# Urbanization and Grain Production Pattern of China: Dynamic Effect and Mediating Mechanism
## 3 Materials and Methods
### Methodology
#### 3.1.2 Standard Deviational Ellipse-Center of Gravity Model (SDE-COG)  
The standard deviational ellipse (SDE) is an effective tool that is capable of accurately revealing the overall spatial distribution characteristics of geographical elements [36; 37]. The SDE describes the spatial distribution characteristics of geographic elements and their spatiotemporal evolution from a global and spatial perspective through a spatial ellipse with the center, major axis, minor axis, and azimuth as the basic parameters [36]. An SDE is established with the center of gravity (COG) of the geographic element distribution as its center, the direction of the main distribution trend as the azimuth (the angle between the major axis and due north), and the standard deviation of the elements in the X and Y directions as the ellipse axes. By constructing ellipses of the spatial distribution of geographic elements, the characteristics of centrality, directionality, and spatial distribution patterns of geographic elements can be described and explained [37]. The center of the SDE marks the relative position of the spatial distribution of an economic phenomenon on the two-dimensional plane--i.e., the COG of its spatial distribution. The SDE can reflect the trajectory and spatial characteristics of the COG of grain production concentration index in a region and offer more intuitive information on the development direction of grain production COG. The main parameters of the SDE-COG model are calculated as follows:  
\[X=\sum_{i=1}^{n}\omega_{i}x_{i}/\sum_{i=1}^{n}\omega_{i},\ Y=\sum_{i=1}^{n} \omega_{i}y_{i}/\sum_{i=1}^{n}\omega_{i} \tag{5}\]  
\[\sigma_{x}=\sqrt{\sum_{i=1}^{n}\left(\omega_{i}x_{i}^{*}\cos\theta-\omega_{i}y _{i}^{*}\sin\theta\right)^{2}/\sum_{i=1}^{n}\omega_{i}^{2}},\ \sigma_{y}=\sqrt{\sum_{i=1}^{n}\left(\omega_{i}x_{i}^{*}\sin\theta-\omega_{i}y _{i}^{*}\cos\theta\right)^{2}/\sum_{i=1}^{n}\omega_{i}^{2}} \tag{6}\]  
\[\tan\theta=\left(\left(\sum_{i=1}^{n}\omega_{i}^{2}{x_{i}^{*2}}^{2}-\sum_{i=1 }^{n}\omega_{i}^{2}y_{i}^{*2}\right)+\sqrt{\left(\sum_{i=1}^{n}\omega_{i}^{2}{ x_{i}^{*2}}^{2}-\sum_{i=1}^{n}\omega_{i}^{2}y_{i}^{*2}\right)^{2}-4\sum_{i=1}^{n} \omega_{i}^{2}x_{i}^{*2}y_{i}^{*2}}\right)/2\sum_{i=1}^{n}\omega_{i}^{2}x_{i}^ {*}y_{i}^{*} \tag{7}\]  
where \((X,Y)\) are the COG coordinates of GPCI, \((x_{i},y_{i})\) are the spatial coordinates of the study area, \(\left(x_{i}^{*},y_{i}^{*}\right)\) are the coordinates of each area relative to the regional COG, \(\omega_{i}\) is the weight, \(\sigma_{x},\sigma_{y}\) are the standard deviation along the \(x\) and \(y\) axes, respectively, and \(\theta\) is the ellipse azimuth--the angle between the major axis and due north. In addition, the distance of COG transfer can be measured as follows:  
\[D_{\alpha\rightarrow\beta}=k\times\sqrt{\left(x_{iu}-x_{i\beta}\right)^{2}+ \left(y_{iu}-y_{i\beta}\right)^{2}} \tag{8}\]  
where \(D_{\alpha\rightarrow\beta}\) is the grain production COG transfer distance (km) from year \(\alpha\) to year \(\beta\), \(\left(x_{i\alpha},y_{iu}\right)\) and \(\left(x_{i\beta},y_{i\beta}\right)\) denote the geographic coordinates of COG transfer over time, and \(k\) is the coefficient of conversion from Earth surface coordinate units (degrees) to plane distance (km), generally equal to 111.111 km.