- We propose a methodological framework to generate the network-based TCAs by walking.
- We propose a method to evaluate the geospatial accuracy of the network-based TCAs.
- We provide an open-source implementation of our framework.
2.1. Problem Definition and Assumptions
Given a road network graph and a set of facilities . Each facilityhas an associated cut-off distance . The aim is to design a method to generate n TCAs for these facilities in an efficient and accurate way.
2.2. Framework of Generating Network-Based Transit Catchment Areas (TCAs)
2.2.1. Subgraph Construction
- Building R-tree. Based on the input road network edges, an R-tree  is built to accelerate the nearest road searching and sub-edge extraction.
- Projecting facilities to the nearest edges. In order to measure the distance from/to a facility, each facility point is projected to its nearest edge. The nearest edge of a facility is retrieved using the nearest neighbor query of R-tree . Then, each facility is projected to its nearest edge by using a linear reference algorithm, which iterates through every segment (i.e., a line connecting two neighboring points of an edge) of the edge to determine the nearest segment . As illustrated in Figure 2, is the corresponding projected point of the facility.
- Extracting sub-edges. Based on the projected points obtained, a set of edges around each projected point can be identified. Given a facility with its projected point and the cut-off distance, a square searching box with a width of and centered at is created (Figure 2). The sub-edges of each facility are then extracted by finding the edges that intersect with its searching box with the assistance of the intersection query of R-Tree. Based on the extracted sub-edges of each facility, the corresponding subgraph for a facility can be constructed. Additionally, the projected point is inserted into as a new node. The parameter setting of needs to satisfy two requirements: (1) all the accessible edges (i.e., edges whose distance to/from are less than or equal to should be included in ; and (2) some edges beyond the distance of need to be included in , which will be used to interpolate additional boundary points of the catchment area (see Section 2.2.3). Therefore, should satisfy the following formula
2.2.2. Extended Shortest Path Tree Construction
- Constructing SP tree. Given a node as the root node, the SP tree starting from a root node can be constructed by employing Dijkstra’s algorithm.
- Finding the non-SP tree edges. As illustrated by an example in Figure 3. Some edges (the red edges in Figure 3b) are not included in the SP tree, which are termed non-SP tree edges. In order to construct an extended SP tree that includes these non-SP tree edges, we need to insert additional points into them. According to , if a given edge is a non-SP tree edge, there must be a point (termed as break point) on this edge that satisfies the following:
- Reconstructing SP tree. Each non-SP tree edge is split into two edges at its break point , namely edge and edge . Although and have the same location, there are regarded as two distinct nodes to avoid circular roads in the graph after inserting the break points . After the insertion of every single non-SP tree edge, an updated graph can be obtained. Then, by re-running Dijkstra’s algorithm on this updated graph, the corresponding extended SP tree can be generated (Figure 3c).
2.2.3. Contour Generation
- Segmenting edges. Given an edge represented by point sequence, where to are intermediate points of edge . We divide the edge into segments and add them as constraints during the triangulation (their endpoints thus act as the vertices of the triangulation). Since every single edge is included in the extended SP tree, the distance from a root node to an intermediate point can be calculated by the following formula.
- Building constrained triangulation. Based on the constrained segments obtained in the previous step, a constrained Delaunay triangulation is built for each extended SP tree by using the Computational Geometry Algorithms Library .
- Generating contour lines. Using the constrained triangulation as input, the contour lines specified at the cut-off distance (i.e., ) are generated by employing a tracing-based contour generation algorithm .
2.3. Generalization of the Framework
2.3.1. Generalization to the Directed Road Network
2.3.2. Generalization to Non-Point Facilities
2.4. Geospatial Accuracy Metrics
3. Case Study and Evaluation
3.1. Data Description and Experimental Set-up
3.2. Case Study
3.2.1. Undirected Graph with Point-Based Facility
3.2.2. Directed Graph and Non-Point Facilities
3.3. Geospatial Accuracy Evaluation
3.3.1. Visual Comparison
3.3.2. Quantitative Evaluation
3.4. Computation Efficiency Evaluation
3.4.1. Point-Based Facility
3.4.2. Non-Point Facility
Conflicts of Interest
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|Average Number of Grid Points in the TCAs||Average Precision||Average Recall||Average F1 Score|
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