## 4 Results
### Comparison between FGWR and MFGWR  
As argued before, spatial dependence should be considered in the process of flow modelling. The flow-focused global Moran Indice (shown in Table 4) was used to calculate the spatial autocorrelation \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Estimate Value** & **Standard Error** & **z-Value** & _p-Value_ \\ \hline k & \(-\)3.059 & 0.243 & \(-\)12.59 & \(<\)0.0001 \\ \(\alpha\) & 1.457 & 0.037 & 39.392 & \(<\)0.0001 \\ \(\beta\) & 1.068 & 0.026 & 41.59 & \(<\)0.0001 \\ \(\gamma\) & \(-\)0.009 & 1.725e-4 & \(-\)52.169 & \(<\)0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameter estimates for the global flow model calibrated by OLS. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Diagnostic** & **OLS** & **FGWR** & **MFGWR** \\ \hline Bandwidth 1  & \(\infty\) & 69 & [53,3463,57,47] 2  \\ SNP 3  & 4 & 725.146 & 678.537 \\ R\({}^{2}\) & 0.626 & 0.832 & 0.841 \\ AIC & 11,481.240 & 9336.018 & 9095.615 \\ AICc & 11,483.257 & 9719.491 & 9425.881 \\ RSS & 5503.605 & 1962.521 & 1881.271 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model fit statistics for OLS, FGWR and MFGWR. of (both global and local) model residuals. The fixed distance band for calculating the weight was 1.39285. The global Moran Indice was 0.08505 for OLS, 0.01154 for FGWR, and 0.00596 for MFGWR (Table 4). Compared to OLS, FGWR significantly reduced spatial autocorrelation, while MFGWR outperformed FGWR. This indicates that FGWR is more efficient than OLS in considering spatial dependence, and MFGWR is the most effective of the three. Another strength of MFGWR over FGWR is its suitability for mapping the estimated parameters across the study area [15], which is applicable to flow modelling. MFGWR and FGWR produce a set of parameter estimations and corresponding \(t\)-statistics (Table 5). The standard deviation of the parameter estimates for each explanatory variable of MFGWR is smaller than the standard deviation of the FGWR parameter estimates. This indicates that the MFGWR parameter estimates were more clustered. It can also be seen from Table 5 that the range of parameter estimates of MFGWR was smaller and more concentrated than FGWR which indicates that the surface of the FGWR parameter estimation was not smooth enough, whereas the surface of the MFGWR parameter estimation value was more uniform and smooth. Only when the \(t\)-statistic value is large enough, the estimated parameter is meaningful and interpretable [39]. At the significance level of 5%, the \(t\)-statistic threshold value is 1.96/\(-\)1.96, although MFGWR may have a slightly varied (higher) threshold value from the traditional significance test [15]. The histograms of the \(t\)-statistic values in response to three explanatory variables (distance, Topp and GDP) are displayed in Table 5 and Figure 6. It is clear to see a proximate normal distribution for all three for FGWR. However, the \(t\)-statistic distribution of the three parameters of MFGWR is obviously not an approximately normal distribution. This is because MFGWR spatially controls the range of influence of each explanatory variable, so the frequency distribution of \(t\)-statistics is not an approximately normal distribution. At the 5% level of FGWR, there were a total of 2970 significant parameter estimations of distance, 2416 significant parameter estimations of Topp and 2995 significant parameter estimations of GDP. At the 5% level of MFGWR, the parameter estimates for all explanatory variables were significant, which means MFGWR is more robust. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Global\_Moran\_I** & **z\_Norm** & **p\_Norm** \\ \hline OLS & 0.08505 & 49.6058 & 0.00000e+00 \\ FGWR & 0.01154 & 6.8723 & 6.31917e-12 \\ MFGWR & 0.00596 & 3.63033 & 2.83059e-04 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of global Moran I of residual between OLS, FGWR and MFGWR.