## 3 Data and Methodology
### MGWR Model  
It is important to note that, in GWR, all the explanatory variables are assigned the same bandwidth parameter, meaning that the spatial influence range is kept consistent across all variables in the modeling process [84]. However, this assumption may not be valid as different variables might have varying spatial effects. To address this limitation, we use the Multi-Scale Geographic Weighted Regression (MGWR) model, which allows different bandwidths for each explanatory variable to better capture the spatial heterogeneity of variable impacts [75; 87], and the model can be mathematically expressed as in Equation (11):  
\[y(u,v)=\beta_{0}(u,v)+\sum_{k}\beta_{k}^{bw^{*}_{k}}(u,v)x_{k}(u,v)+\varepsilon (u,v) \tag{11}\]  
where \(\beta_{bw^{*}_{k}}^{k}(u,v)\) is the regression coefficient of the explanatory variable \(k\) at location \((u,v)\) based on a specific bandwidth \(bw^{*}_{k}\), which is used in determining the regression coefficient of the explanatory variable \(k\). In the GWR framework, the local coefficients are estimated using a locally weighted-least-squares approach, and the weights for all the explanatory variables are derived from a single bandwidth. In contrast, each explanatory variable in MGWR model is permitted its own optimal bandwidth \(bw^{*}_{k}\) and constructs the variable-specific weight matrices. These individual weight matrices are then combined to form the overall weight matrix \(W(u,v)\) used in the local parameter estimation as in Equation (9). For the \(h\)th observation at location, the weight function of the explanatory variable \(k\) in MGWR is defined as in Equation (12):  
\[w^{h}_{k}(u,v)= \begin{cases}\left[1-\left(\frac{d((u,v),(u^{h},v^{h}))}{bw^{* }_{k}}\right)^{2}\right]^{2},&d((u,v),(u^{h},v^{h}))\leq bw^{*}_{k},\\ 0,&\text{otherwise}.\end{cases}  \tag{12}\]  
where the optimal bandwidth \(bw^{*}_{k}\) for the variable \(k\) is determined using the Golden Section Search, just as in GWR, to minimize the \(AICc_{k}(bw_{k})\). To operationalize the MGWR estimation, we employ a back-fitting procedure in which the optimal bandwidth for each explanatory variable is iteratively updated using Golden Section Search to minimize the corrected \(AICc_{k}\), as in GWR. This process is summarized in Algorithm 1:```
Input: Y: Dependent variable vector (\(n\times 1\)); \(\mathbf{X}\): Matrix of explanatory variables (\(n\times k\)); \(\mathbf{h}^{(0)}\): Initial bandwidth vector (using conventional GWR); Output: \(\mathbf{h}^{\text{opt}}\): Optimal bandwidth vector; \(\hat{\boldsymbol{\beta}}(u,v)\): Estimated local regression coefficients;
1 Step 1: Initialize the bandwidth vector \(\mathbf{h}^{(0)}\) using standard GWR estimation;
2 whileconvergence criterion not metdo
3 foreach variable \(k=1\) to \(p\)do
4 Fix all bandwidths except \(h_{k}\);
5 Optimize \(h_{k}\) using Golden Section Search: \[h_{k}^{\text{new}}=\arg\min_{h_{k}}AIC_{k}(h_{1},...,h_{k},...,h_{p})\] Update bandwidth: \(h_{k}\gets h_{k}^{\text{new}}\);
6 if\(\max|\mathbf{h}^{(t+1)}-\mathbf{h}^{(t)}|<\epsilon\)then
7 Convergence achieved;
8
9 else
10\(\mathbf{h}^{(t)}\leftarrow\mathbf{h}^{(t+1)}\);
11 return\(\mathbf{h}^{opt}=\mathbf{h}^{(t)}\), \(\hat{\boldsymbol{\beta}}(u,v)\)
```

**Algorithm 1**Back-Fitting Procedure for MGWR Bandwidth Optimization

This method allows each variable to have a distinct spatial influence range, adapting more effectively to the spatial scale of relationships. ## 4 Results

### Temporal and Spatial Pattern of Ride-Hailing Services in Haikou's Urban Area

This section analyzes the temporal variation and spatial distribution of ride-hailing demand in Haikou. Figure 2 presents (a) the average number of trips and (b) the average travel duration of ride-hailing in Haikou. The x-axis, shared by both panels, represents the time of day. A distinct morning peak occurs between 7:00 and 8:00, followed by the highest daily volume between 17:00 and 18:00. The morning peak is more pronounced on workdays, whereas holidays tend to show higher average hourly trip volumes during non-peak periods.