## 3 Methods
### Regression Model  
The GWR model takes into account the spatial heterogeneity of the distribution of each rail transit station (as shown in Figure 1), and the regression coefficients for each subway station take different values due to the differences in geographic location, and the model expression is shown in the following Equation (3):  
\[\hat{y}_{i}=\beta_{0}(u_{i},v_{i})+\sum\limits_{t=1}^{m}{\beta_{it}(u_{i},v_{i })x_{it}+\varepsilon_{i}} \tag{3}\]  
where \(\hat{y}_{i}\) is the fitted value of the dependent variable passenger flow y at the station i, \((u_{i},v_{i})\) are the spatial coordinates of the station i, and \(\beta_{it}(u_{i},v_{i})\) is the coefficient of the regression analysis of the \(t\)th independent variable of the continuous function \(\beta(u,v)\) at station \(i\). The GWR model provides a measurable measure of spatial heterogeneity.  
Figure 1: Map of the administrative area of Shenzhen, China.  
In the GWR regression analysis, the basic principle of calculating the weights is as follows: the closer the distance, the higher the weight value or, conversely, the lower the weight value. The weights that can realize the monotonic reduction function of spatial distance are called kernel functions. The two most commonly used kernel functions are the Gaussian function and the Bi-square function. Considering the overall spatial distribution of urban rail transit stations in Shenzhen, the kernel function of the GWR model adopts the Bi-square function, which is shown in the following Equation (4):  
\[\omega_{ij}=\begin{cases}\left[1-\left(d_{ij}/b\right)^{2}\right]^{2},&d_{ij} \leq b\\ 0,&d_{ij}>b\end{cases} \tag{4}\]  
where \(\omega_{ij}\) is the spatial weight of the estimated point \(j\) when fitting the model for subway station \(i\); \(d_{\text{ij}}\) is the distance between the two; and b is the kernel bandwidth, which is the key control parameter and can be specified either by a fixed distance (i.e., fixed bandwidth) or a fixed number of nearest neighbors (i.e., adaptive bandwidth). The setting of the bandwidth plays a decisive role in the regression model in terms of both prediction accuracy and stability.  
The optimal bandwidth can be found by minimizing a number of model fit goodness-of-fit diagnostics, and the choice of bandwidth values can be made by either CV (cross-validation) [31] or the AIC (Akaike information criterion) [32; 33]. CV generally considers only the accuracy of the prediction, whereas the AIC will consider parsimony, which balances accuracy with complexity. It is mainly to test and calibrate the level of fit of a regression model, and in experiments, a modified AIC, AIC\({}_{C}\), is usually used, which, unlike the basic AIC, is a function of sample size [34], i.e., the AIC\({}_{C}\) will be infinitely close to the AIC when the sample size is large. The AIC\({}_{C}\) formula [35] is shown in Equation (5) below:  
\[AICC=2n\ln(\hat{\sigma})+n\ln(2\pi)+n\left\{\frac{n+tr(S)}{n-2-tr(S)}\right\} \tag{5}\]  
where \(n\) is the sample size; \(\hat{\sigma}\) is the estimated standard deviation of the error term; and \(tr(S)\) denotes the trajectory of the hat matrix \(S\). The hat matrix is the projection matrix from the observation y to the fitted value \(\hat{y}\), where for the GWR, each row \(r_{i}\) of this hat matrix is shown in Equation (6):  
\[r_{i}=X_{i}(X^{T}W_{i}X)^{-1}X^{T}W_{i} \tag{6}\]  
where \(X_{i}\) is the \(i\)th row of the matrix of independent variables \(X\), and \(W_{i}\) is a diagonal matrix indicating the geographic weight of each observation at station \(i\).