## 3 Research Ranges, Objects and Methods
### Research Methods
#### 3.3.1 Weighted Standard Deviation Ellipse  
The standard deviational ellipse method (SDE) is a statistical method that can accurately reveal the spatial distribution characteristics of research elements, and is widely used to explore and analyze the spatial variation trends of geographic elements such as the gravity center, distribution, orientation and shape [35]. Generally, the SDE model contains ellipse center coordinates, rotation angle, major and minor axe standard deviations for four basic parameters, and displays the parameter calculation results in ellipses, which more intuitively describes the spatial and temporal evolution trends of geographic elements from multiple angles [36]. In this work, the overall contour and evolution direction of the spatial distribution of green residences are reflected by the moving trajectory of the center of gravity, the distribution direction and the diffusion range of the ellipse [37]. This paper employed the weighted ellipse center coordinates and took the number of green residences owned by each city as the weight. The moving trajectory of the center of gravity was observed to understand the direction of green residences in China. The formula used to calculate the weighted ellipse center coordinates is as follows:  
\[(\overline{x},\overline{y})=\left[\left(\sum_{i=1}^{n}w_{i}x_{i}/\sum_{i=1}^{n }w_{i}\right),\left(\sum_{i=1}^{n}w_{i}y_{i}/\sum_{i=1}^{n}w_{i}\right)\right] \tag{2}\]  
where, \((x_{i},y_{i})\) represents the spatial coordinates of city \(i\), \(w_{i}\) denotes the weight of green residences' number of city \(i\). The rotation angle represents the angle at which the major axis rotates clockwise from due north and reflects the main trend direction of green residences' distribution. The calculation formula is as follows: \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Dimension** & **Factor** & **Unit** & **Obs.** & **Mean** & **Min.** & **Max.** \\ \hline \multirow{3}{*}{Economic Factor} & GDP & CNY 100 million & 336 & 5314.3 & 439.94 & 25,643.47 \\  & RGPB & CNY 100 million & 336 & 616.18 & 29.46 & 5519.5 \\ \hline \multirow{6}{*}{Social Environment} & EECEP & CNY 100 million & 336 & 32.98 & 0.5 & 708.83 \\  & AQI & \% & 336 & 81\% & 12\% & 100\% \\  & RPP & 10,000 & 336 & 827.89 & 165.43 & 3016.55 \\  & UR & \% & 336 & 66\% & 26\% & 100\% \\  & UCDI & CNY & 336 & 25,827.19 & 11,290.5 & 52,962 \\  & UCHA & sq. m & 336 & 30.27 & 12.81 & 47.6 \\  & PPEU & \% & 336 & 18\% & 5\% & 37\% \\ \hline \multirow{6}{*}{The Real Estate Market} & COV & CNY 100 million & 336 & 1544.26 & 73.11 & 8436.7 \\  & ICR & CNY 100 million & 336 & 556.09 & 22.7 & 2451.37 \\ \cline{1-1}  & CRCNA & 10,000 sq. m & 336 & 3986.63 & 461.1 & 20,294.49 \\ \cline{1-1}  & CRSMA & 10,000 sq. m & 336 & 683.12 & 28.13 & 3386.35 \\ \cline{1-1}  & CRBP & CNY/sq. m & 336 & 903.42 & 100.78 & 4477.71 \\ \cline{1-1}  & CP & & 336 & 0 & 1 \\ \cline{1-1}  & EIP & & 336 & 0 & 1 \\ \cline{1-1}  & NEIP & & 336 & 0 & 1 \\ \cline{1-1}  & DE & & 336 & 0 & 1 \\ \cline{1-1}  & GE & & 336 & 0 & 1 \\ \hline Climatic Condition & CRA & 42 & 1 & 5 \\ \hline \end{tabular} Obs. = number of observed variables; Min. = minimum value; Max. = maximum value; sq. m = square meter.  
\end{table} 
Table 2: Descriptive statistics on the driving factors and variables of green residences in China. \[\tan\theta=\frac{\left(\sum\limits_{i=1}^{n}w_{i}^{2}\widetilde{x}_{i}^{2}-\sum \limits_{i=1}^{n}w_{i}^{2}\widetilde{y}_{i}^{2}\right)+\sqrt{\left(\sum\limits_ {i=1}^{n}w_{i}^{2}\widetilde{x}_{i}^{2}-\sum\limits_{i=1}^{n}w_{i}^{2} \widetilde{y}_{i}^{2}\right)^{2}+4\sum\limits_{i=1}^{n}w_{i}^{2}\widetilde{x}_ {i}^{2}\widetilde{y}_{i}^{2}}}{2\sum\limits_{i=1}^{n}w_{i}^{2}\widetilde{x}_{i} \widetilde{y}_{i}^{2}}} \tag{3}\]  
where, \(\widetilde{x}_{i}\) and \(\widetilde{y}_{i}\) denote the coordinate deviations from the spatial coordinates of city \(i\) to the weighted ellipse center, respectively. The other symbols are the same as Equation (2).