_{i}^{2}=\frac{\sum_{j=1,j\
eq 1}^{N}\omega_{ij}}{N-1}-\vec{y}^{2}\]  
where \(N\) represents the number of spatial grids (observations), \(s_{0}\) represents the sum of all elements in the spatial weight matrix, \(y_{i}\) represents the observed value of variable y in spatial unit I, and \(\omega_{ij}\) represents the element in the spatial weight matrix. 2. Spatial econometrics models (1) The Ordinary Least Squares (OLS) model  
The OLS is a statistical method used to estimate the relationship between a dependent variable and one or more independent variables. It is widely used in regression analysis. The OLS models aim to minimize the sum of squared differences between observed and predicted values. The formula for the OLS model can be presented as  
\[\mathrm{Y}=\beta_{0}+\beta_{1}\mathrm{X}_{1}+\beta_{2}\mathrm{X}_{2}+\ldots+ \beta_{n}\mathrm{X}_{n}+\varepsilon\]  
where Y is the dependent variable; \(\mathrm{X}_{1}\), \(\mathrm{X}_{2}\), \(\ldots\), \(\mathrm{X}_{n}\) are the independent variables; \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\), \(\ldots\), \(\beta_{n}\) are the coefficients representing the relationship between the dependent and independent variables; and \(\varepsilon\) is the error term representing the unexplained variation in the dependent variable. (2) Spatial weight matrix:  
A spatial weight matrix [508; 508] was constructed using the First Law of Geography, which states that the relationship between geographic regions weakened as the geographic distance increased. The Geoda software was used to construct a binary spatial weight matrix, which was then converted to a standardized spatial weight matrix with 508 \(\times\) 508 grids using Stata. The equation for spatial econometrics modeling is as follows:  
\[\mathrm{w}= \begin{bmatrix}w_{11}w_{12}\cdots w_{1n}\\ w_{21}w_{22}\cdots w_{2n}\\ \vdots\vdots\vdots\\ w_{n1}w_{n2}\cdots w_{nm}\end{bmatrix} \]  
Spatial econometric models are statistical models used to analyze spatial correlation and dependence.