## 3 Methodology
### Multiscale Flow-Focused Geographically Weighted Regression
The FGWR model of spatial interaction data is defined as follows (Equation (6)). \[\ln\big{(}\mathrm{M}_{\mathrm{ij}}\big{)}=\mathrm{k}_{\mathrm{ij}}+\alpha_{ \mathrm{ij}}\ast\ln\big{(}\mathrm{Tpop}_{\mathrm{i}}\big{)}+\beta_{\mathrm{ij} }\ast\ln\big{(}\mathrm{GDP}_{\mathrm{j}}\big{)}+\gamma_{\mathrm{ij}}\ast \mathrm{d}_{\mathrm{ij}}+\epsilon_{\mathrm{ij}} \tag{6}\]  
where \(\mathrm{k}_{\mathrm{ij}}\), \(\alpha_{\mathrm{ij}}\), \(\beta_{\mathrm{ij}}\), \(\gamma_{\mathrm{ij}}\) are local parameters to be calibrated by GWR using flow-based distance and spatial weight described in Sections 3.1 and 3.2. Since there is only one bandwidth in FGWR, FGWR limits the local relationships in each model to the same spatial scale and does not take into account the conditional relationships between each independent variable nor does it account for the variation of across different spatial scales. The MGWR method allows conditional relationships between response variables and for predictors to vary across different spatial scales. Contrasting with the global spatial interaction models in Equations (4)-(6), a local model specification of the flow data using MFGWR, which is the application of MGWR into flow data, is defined in Equation (7). \[\ln\big{(}\mathrm{M}_{\mathrm{ij}}\big{)}=\mathrm{k}_{\mathrm{ij}(\mathrm{b} _{0})}+\alpha_{\mathrm{ij}(\mathrm{b}_{1})}\ast\ln\big{(}\mathrm{Tpop}_{ \mathrm{i}}\big{)}+\beta_{\mathrm{ij}(\mathrm{b}_{2})}\ast\ln\big{(}\mathrm{ GDP}_{\mathrm{j}}\big{)}+\gamma_{\mathrm{ij}(\mathrm{b}_{3})}\ast\mathrm{d}_{\mathrm{ij}}+ \epsilon_{\mathrm{ij}} \tag{7}\]  
where \(\mathrm{b}_{1}\) indicates the bandwidth for calibration of the lth conditional relationship for \(1~{}=~{}\{1,\,2,\,3,\,\dots,\,\mathrm{n}\}\) and ij represents the location of flow from origin site i to destination site j. That is to say, there are four explanatory variables (including a constant) in the MFGWR model of transport flow, so each explanatory variable has its own spatial bandwidth in the regression analysis. Each flow has a set of parameters together with other local statistics, such as residuals and \(t\)-statistics. In the case of MFGWR, its parameters are calibrated as follows (Equation (8));  
\[\beta_{\mathrm{ij}}=\big{[}\mathrm{X}^{\mathrm{T}}\mathrm{W}_{\mathrm{ij}} \mathrm{X}\big{]}^{-1}\mathrm{X}^{\mathrm{T}}\mathrm{W}_{\mathrm{[ij]}} \mathrm{Y} \tag{8}\]  
where y is a \(\mathrm{n}\times 1\) vector of observations for the dependent variable, \(\beta_{\mathrm{[ij]}}\) is an \(\mathrm{m}\times 1\) vector of parameter estimates and X is an \(\mathrm{n}\times\mathrm{m}\) matrix of explanatory variables as follows (Equation (9)). \[\mathrm{X}=\left[ \begin{array}{cccc}1&\mathrm{x}_{12}&\ldots&\mathrm{x}_{1 \mathrm{m}}\\ 1&\mathrm{x}_{22}&\ldots&\mathrm{x}_{2\mathrm{m}}\\ \vdots&\vdots&\ddots&\vdots\\ 1&\mathrm{x}_{n2}&\ldots&\mathrm{x}_{nm}\end{array} \right] \tag{9}\]  
Each row in the hat matrix \(\mathrm{S}\) in the Gaussian GWR process is defined as Equation (10). \[\mathrm{r}_{\mathrm{i}}=\mathrm{X}_{\mathrm{i}}\Big{[}\mathrm{X}^{\mathrm{T}} \mathrm{W}_{\mathrm{[ij]}}\mathrm{X}\Big{]}^{-1}\mathrm{X}^{\mathrm{T}} \mathrm{W}_{\mathrm{[ij]}} \tag{10}\]  
where \(\mathrm{X}_{\mathrm{i}}\) is the ith row of \(\mathrm{X}\). In equation (8), \(\mathrm{W}_{\mathrm{[ij]}}\) is an n\(\times\)n diagonal matrix of spatial weight for flow [ij] (Equation (11)). The spatial weight matrix is a data-borrowing scheme that allows the flow data closer to the flow [ij] position to have a stronger influence on the local regression. \[\mathrm{W}_{\mathrm{[ij]}}=\left[ \begin{array}{cccc}\omega_{\mathrm{i}1}&0& \ldots&0\\ 0&\omega_{\mathrm{i}2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&\omega_{\mathrm{in}}\end{array} \right] \tag{11}\]  
There are two popular methods of calculating spatial weight \(\mathrm{W}\Big{(}\mathrm{d}_{\mathrm{ij}}\Big{)}\): fixed bandwidth and adaptive bandwidth.