# Analysis of Ground Subsidence Vulnerability in Urban Areas Using Spatial Regression Analysis
## 4 OLS Analysis  
To develop a model for predicting ground subsidence vulnerability in urban areas, a multiple linear regression analysis was conducted to determine the influence of specific ground subsidence influencing factors (underground utility attribute information) on whether ground subsidence occurs. Multiple linear regression analysis represents the relationship between multiple explanatory variables and the dependent variable (whether or not ground subsidence occurred) as a regression equation, as presented in Equation (4), and the coefficient of the regression equation is estimated through the OLS that minimizes the sum of the squares of the residuals [22; 23]. \[\mathrm{y}=\beta_{0}+\beta_{1}\mathrm{x}_{1}+\beta_{2}\mathrm{x}_{2}+\cdots+ \beta_{n}\mathrm{x}_{n}+\varepsilon \tag{4}\]  
\(\mathrm{y}\): Dependent variable, \(\mathrm{x}_{\mathrm{i}}\): independent variable (explanatory variable), \(\beta_{\mathrm{i}}\): estimated regression coefficient, and \(\varepsilon\): error term. To build a multiple regression model, it is necessary to check for multicollinearity, which is the correlation between the explanatory variables. This is because the presence of multicollinearity will distort the estimated regression coefficients of the multiple regression model. Thus, multicollinearity is measured by the VIF. The VIF is generally based on a value of 10; if it is less than 10, it can be considered that multicollinearity is not present. Equation (5) presents the VIF calculation for the j-th regression coefficient estimate \(\beta_{\mathrm{j}}\)[24; 25]. \[\mathrm{VIF_{j}}=\frac{1}{1-\mathrm{R_{j}}^{2}} \tag{5}\]  
The coefficient of determination calculated from a regression model with \(\mathrm{R_{j}}^{2}:\mathrm{X_{j}}\) as the dependent variable. Table 3 presents VIFs of the attribute information of underground utilities used in this study. The diameter, average depth, age of six types of underground utilities, and density of all six types of underground utilities exhibited a VIF of 10 or less. Accordingly, all attribute information was used to perform the OLS analysis. The Jarque-Bera and Breusch-Pagan statistics were used to test for non-normality and heteroscedasticity in the OLS model. Table 4 presents the OLS analysis results. When performing an OLS analysis, it is necessary to consider assumptions about the presence of spatial effects, such as spatial autocorrelation. To do this, the distribution of residuals is identified using Moran's I value. Moran's I is a measure to verify the spillover effects that occur in spatial proximity. It is an indicator to check the correlation between events in a specific region and events in neighboring regions based on a spatially weighted matrix, such as Equation (6) [26]. \[\mathrm{I}=\frac{\mathrm{N}\sum_{i=1}^{n}\sum_{j=1}^{n}w_{\mathrm{ij}}\bigg{(} Y_{i}-\bar{Y}\bigg{)}\bigg{(}Y_{j}-\bar{Y}\bigg{)}}{\left(\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ \mathrm{ij}}\right)\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}} \tag{6}\]  
N: No. of regions, \(Y_{\mathrm{i}}\): attribute of region i, \(Y_{\mathrm{j}}\): attribute of region, \(\bar{Y}\): mean value, and \(w_{\mathrm{ij}}\): weigh value. \begin{table}
\begin{tabular}{c c} \hline \hline
**Category** & **Ground Subsidence** \\ \hline Gas pipe presence & 0.252 *** \\ Gas pipe density & 0.232 *** \\ Gas pipe diameter & 0.133 *** \\ Gas pipe average depth & 0.018 \\ Gas pipe age & 0.096 *** \\ All six types density & 0.240 *** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._Values of Moran's I usually range from \(-1\) to \(+1\). A value close to \(1\) indicates a positive spatial autocorrelation, while a value close to \(-1\) indicates negative spatial autocorrelation [27]. The value of Moran's I in the model proposed in this study is 0.127. This means that there is a positive spatial autocorrelation that increases the occurrence of ground subsidence as the number of six types of underground facilities increases. In addition, the z-score was 44.359 (\(p\)-value \(<0.001\)).