## 3 Materials and Methods
### LBS Missing Path Completion  
#### 3.5.1 Completion of Missing Paths for LBS Sparse Trajectory Points  
Completing missing personal travel path information can improve the accuracy of LBS data and enhance the robustness of travel decision-making studies. As discussed in Section 3.3, Type B data exhibit significant spatial-temporal gaps between trajectory points, resulting in multiple possible paths. Therefore, conducting missing path completion for Type B data becomes necessary. The known information in Type B LBS data mainly consists of the spatial-temporal information of OD (i.e., the coordinates and time of departure and arrival), with some containing travel mode information. Through a series of steps, including traffic assignment, time point alignment, and others, we have established a core process to complement the missing travel paths, forming the essence of this research. The entire LBS missing path completion process is shown in Figure 6, demonstrating every step from plan generation, through screening and filtering, to the final path completion. Step 1: Generating travel plans with different route choices for travel demands. Since the original travel demands generated from these Type B LBS travel trajectory data do not specify the travel route information, we first generate the shortest S travel plans containing various possible shortest travel routes as the possible solution set for completing the missing paths based on simulation. When the value of S is too small, the travel solution set may not include the actual route selection. Conversely, when the value of S is too large, the efficiency of completion will decrease. Therefore, in this study, the value of S is set to 20 for paths with different degrees of deletion. In these travel plans, the planned departure time and location are consistent with the information of the starting O point in the Type B LBS travel demands. Similarly, the planned destination location is the same as the coordinate of the destination D point in the Type B LBS travel demands. The planned travel mode is consistent with the travel mode attribute of the Type B LBS travel demands. When the travel mode information in the original travel demand is unknown, the 20 possible travel routes generated include different route choices under various travel modes (private car, bus, and subway). Step 2: Obtaining travel time for each route through simulation. The different travel routes under the travel demands are input into the calibrated simulation model, and the travel time corresponding to each travel route in the feasible solutions of the travel demands is obtained through simulation execution:  
\\[\\left\\{(p_{i,k,1},T^{\\prime}_{\\text{O}_{i}D_{k}1}),\\ (p_{i,k,2},T^{\\prime}_{\\text{O}_{i}D_{k }2}),\\ldots,\\ (p_{i,k,m},T^{\\prime}_{\\text{O}_{i}D_{k}n})\\right\\} \\tag{9}\\]  
where \\(p_{i,k,1}\\) represents the \\(n\\)-th shortest path between the origin \\(\\text{O}_{i}\\) and the destination \\(\\text{D}_{k}\\), and \\(T^{\\prime}_{\\text{O}_{i}D_{k}n}\\) corresponds to the travel time of the path. Step 3: Matching and completion of feasible travel paths. The travel time between the origin and destination of the original travel demand is calculated as the actual travel duration on the vehicle's true path, shown as follows:  
\\[\\text{T}_{\\text{O}_{i}D_{k}}=\\text{t}_{\\text{D}_{k}}-\\text{t}_{\\text{O}_{i}} \\tag{10}\\]  
where \\(\\text{T}_{\\text{O}_{i}D_{k}}\\) represents the actual travel time of the trip between the origin \\(\\text{O}_{i}\\) and the destination \\(\\text{D}_{k}\\) corresponding to the LBS data, \\(\\text{t}_{\\text{D}_{k}}\\) is the time at which the travel demand is at \\(\\text{D}_{k}\\), and \\(\\text{t}_{\\text{O}_{i}}\\) is the time at which the travel demand starts at \\(\\text{O}_{i}\\). Figure 6: Path completion method.