## 3 Results
### Results of MLR Models and Evaluation of Multi-Scale BCEs
#### 3.1.2 The Validity of Multi-Scale Estimation Results  
Based on the regression results in Table 2, BCEs with a 1 km resolution can be calculated using effective factors of remote sensing data. To verify the results, the estimation results at the prefecture and county (municipality) levels in 2000, 2005, 2010, 2015, and 2019 were compared to emissions from the building sector provided by the Emissions Database for Global Atmospheric Research (EDGAR). EDGAR is a global database of the total CO\({}_{2}\) emissions and sector CO\({}_{2}\) emissions at an approximately 10 km \(*\) 10 km resolution. The building sector's CO\({}_{2}\) emissions from EDGAR focus on emissions caused by the combustion of fossil fuel at a country level, and the spatial proxies of it are fishing, the rural population, and urban population [69]. Although the accuracy of grid emissions from EDGAR's building sector is affected by the population [70], it largely reflects the activity intensity of the building sector and, is appropriate for comparison with the estimated results in this study since the population is an important parameter in downscaling BCEs. The fitting results at the prefecture level in the BTH showed that R\({}^{2}\) and RMSE were 0.81 and 11.91 million tons, respectively (Figure 3a). The R\({}^{2}\) and RMSE between estimation and EDGAR at the prefecture level in the NCRJ were 0.85 and 12.11 million tons, respectively, which were higher in the BTH (Figure 3c). Counties in the BTH and municipalities more than 100 square kilometers in the NCRJ were selected to verify the estimation results of BCEs at a finer scale. We found that the R\({}^{2}\) of county-level BCEs and EDGAR in the BTH urban agglomeration was 0.67, lower than that of the prefecture level, and the RMSE was about 0.95 million tons (Figure 3b). According to the fitting results, the R\({}^{2}\) and RMSE of the municipality level in the NCRJ were higher at 0.92 and 1.78 million tons, respectively (Figure 3d). The fitting results at different scales indicated that this study's estimation results of BCEs are highly consistent with the scale trend in emissions from the building sector of the known database, among which the fitting effect of the NCRJ is better at a multi-scale. However, considering RMSE, the RMSE of the BTH urban agglomeration is lower. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **BTH** & **NCRJ** \\ \hline Public BCE & \(BCEP=12.076* POP+9.464*PLU+0.081*ST+0.052*WT\) & \(BCEP=-1.346*NTL+1.893*POP-0.538*PLU+1.655*ST\) \\ Residential BCE & \(BCER=0.094*NTL+87*POP\) & \(BCER=-0.439*NTL+1.189*POP-0.332*EV1+0.604*ST\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Models for downscaling BCEs in the BTH and NCRJ. Figure 3: Fitting results between this study’s estimations and EDGAR at a multi-scale. Since the existing grid-scale BCE dataset with a 1 km resolution is limited, the grid-scale emissions of this study are resampled to a 10 km resolution to compare with EDGAR's building sector (Figure A4a,c). Existing studies use other data related to building to verify their estimation results. Therefore, we compared grid-scale BCEs in 2019 with the grid-scale building volume in 2020, which is calculated using the building height in 2018 and built-up surface area in 2020 provided by the GHSL (Figure A4b,d). Although EDGAR's building sector and building volume both show a linear correlation with the grid-scale estimation value of the BTH, the explanation for the fitting results is relatively lower compared with that at the prefecture and county levels, indicating the limitation of grid-scale estimation results in this study to a certain extent. Similarly, grid-scale fitting results in the NCRJ also show a significant linear correlation with EDGAR's sector and building volume. However, the fitting results show a higher correlation between BCEs and EDGAR's building sector, the R\({}^{2}\) of which is 0.86.