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Deep learning-based multivariate and multistep-ahead traffic forecasting models are typically trained with the mean squared error (MSE) or mean absolute error (MAE) as the loss function in a sequence-to-sequence setting, simply assuming that the errors follow an independent and isotropic Gaussian or Laplacian distributions. However, such assumptions are often unrealistic for real-world traffic forecasting tasks, where the probabilistic distribution of spatiotemporal forecasting is very complex with strong concurrent correlations across both sensors and forecasting horizons in a time-varying manner. In this paper, we model the time-varying distribution for the matrix-variate error process as a dynamic mixture of zero-mean Gaussian distributions. To achieve efficiency, flexibility, and scalability, we parameterize each mixture component using a matrix normal distribution and allow the mixture weight to change and be predictable over time. The proposed method can be seamlessly integrated into existing deep-learning frameworks with only a few additional parameters to be learned. We evaluate the performance of the proposed method on a traffic speed forecasting task and find that our method not only improves model performance but also provides interpretable spatiotemporal correlation structures., Comment: 11 pages, 4 figures, 2 table

Modeling the relationship between vehicle speed and density on the road is a fundamental problem in traffic flow theory. Recent research found that using the least-squares (LS) method to calibrate single-regime speed-density models is biased because of the uneven distribution of samples. This paper explains the issue of the LS method from a statistical perspective: the biased calibration is caused by the correlations/dependencies in regression residuals. Based on this explanation, we propose a new calibration method for single-regime speed-density models by modeling the covariance of residuals via a zero-mean Gaussian Process (GP). Our approach can be viewed as a generalized least-squares (GLS) method with a specific covariance structure (i.e., kernel function) and is a generalization of the existing LS and the weighted least-squares (WLS) methods. Next, we use a sparse approximation to address the scalability issue of GPs and apply a Markov chain Monte Carlo (MCMC) sampling scheme to obtain the posterior distributions of the parameters for speed-density models and the hyperparameters (i.e., length scale and variance) of the GP kernel. Finally, we calibrate six well-known single-regime speed-density models with the proposed method. Results show that the proposed GP-based methods (1) significantly reduce the biases in the LS calibration, (2) achieve a similar effect as the WLS method, (3) can be used as a non-parametric speed-density model, and (4) provide a Bayesian solution to estimate posterior distributions of parameters and speed-density functions.

Accurate forecasting of bus travel time and its uncertainty is critical to service quality and operation of transit systems; for example, it can help passengers make better decisions on departure time, route choice, and even transport mode choice and also support transit operators to make informed decisions on tasks such as crew/vehicle scheduling and timetabling. However, most existing approaches in bus travel time forecasting are based on deterministic models that provide only point estimation. To this end, we develop in this paper a Bayesian probabilistic forecasting model for bus travel time. To characterize the strong dependencies/interactions between consecutive buses, we concatenate the link travel time vectors and the headway vector from a pair of two adjacent buses as a new augmented variable and model it with a constrained Multivariate Gaussian mixture distributions. This approach can naturally capture the interactions between adjacent buses (e.g., correlated speed and smooth variation of headway), handle missing values in data, and depict the multimodality in bus travel time distributions. Next, we assume different periods in a day share the same set of Gaussian components but different mixing coefficients to characterize the systematic temporal variations in bus operation. For model inference, we develop an efficient Markov chain Monte Carlo (MCMC) sampling algorithm to obtain the posterior distributions of model parameters and make probabilistic forecasting. We test the proposed model using the data from a twenty-link bus route in Guangzhou, China. Results show our approach significantly outperforms baseline models that overlook bus-to-bus interactions in terms of both predictive means and distributions. Besides forecasting, the parameters of the proposed model contain rich information for understanding/improving the bus service.

Forecasting the short-term ridership among origin-destination pairs (OD matrix) of a metro system is crucial in real-time metro operation. However, this problem is notoriously difficult due to the high-dimensional, sparse, noisy, and skewed nature of OD matrices. This paper proposes a High-order Weighted Dynamic Mode Decomposition (HW-DMD) model for short-term metro OD matrices forecasting. DMD uses Singular Value Decomposition (SVD) to extract low-rank approximation from OD data, and a low-rank high-order vector autoregression model is established for forecasting. To address a practical issue that metro OD matrices cannot be observed in real-time, we use the boarding demand to replace the unavailable OD matrices. Particularly, we consider the time-evolving feature of metro systems and improve the forecast by exponentially reducing the weights for old data. Moreover, we develop a tailored online update algorithm for HW-DMD to update the model coefficients daily without storing historical data or retraining. Experiments on data from a large-scale metro system show the proposed HW-DMD is robust to the noisy and sparse data and significantly outperforms baseline models in forecasting both OD matrices and boarding flow. The online update algorithm also shows consistent accuracy over a long time when maintaining an HW-DMD model at low costs., Comment: Transportation Science (2022)

One of the major goals of transport operators is to adapt the transport supply scheduling to the passenger demand for existing transport networks during each specific period. Another problem mentioned by operators is accurately estimating the demand for disposable ticket or pass to adapt ticket availability to passenger demand. In this context, we propose generic data shaping, allowing the use of well-known regression models (basic, statistical and machine learning models) for the long-term forecasting of passenger demand with fine-grained temporal resolution. Specifically, this paper investigates the forecasting until one year ahead of the number of passengers entering each station of a transport network with a quarter-hour aggregation by taking planned events into account (e.g., concerts, shows, and so forth). To compare the models and the quality of the prediction, we use a real smart card and event data set from the city of Montr\'eal, Canada, that span a three-year period with two years for training and one year for testing., Comment: 18 pages, 13 figures, 7 tables

Accurate forecasting of passenger flow (i.e., ridership) is critical to the operation of urban metro systems. Previous studies mainly model passenger flow as time series by aggregating individual trips and then perform forecasting based on the values in the past several steps. However, this approach essentially overlooks the fact that passenger flow consists of trips from each individual traveler. For example, a traveler's work trip in the morning can help predict his/her home trip in the evening, while this causal structure cannot be explicitly encoded in standard time series models. In this paper, we propose a new forecasting framework for boarding flow by incorporating the generative mechanism into standard time series models and leveraging the strong regularity rooted in travel behavior. In doing so, we introduce returning flow from previous alighting trips as a new covariate, which captures the causal structure and long-range dependencies in passenger flow data based on travel behavior. We develop the return probability parallelogram (RPP) to summarize the causal relationships and estimate the return flow. The proposed framework is evaluated using real-world passenger flow data, and the results confirm that the returning flow -- a single covariate -- can substantially and consistently improve various forecasting tasks, including one-step ahead forecasting, multi-step ahead forecasting, and forecasting under special events. And the proposed method is more effective for business-type stations with most passengers come and return within the same day. This study can be extended to other modes of transport, and it also sheds new light on general demand time series forecasting problems, in which causal structure and long-range dependencies are generated by the user behavior.

The Clustered Traveling Salesman Problem with a Prespecified Order on the Clusters, a variant of the well-known traveling salesman problem is studied in literature. In this problem, delivery locations are divided into clusters with different urgency levels and more urgent locations must be visited before less urgent ones. However, this could lead to an inefficient route in terms of traveling cost. This priority-oriented constraint can be relaxed by a rule called d-relaxed priority that provides a trade-off between transportation cost and emergency level. Our research proposes two approaches to solve the problem with d-relaxed priority rule. We improve the mathematical formulation proposed in the literature to construct an exact solution method. A meta-heuristic method based on the framework of Iterated Local Search with problem-tailored operators is also introduced to find approximate solutions. Experimental results show the effectiveness of our methods.