# Analysis of Ground Subsidence Vulnerability in Urban Areas Using Spatial Regression Analysis
## 5 Spatial Regression Analysis  
The aim of this study was to develop a model to predict the vulnerability of ground subsidence in the target area using the attribute information of six types of underground utilities and the history information of ground subsidence. The OLS analysis results demonstrate that there was spatial autocorrelation. Thus, a spatial regression model was analyzed. To estimate a spatial regression model, spatial data were built first. Then, a spatial distance was defined and a spatial weight matrix that set the relationship between regions was constructed using the GeoDa 1.20 software [28]. For spatial distance, this study chose the Queen method, which determines that in gridded data, neighbors are adjacent if they share a side or corner of a given grid cell. The Queen method is used in geospatial data consisting of administrative districts, and is suitable for considering the adjacency of regions that share boundaries with each other [29]. The spatial regression analysis in this study employed SEM and SLM. SEM is a model that considers the covariance of errors in the regression model when spatial autocorrelation exists in the errors. SLM is a model that analyzes spatial dependence (spatial lagged variables) by including them in the regression model as one explanatory variable [30]. In addition, SEM and SLM were estimated through maximum likelihood. The SEM and SLM are produced using Equations (7) and (8). \[\mathrm{y}=\delta_{0}+\delta_{1}\mathrm{X}_{1}+\delta_{2}\mathrm{X}_{2}+\dots+ \mathrm{u}\;\mathrm{u}=\lambda\mathrm{W}_{\mathrm{u}}+\mathrm{e} \tag{7}\]  
\(\mathrm{y}\): Dependent variable, \(\mathrm{X}\): independent variable, \(\mathrm{U}\): error term with spatial autocorrelation, \(\mathrm{W}\): spatial weight matrix, \(\Delta\), \(\lambda\) : estimation coefficient, and e:error term without spatial autocorrelation. \[\mathrm{y}=\rho\mathrm{W}_{\mathrm{y}}+\delta_{0}+\delta_{1}\mathrm{X}_{1}+ \delta_{2}\mathrm{X}_{2}+\dots+\varepsilon \tag{8}\]  
\(\mathrm{y}\): Dependent variable, \(\mathrm{X}\): independent variable, \(\mathrm{W}\): spatial weight, \(\rho\), \(\delta\): estimation coefficient, and \(\varepsilon\) : error term. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Underground Facility Information**} & \multicolumn{2}{c}{**Linear Regression Model**} \\  & Heating pipeline diameter (cm) & 0.000183 \\  & Heating pipeline average depth (m) & 0.000090 *** \\  & Heating pipeline age (years) & \(-\)0.021801 *** \\  & Heating pipeline diameter (cm) & \(-\)0.001326 *** \\  & Gas pipe average depth (m) & 0.000210 *** \\  & Gas pipe age (years) & 0.033765 *** \\  & Density of all six types & 0.000888 *** \\ \hline Spatial autocorrelation of standardized residuals & Global Moran’s I & 0.326620 \\  & z-score & 0.127047 \\ \hline Explanatory power of the model & \(\mathrm{R}^{2}\) & 44.358796 *** \\ \hline  & log\(-\)likelihood & 0.130332 \\ Fit of the model & AIC & \(-\)6183.22 \\  & SC & 12406.4 \\ \hline Non-normality & Jarque\(-\)Bera & 12587.0 \\ \hline Heteroscedasticity & Breusch\(-\)Pagan & 104,760.6103 *** \\ \hline Multicollinearity & Multicollinearity conditional number & 31,757.1448 *** \\ \hline  & LM\(-\)lag & 11.309430 \\ Spatial autocorrelation in spatial regression models & Robust LM\(-\)lag & 2869.9710 *** \\  & LM\(-\)error & 84.5001 ** \\  & Robust LM\(-\)error & 2807.7994 *** \\ \hline  & \(\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text\(\text{R}^{2}\) was used as a measure of explanatory power to compare spatial regression models. To determine the fit of the model, log-likelihood, Akaike information criterion (AIC), and Schwarz criterion (SC) were used. To determine the significance of the spatial autoregressive coefficient (\(\rho\), \(\lambda\)), the likelihood ratio test was used [31,32]. Table 5 presents the results of SEM and SLM. Statistics were checked to determine which model out of SLM or SEM was more appropriate for predicting ground subsidence vulnerability.