## 3 Modelling and Parameterisation
### Gradient Boosting Decision Tree (GBDT) Model  
The Gradient Boosting Decision Tree (GBDT) model is a major class of algorithms in Boosting. It draws on the idea of gradient descent, with the basic principle of training new weak classifiers based on the negative gradient information of the current model's loss function and then combining the trained weak classifiers additively into the existing model. In other words, it makes decisions through the iteration of multiple trees. The core of GBDT is that each tree learns the residual of the sum of the conclusions of all previous trees, which is an additive amount that can yield the true value after adding the predicted value. It can make up for the shortcomings of the MGWR model's multivariate linear regression prediction and capture the nonlinear relationships between independent variables [23]. The algorithm process of this model is as follows [11]:  
Step 1: For the loss function \(L(P,f(x))\), set an initial constant model to minimise it, as shown in Equation (4) below:  
\[f_{0}(x)=\text{arg min}\sum_{i=1}^{N}L(P_{i},c) \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter Category** & **Parameter Setting** & **R2 (Inbound)** & \begin{tabular}{c} **Adjusted R2** \\ **(Inbound)** \\ \end{tabular} & **R2 (Outbound)** & \begin{tabular}{c} **Adjusted R2** \\ **(Outbound)** \\ \end{tabular} \\ \hline Bandwidth Selection & AICc criterion & 0.792 & 0.737 & 0.765 & 0.714 \\ Method & Cross-validation (CV) & 0.778 & 0.721 & 0.752 & 0.698 \\ Kernel Function Type & Quadratic function & 0.792 & 0.737 & 0.765 & 0.714 \\  & Gaussian kernel function & 0.785 & 0.729 & 0.758 & 0.705 \\ Convergence Threshold & \(1\times 10^{-5}\) & 0.792 & 0.737 & 0.765 & 0.714 \\  & \(1\times 10^{-3}\) & 0.788 & 0.732 & 0.76 & 0.709 \\ Variable Collinearity & \begin{tabular}{c} Exclude high collinearity \\ variables \\ \end{tabular} & 0.792 & 0.737 & 0.765 & 0.714 \\ Handling & \
\begin{tabular}{c} Include all original variables \\ \end{tabular} & 0.752 & 0.685 & 0.728 & 0.663 \\ Initial Value Setting &
\begin{tabular}{c} Initialise with MGWR \\ results \\ \end{tabular} & 0.792 & 0.737 & 0.765 & 0.714 \\  & Initialise with GWR results & 0.78 & 0.724 & 0.755 & 0.702 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sensitivity analysis of MGWR model parameters. where \(P_{i}\) is the dependent variable, \(L(P_{i},c)\) is the expression of the loss function, and c is the initial constant value that minimises \(f_{0}(x)\). Step 2: Calculate the negative gradient direction, as shown in Equation (5) below:  
\[r_{im}=-[\partial L(P_{i},f(x_{i})/\partial f(x_{i}))]_{f(x)=f_{m-1}(x)} \tag{5}\]  
where \(r_{im}\) is the residual obtained by calculating the negative gradient direction of the loss function at the \((m-1)\)-th iteration. Step 3: Calculate the optimal step size for gradient descent, as shown in Equation (6) below:  
\[\rho_{m}=\arg\min L[y_{i},f(x_{i})+\rho h_{m}(x_{i})] \tag{6}\]  
where \(h_{m}(x_{i})\) is the estimated value corresponding to the \(m\)-th tree, \(\rho\) is the change step length of the model's gradient iterative descent, and \(\rho_{m}\) is the step length that minimises the loss function corresponding to this iteration. Step 4: Iteratively calculate f\({}_{\rm m}\)(x), where the learning rate is \(\eta\), as shown in Equation (7) below:  
\[f_{m}(x)=f_{m-1}(x)+\varepsilon\rho_{m}h_{m}(x) \tag{7}\]  
where \(f_{\rm m}\)(x) represents the value obtained after the \(m\)-th iteration, \(\eta\) is the iterative learning rate, and 0 \(<\eta<\) 1. Step 5: Determine whether the model result meets the precision requirement. If the precision requirement is met, stop the calculation; otherwise, return to step 1.