## 5 Analysis of Influencing Factors of Urban Vitality
### Exploration of the Spatial Relationship Between Urban Vitality and Influencing Factors  
Prior to regression analysis, the selection of an appropriate spatial econometric model is essential. A multicollinearity assessment, employing the variance inflation factor (VIF), was conducted for urban vitality and 14 influencing factors within the central area of Qingdao. The results indicate VIF values exceeding 10 for building density, functional density and bus accessibility at the subdistrict scale, as well as for functional density at the block scale, thereby confirming the presence of multicollinearity among the variables. Following the removal of the aforementioned variables, the VIF test was repeated. The subsequent results indicated the absence of multicollinearity among the remaining factors. The details are presented in Table 2.  
After passing the multicollinearity test, the ordinary least squares (OLS) method was used for analysis, and only the built environment variables with significant influence were retained, effectively enhancing the robustness of the model. Goodness-of-fit measures were then compared with those from the MGWR model (Table 3), which demonstrated the highest adjusted R\({}^{2}\) and strongest fit. Moreover, compared with the classic GWR model, the MGWR model considered the scale of the independent variables' effects, providing support for exploring the spatial effects of various influencing factors. Therefore, the MGWR model was selected to study the spatial heterogeneity of driving factors in different regions (Tables 4 and 5). The R\({}^{2}\) values are 0.933 at the subdistrict scale and 0.824 at the block scale, while the adjusted R\({}^{2}\) values are 0.923 and 0.804, respectively, indicating a significant association between the potential factors and urban vitality. Figure 8: Interaction detection results of urban vitality influencing factors, * indicates bi-factor enhancement, ** indicates non-linear enhancement. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable** & **Max** & **Median** & **Min** & **Average** & **Standard Deviation** \\ \hline Resident population density & 0.413 & 0.208 & \(-\)0.075 & 0.210 & 0.162 \\ Building height & 0.318 & 0.315 & 0.303 & 0.314 & 0.003 \\ Commercial facility & 0.415 & 0.411 & 0.407 & 0.411 & 0.002 \\ accessibility & & & & & \\ Cultural and leisure facility accessibility & 0.332 & 0.330 & 0.325 & 0.330 & 0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Descriptive statistics of MGWR regression coefficients for the urban vitality at the subdistrict scale. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Modified Covariance Test (Subdistrict Level)** & **Modified Covariance Test (Block Level)** \\ \hline
**Variant** & **VIF Value** & **Variant** & **VIF Value** \\ \hline Resident population density (X1) & 2.677 & Resident population density (X1) & 1.485 \\ Unit area (X4) & 1.728 & Building density (X2) & 1.878 \\ Building height (X5) & 3.849 & Unit area (X4) & 1.865 \\ Greening rate (X6) & 2.751 & Building height (X5) & 1.612 \\ Morphological compactness (X7) & 1.765 & Greening rate (X6) & 1.644 \\ Functional mixing degree (X8) & 3.270 & Morphological compactness (X7) & 1.457 \\ Metro accessibility (X9) & 5.188 & Functional mixing degree (X8) & 2.106 \\ Road network density (X11) & 7.663 & Metro accessibility (X9) & 1.683 \\ Commercial facilities Accessibility (X12) & 4.619 & Bus accessibility (X10) & 1.696 \\ Public facilities Accessibility (X13) & 9.130 & Road network density (X11) & 2.796 \\ Cultural and leisure facilities & 4.053 & Commercial facilities Accessibility (X12) & 1.824 \\ Accessibility (X14) & & Public facilities Accessibility (X13) & 2.283 \\  & & & Cultural and leisure facilities \\  & & & Accessibility (X14) & 1.325 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Covariance test results.