## 3 Research Ranges, Objects and Methods
### The Spatial and Temporal Evolution of Two-Star Green Residences  
As observed from Figure 3, the center of gravity of two-star green residences first appeared at the junction of Beijing and Tianjin in 2009, but the weighted standard deviation ellipse began in 2010. This is because that Beijing and Tianjin became the first cities to have two-star green residences in 2009, but two points could not accurately determine the position of the ellipse.  
From 2010 to 2016, the center of gravity was located in Bengbu and Suzhou in Anhui Province in 2011 and 2012, while the rest of the years were located in Shangqiu in Henan Province. The trajectory of the center of gravity can be clearly divided into a fluctuating period and a stable period. Before 2013, the center of gravity mainly shifted 1027.61 km to the south, and then moved 97.96 km to the northeast. After moving rapidly to the northwest for 140.03 km in 2013, the center of gravity began to stabilize in Shangqiu, and only moved 61.03 km in the three years.  
The areas of the weighted standard deviation ellipse decreased first and then increased, which decreased from 1.99 million km\({}^{2}\) in 2010 to 1.37 million km\({}^{2}\) in 2014, and then increased to 1.69 million km\({}^{2}\) in 2016. Among them, the largest decrease and increase occurred in 2012 and 2015, when they varied by 0.56 and 0.38 million km\({}^{2}\), respectively. In 2010, the ellipse covered most of the central and eastern regions such as Shanghai, Jiangsu, Shandong, Hebei, Henan, Anhui, and Hubei, as well as some of the western regions such as Shanxi, Shaanxi and Ningxia, indicating that two-star green residences appeared in many provinces in that year. From 2011 to 2014, the ellipses gradually shrunk towards the central and eastern regions, especially in the north-south direction, which was much faster than that in the east-west direction. From 2014 to 2016, the areas of the ellipse begun to expand, and the scopes begun to expand to the western region, indicating that the two-star green residences in the western region such as Chongqing, Shaanxi and Xinjiang began to grow rapidly, but the central and eastern regions still took the lead.  
The ellipse showed a spatial pattern of a northwest-southeast direction in 2010, and the rotation angle was 122.86\({}^{\circ}\). This is because many provinces, including Xinjiang and Ningxia in the northwest regions, began to develop two-star green residences in 2010, and there was little difference in the number. The ellipse distribution direction approached due north and due south and the rotation angle was 178.70\({}^{\circ}\) due to the rapid increase in the number of two-star green residences in northern regions such as Hebei and Shandong in 2013. In other years during the study period, the ellipses made clear a spatial pattern of a northeast-southwest direction, and the rotation angle increased from 15.58\({}^{\circ}\) in 2011 to 29.28\({}^{\circ}\) in 2016.  
The major axis standard deviation lessened 395.38 km from 1970.71 km in 2016 to 1575.33 km in 2016, and the minor axis standard deviation added 79.79 km from 1286.33 km in 2010 to 1366.12 km in 2016. It is obvious that the shrinkage amplitude of the major axis standard deviation is larger than the expansion amplitude of the minor axis standard deviation. Similarly, the axial ratio increased from 0.65 in 2010 to 0.87 in 2016, which also indicates that the directionality of the change trend of two-star green residences is more pronounced in the western region than in the western region.  
residence is weakening, and the weighted standard deviation ellipse gradually tended to be circular.