## 3 Methodology
### Multiscale Flow-Focused Geographically Weighted Regression  
Spatial interaction modelling has been widely deployed to analyze various geographic mobilities (e.g., commuting, travel and migration). Currently, geographically weighted regression (GWR) [15; 32] is the most commonly used model for geographical-weighted regression analysis. In this paper, the MGWR model was applied to the modelling of transport flow for the first time. A simplified form of spatial interaction model is represented as follows (Equation (4)). \[\mathrm{M}_{\mathrm{ij}}=\mathrm{K}*\mathrm{P}_{\mathrm{i}}^{\mathrm{a}}* \mathrm{P}_{\mathrm{j}}^{\mathrm{b}}/\mathrm{f}\mathrm{f}\mathrm{d}_{\mathrm{ ij}} \tag{4}\]  
where \(\mathrm{M}_{\mathrm{ij}}\) is the interaction intensity between i and j, \(\mathrm{P}_{\mathrm{i}}\) and \(\mathrm{P}_{\mathrm{j}}\) represent the push and pull forces at the origin place i and destination place j respectively, \(\mathrm{d}_{\mathrm{ij}}\) denotes the distance between places i and j, and \(\mathrm{f}\mathrm{f}\mathrm{d}_{\mathrm{ij}}\) is a distance decay function of \(\mathrm{d}_{\mathrm{ij}}\). In practice, \(\mathrm{P}_{\mathrm{i}}\) and \(\mathrm{P}_{\mathrm{j}}\) are often replaced by the scale of two places, e.g., population of origin and GDP of destination. Many studies argue that the distance decay function varies with spatial scales [26]. The non-linear functions of quantifying the distance decay effect are dominated by an inverse power function and a negative exponential function. The former is  
Figure 4: The Euclidean distance between flow(ij) and flow(i\({}^{\prime}\)). more suitable for analyzing short distance interactions on the urban scale and the latter more suitable for analysing longer distance interactions (such as migration flow) on regional or national scales [33]. For example, at urban scales, this distance decay function \(\mathsf{f}\big{(}\mathrm{d}_{\mathrm{ij}}\big{)}\) is often expressed as an inverse power function, which is evidentially confirmed by ref. [26]. At the regional scale (e.g., province in this study) the function \(\mathsf{f}\big{(}\mathrm{d}_{\mathrm{ij}}\big{)}\) is better expressed as negative exponential function (e.g., \(\mathrm{e}^{-\beta\mathrm{d}}\)), where \(\beta\) is the distance friction coefficient. The higher the value of \(\beta\) is, the more sensitive to distance the flow is. In the case of transport flows at county level across a province, the pushing force at the county of origin is represented by its total population, \(\mathrm{Tpop}_{\mathrm{i}}\), and the pulling force at the destination county by its total GDP, \(\mathrm{GDP}_{\mathrm{j}}\). A negative exponential function is suitable for the distance decay at the regional scale. The expressway network distance used for this case study was calculated using ArcGIS 10.5 Network Analyst. The global model based on transport flow space interaction data is as follows (Equation (5)). \[\ln\big{(}\mathrm{M}_{\mathrm{ij}}\big{)}=\mathrm{k}+\alpha\ast\ln\big{(} \mathrm{Tpop}_{\mathrm{i}}\big{)}+\beta\ast\ln\big{(}\mathrm{GDP}_{\mathrm{j} }\big{)}+\gamma\ast\mathrm{d}_{\mathrm{ij}}+\epsilon_{\mathrm{ij}} \tag{5}\]  
where \(\mathrm{k}\) is the estimated intercept, \(\alpha\), \(\beta\) and \(\gamma\) are the parameters to be calibrated, \(\epsilon_{\mathrm{ij}}\) is the random error term, \(\mathrm{Tpop}\) and GDP are explanatory variables where \(\mathrm{Tpop}\) is the population of the county of origin and GDP is the GDP of the destination county. Other parameters are as described above. The global models mentioned above take no account of spatial heterogeneity in the spatial interaction modelling. The spatial non-stationarity, a form of heterogeneity, which refers to the varying relationships between dependent and independent variables across the study area, can be explored by the innovative method of geographically weighted regression [15; 32]. FGWR allows for spatial changes in parameters, and parameter calibration of the data in the vicinity of the data by \"borrowing\" nearby data.