## 2 Literature Review
### LBS Data Completion  
In practice, the availability and efficiency of LBS data are hindered by issues such as detector malfunctions, adverse weather conditions, and communication system failures. The partial absence of movement trajectory is a common occurrence. Specifically, within a specific time period, intermediate details in the travel (positional information and timestamp data) are missing, and only the OD information is retained. This situation seriously disrupts data continuity, making it impossible to accurately reconstruct the traveler's path choice and travel mode. Hence, it becomes necessary to employ specific technical methods to supplement the incomplete data, a process known as LBS data completion.  
Based on the idea of the shortest path method, the study proposed a macro-micro-integrated framework, combining particle filter (PF) and path flow estimation (PFE) to reconstruct the running path of vehicles. Their experiments with the VISSIM simulation model showed outstanding performance over a single PFE model [38]. The study proposed a trajectory-matching algorithm for emission-exceeding vehicles based on network topology and weights. Researchers select a group of adjacent matching trajectory candidate road segments using topological and spatial constraints. A composite weight, considering factors such as distance, direction, and relative position relationships, is then calculated for each candidate segment as its new evaluation criterion. Finally, the Dijkstra algorithm is employed to obtain the optimal matching path sequence based on these weights [39]. A researcher used the topological optimization based on the speed constraint (SC + TO) method combined with the technique for order preference by similarity to the ideal solution method (TOPSIS) to complete the extraction and reconstruction of vehicle travel paths [40]. The study constructed the feasible solution set of travel paths based on the K-th shortest paths algorithms (KSP). By establishing indicators such as path distance, travel time consistency, the number of signalized intersections, the degree of path preference, number of turns, and other decision indicators, the gray relational algorithm (GRA) determines the optimal completion path with a comprehensive accuracy of 92% [41]. Based on the idea of semi-supervised learning, the study constructed a path selection model based on sparse data by estimating the parameters of the maximum likelihood function. The experimental results showed that the method effectively improved the accuracy of the path selection model [42].  
In the era of transportation big data, traditional methods (e.g., shortest path method and topology optimization method based on speed limit) are not satisfactory for LBS travel path completion. Machine learning algorithms can explore the complex relationships behind data, but due to their black-box nature, subsequent analysis often lacks sufficient theoretical and model support, limiting the ability to understand and optimize model behavior. The emergence of multi-modal transportation (including bike-sharing and ride-sharing) has posed challenges in finding the most efficient path, as traditional methods only consider one travel mode. Completing a travel path using LBS involves various factors, e.g., traffic conditions, weather, road closures, and construction. Traditional methods that rely solely on speed limits or road topology are insufficient to address these complex scenarios effectively. Therefore, it is of great significance to establish a new modeling approach to improving imputation accuracy for missing travel paths in large-scale LBS data.