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Transportation simulation and analysis projects that utilize maps with inappropriate fidelity levels carry a significant risk of having poor runtime or poor prediction performance. To address this, researchers use map abstraction method to abstract out a simplified map with fewer links and nodes based on the original full detailed map. Traditional static abstraction methods produce analysis maps with a single fidelity across the entire planning horizon, which cannot reflect the dynamic changes of daily traffic. This paper proposes a spatiotemporal dynamic map abstraction approach that adopts a time series clustering method to segment the analysis time horizon adaptively based on a Macroscopic Fundamental Diagram (MFD) curve, which describes network-wide dynamic traffic states. Time periods with similar macro-performance are grouped into one subinterval. A map with a dedicated fidelity is produced for each subinterval. Furthermore, a simulation is run on multiple abstracted maps with different fidelities in a sequence according to their temporal order. A numerical experiment ascertains that the proposed approach has promising results in both analysis accuracy and efficiency for resource-constrained modeling agents.

Fidelity in traffic analysis refers to either simulation fidelity or network fidelity. The simulation fidelity is embodied in simulation models, such as macro-, meso-, and microscopic simulations. Network fidelity can be interpreted as the degree of detailed expression of a road network, which includes both map topological fidelity and link representation fidelity. The efficiency and accuracy of traffic analysis hinge on the topological fidelity of the network. A traffic network is essentially a spatially simplified and temporally discretized representation of a transportation system. The network topology fidelity level is often chosen at the modeling agents’ discretion, based on the trade-off between runtime efficiency and prediction power.

The impact of the topological fidelity on efficiency and accuracy is indicated in Figure

The impact of topological fidelity on the efficiency and accuracy of traffic analysis.

Although existing event-based mesoscopic traffic simulators are able to achieve a faster-than-real-time simulation performance in simulating city-scale networks [

The idea of aggregation has long been used in traditional map abstraction. A typical aggregation approach uses elements extraction [

In relevant map generalization and cartography studies, the concept of map abstraction is more about generating a legible and scale reduced map. For this, methods of network facility selection and deletion have been widely discussed [

In the field of traffic analysis and modeling, early studies divided the traffic network into several research zones and took these zones as analysis units to model the network. Smith [

In response to the problems of disconnectivity and vehicle routing selection, the Connectivity Enhancement Algorithm [

However, all these approaches are static, which means that temporal traffic dynamics caused by vehicle movement and time-varying traffic demand is omitted. Static methods, such as the ones used to filter out low-class links, may also introduce prediction problems. As shown in Figure

Different types of road network topology.

Dense network (London, UK)

Broad network (Beijing, China)

Sparse network (Xiamen, China)

A hybrid simulation idea incorporating macro-, meso-, and micro-counterparts in one single map has been raised recently. The motivation of using hybrid simulations is simple and straightforward. A single analysis map with fixed fidelity, like Figures

Conceptualization of simulations in spatial–temporal fidelity domains: (a) mesoscopic, (b) microscopic, (c) hybrid, and (d) adaptive [

However, the spatial-dependent methods solely care about the spatial heterogeneity of traffic patterns and omit the temporal variations of traffic within a day to some extent. Figure

Temporal and network heterogeneity of traffic patterns.

EI Paso, TX

Tucson, AZ

Austin, TX

A MFD is a type of fundamental diagram [

The proposal of MFD inspired many scholars to study its shape, properties, and applications, and design various MFD-based traffic management and control strategies. In the Yokohama experiment, Geroliminis and Daganzo [

Because real-life large-scale traffic networks have multiple levels of roads, network heterogeneity may impact the shape of the MFD. Therefore, the idea of network partitioning has been raised by some researchers [

Considering the evidently temporal variability of traffic dynamics, there have also been studies that temporally clustered the planning time horizon based on MFD to help detect homogeneous time periods. Pascale, Mavroeidis [

As the concept of MFD is relatively new, to the best of the author’s knowledge, existing applications of MFD are limited. In recent years, the practical application of MFDs has become a hot research area. Ampountolas, Zheng [

To overcome the shortcomings of the static and semidynamic (spatial dynamic) map abstraction approaches, we propose a full-dynamic, or adaptive (both temporal and spatial) map abstraction framework that can adjust the fidelity of a map with reference to both temporal and spatial traffic dynamics. Figure

Full-dynamic map.

The paper is organized as follows: Section

The purpose of the proposed spatiotemporal dynamic map abstraction method is to abstract a series of abstracted maps with varying fidelities to fit the traffic dynamics of different time periods.

The overall workflow of the proposed spatiotemporal dynamic map abstraction method is presented in Figure

Workflow of the spatiotemporal dynamic map abstraction method.

The abstraction starts from a detailed original map, and the sketch map with only high-class roads retained is directly extracted from the original map. What needs to be guaranteed is that there must exist at least one generation link and one destination nodes in each TAZ. Thus, the vehicles entering can at least find one path out. The MFD chart can be plotted after calculating the network-wide average flow, density, and speed. According to the macro performance of the analysis traffic network, the MFD-based time series clustering method, Advanced Toeplitz Inverse Covariance-Based Clustering (ATICC) then clusters the MFD curve and segments the time horizon into multiple subintervals. For each subinterval

Once a series of abstracted maps is obtained, they are integrated and applied to SBDTA. SBDTA provides a network-wide Dynamic User Equilibrium (DUE) flow pattern, which is adopted as the input to CEA to embody traffic dynamics. It is worth mentioning that three unique strategies are adopted herein to cope with the boundary issues caused by temporal segmentation between the sequentially abstracted maps.

The iteration of CEA and SBDTA continuously adds critical links to the sketch map and will stop once the abstracted map reaches stability; i.e., the map cannot be expanded any more. With the expanded abstracted map generated, the postprocessing procedure is then performed to remove misclassified links with almost no traffic flow and to deal with the dis-connectivity problem. After the completion of the postprocessing procedure, the final abstracted dynamic map can be exported.

The major steps of the proposed adaptive map abstraction method are outlined below.

Prepare the original map and extract the sketch map, run SBDTA on the original map, and calculate the network average traffic flow, density, and speed.

Run ATICC to cluster the MFD curve and get the planning horizon segmentation scheme.

Run CEA on the sketch map for each subinterval to expand the map.

Combine a series of sequential abstracted maps with different fidelities and label the links against the original map.

Handle the boundary issues. Run SBDTA to get the DUE solution.

Check whether new links can be added to the abstracted maps, and if there are none, stop criterion is satisfied. Otherwise, go back to Step

Postprocessing procedure is performed, and selectively remove the noncritical links and guarantee network connectivity.

The following segment introduces the MFD-based time series cluster (Step

In consideration of the temporal continuity of the MFD curve, it is reasonable to consider the traffic flow, density, and speed as time series and to encourage adjacent sequences to be clustered together. The clustering algorithm adopted here is Advanced Toeplitz Inverse Covariance-Based Clustering (ATICC).

The definitions of space-mean traffic flow, density, and speed are

where

ATICC was developed based on the fundamental multivariate time series data clustering approach, Toeplitz Inverse Covariance-Based Clustering (TICC) [

Suppose that

where

Thus,

The overall TICC problem can be expressed as follows:

where

In adopting TICC to perform time horizon segmentation in the map abstraction experiment, the unique features of traffic flow data, i.e., the existence of morning and evening peak hours, should be taken into consideration. The original TICC needs to be modified, and we propose the updated clustering method, ATICC.

The MFD-based temporal clustering of traffic analysis networks is the focal point of ATICC. To facilitate the analysis of traffic pattern differences between peak and nonpeak hours, the tendency is to separate out peak hours. As such, another penalty parameter

where

A smaller

where

The traffic analysis map of each subinterval needs to be abstracted separately based on the division plan of ATICC. The Connectivity Enhancement Algorithm (CEA) is an efficiency dynamic map abstraction method, which helps to catch critical links. It contains two main procedures: Topological Nearest Neighbors Search (TNNS) [

TNNS is an efficiency node search algorithm, which considerably reduces the search scope. Each node of the sketch map acts as a search node for the first round of searching. Starting from one search node, it performs Breadth-First-Search (BFS) [

The SP comparison procedure then computes the shortest path costs (DUE travel time) from each search node to its TNNs and compares the costs of the sketch map and original map, respectively. For the same OD pair, if the SP cost ratio between the original map and sketch map is less than or equal to a given constant (greater than one), the original map’s shortest path and the corresponding nodes will be expanded to the sketch map. The nodes added after SP comparison continue as the search nodes and are passed to the next iteration. By taking DUE travel time as the link cost, this SP comparison procedure can accurately capture traffic dynamics.

By alternating TNNS and SP comparison procedures, the sketch map is expanded. CEA stops when all search nodes are visited, and no more links can be added. The ultimate expanded sketch map is then exported as an initial abstracted map.

In order to demonstrate the feasibility and effectiveness of the proposed method, the spatiotemporal dynamic map abstraction approach is applied to a real transportation network–the Alexandria network. DynusT [

The Alexandria network is a high-fidelity digital map with detailed road information, and was downloaded from Open Street Map [

Analysis traffic maps: (a) original map, (b) sketch map.

The planning time horizon covers 24 hours (1,440 minutes), which is consistent with the OD demand matrix time range. The traffic demand amount file is derived from the US Census Bureau Public Use Microdata Sample. We derived the network MFD curve based on the dynamic traffic flow and density patterns obtained from the initial SBDTA run. Figure

Network MFD (the projection in green is the time-dependent flow pattern, the projection in red is the time-dependent density pattern, and the projection in yellow is the relationship between flow and density).

The planning horizon segmentation result of ATICC and the statistical information of each abstracted map is summarized and elaborated in the following parts.

ATICC is performed here to simultaneously cluster the MFD data points and further segment the planning time horizon. Based on the macro traffic performance presented by the MFD, it intends to segment the time horizon into six subintervals, i.e., the subinterval number

ATICC clustering result. Each color stands for one cluster with peak hours identified and captured.

To demonstrate the rationality of the clustering scheme, the traffic demand distribution of each analysis period is given in Figure

Subinterval traffic demand distribution.

Compared with the nonsegmentation MFD, the main advantage of the time series data segmentation scenario is that the traffic behavior remains nearly identical over the network in the same cluster, with the standard deviations for each cluster at 0.016, 0.451, 1.110, 0.097, 0.550, and 0.436, respectively. To interpret the difference of network density between adjacent subintervals, Student’s t test was adopted. The null hypothesis suggests that there exists no statistical significance in two sets of given observations. With a significance level of 0.05, the P-value is equal to 0.000 in each test between adjacent subintervals, which is less than 0.05. Therefore, we reject the null hypothesis.

Once the time horizon segmentation scheme is obtained, the CEA and SBDTA are iteratively executed for each subinterval, and 6 initial abstracted traffic analysis maps with different fidelities can be exported.

Table

Statistical information of the maps.

sub_1 | sub_2 | sub_3 | sub_4 | sub_5 | sub_6 | original map | |
---|---|---|---|---|---|---|---|

period/min | 0-255 | 256-375 | 376-595 | 596-980 | 981-1160 | 1161-1440 | 0-1440 |

Duration/min | 255 | 120 | 220 | 385 | 180 | 280 | 1440 |

Node.Num | 855 | 967 | 1291 | 1103 | 1236 | 970 | 2573 |

Link.Num | 1675 | 1877 | 2715 | 2205 | 2608 | 1883 | 6724 |

Avg.dense/(veh/mile/lane) | 0.025 | 0.598 | 3.848 | 2.102 | 3.107 | 0.566 | 1.702 |

Std.dense | 0.016 | 0.451 | 1.110 | 0.097 | 0.550 | 0.436 | 1.458 |

(Note: Avg.dense = Average density, Std.dense = standard deviation of density).

Figure

Abstracted map combinations.

sub_1

sub_2

sub_3

sub_4

sub_5

sub_6

What is more, the spatial distribution can also be captured. The eastern area of the map is apparently denser and has more links added. Therefore, the fidelity of the analysis map is adjustable in both spatial and temporal dimension, which implies the realization of adaptive topological map representation.

The final series of abstracted maps is fed into the SBDTA model. The performance statistics of SBDTA on the dynamic map and the original full static map with identical traffic demand are shown in Table

Experiment validation results.

Avg.time/min | Avg.dist/mile | Aff.time/min | CPU/sec | |
---|---|---|---|---|

Abstracted map | 5.869 | 5.044 | 5.857 | 10389.38 |

Original map | 5.607 | 4.818 | 6.010 | 13833.80 |

Sketch map | 11.641 | 5.533 | — | 11664.33 |

Error (%) | 4.67 | 4.69 | 2.55 | — |

En.eff(%) | — | — | — | 24.90 |

The prediction performance of the abstracted map scenario is close to that of the full static map scenario. The errors of the predicted traffic pattern, which are described by the average travel time and the average travel distance, are both less than 5%. In Jafari’s contraction simulation experiment [

Meanwhile, among the total 529,336 vehicles, 194,121 were affected by the map abstraction. The error of the affected vehicles’ average travel times between two maps was 0.15 minute, and the difference ratio was less than 3%. The impact of boundary issues is therefore manageable, as the travel time error is small compared with the full static map.

On the other hand, the CPU time savings were roughly 25%, which proves that the abstracted map can alleviate computational burden to a great extent. Furthermore, the comparison of the CPU time with the sketch map validates that the map with the lowest fidelity may contradictorily produce the worst efficiency. The substantial growth of the vehicle average travel time confirms that the excessive deletion of links, especially critical links, leads to additional congestion and takes extra computational time.

Figure

CPU times of simulation and assignment over iterations in SBDTA. A comparison between the original detailed map scenario, abstracted dynamic map scenario, and sketch map scenario.

It can also be found that the sketch map takes more CPU time in simulation and assignment than both the abstracted dynamic map and the original detailed map. The extension of simulation CPU time is closely related to the additional vehicles caused by congestion. The impact of massive link reduction on assignment can be summarized in two aspects. On one hand, fewer links need to be calculated in TDSP, which saves computational time in the spatial dimension. On the other hand, the congestion caused by the sketch map leads to longer search times in TDSP for the same OD pair. The interplay of the computational time reduction in the spatial dimension and the corresponding increment in the temporal dimension eventually reaches a balance condition, resulting in almost identical assignment computational times.

Figure

The CDF of travel time frequency in original detailed map and abstracted dynamic map.

The relative gap for the DUE condition is usually adopted as the convergence criterion of SBDTA. Figure

Convergence results.

Comparing the accumulation volume of the abstracted dynamic map with the original detailed map, we see the equilibrium flow pattern shown in Figure

Fitting results.

coefficient | R-square | |
---|---|---|

sub_1 | 0.869 | 0.457 |

sub_2 | 1.020 | 0.873 |

sub_3 | 1.013 | 0.927 |

sub_4 | 1.034 | 0.910 |

sub_5 | 1.023 | 0.934 |

sub_6 | 1.058 | 0.859 |

accu_vol | 1.044 | 0.963 |

Equilibrium flow pattern.

This paper presented an innovative MFD-based spatiotemporal dynamic map abstraction method that can adaptively segment the planning horizon and carry out abstraction in each subinterval to obtain a proper-fidelity map for traffic analysis. The proposed abstraction method is able to balance the efficiency and accuracy of traffic analysis. The main conclusions are summarized below:

The abstraction of an analysis map only needs to be performed once and the resulting abstracted map can be applied to future scenarios. However, the MFD used in our experiment is actually an artificial MFD, which is calculated from a simulation run. It would be more reasonable to use a MFD calculated from detector data, and it remains to be further explored.

The Alexandria network data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This research was sponsored by the National Key Research and Development Plan of China (2018YFB1600800), the Shanghai “Sailing” talent program (19YF1451200), the Shanghai Science and Technology Committee project (19692108700), and Fundamental Research Funds for the Central Universities (22120180622).