Low-Rank Hankel Tensor Completion for Traffic Speed Estimation

Abstract

This paper studies the traffic state estimation (TSE) problem using sparse observations from mobile sensors. TSE can be considered a spatiotemporal interpolation problem in which the evolution of traffic variables (e.g., speed/density) is governed by traffic flow dynamics (e.g., partial differential equations). Most existing TSE methods either rely on well-defined physical traffic flow models or require large amounts of simulation data as input to train machine learning models. Different from previous studies, in this paper we propose a purely data-driven and modelfree solution. We consider TSE as a spatiotemporal matrix completion/interpolation problem, and apply spatiotemporal Hankel delay embedding to transforms the original incomplete matrix to a fourth-order tensor. By imposing a low-rank assumption on this tensor structure, we can approximate and characterize both global patterns and the unknown and complex local spatiotemporal dynamics in a data-driven manner. We use the truncated nuclear norm of the spatiotemporal unfolding (i.e., square norm) to approximate the tensor rank and develop an efficient solution algorithm based on the Alternating Direction Method of Multipliers (ADMM). The proposed framework only involves two hyperparameters-spatial and temporal window lengths, which are easy to set given the degree of data sparsity. We conduct numerical experiments on both synthetic simulation data and real-world high-resolution trajectory data, and our results demonstrate the effectiveness and superiority of the proposed model in some challenging scenarios.

Uncertainty Quantification of Sparse Travel Demand Prediction with Spatial-Temporal Graph Neural Networks

“…Missing data, on the other hand, are unobservable, potentially due to sensor malfunction. Wang et al [35] applied the spatial-temporal Hankelization and tensor factorization to estimate traffic states using only 17% of the matrix entries. However, tensor factorization is a transductive method, which needs recalculation when new data come in.…”

Hankel-structured Tensor Robust PCA for Multivariate Traffic Time Series Anomaly Detection

“…Correspondingly, the inverse Hankelization operation H −1 τ is to transform a Hankel tensor X to a matrix X by averaging the corresponding entries in the Hankel tensor [16]. In other words, we aggregate the sub-matrix X :,:,t for t = 1, • • • , τ along temporal dimension by shifting one step each time to obtain X and then we divide the corresponding repeat times of each entry.…”

“…In light of this, the N × T matrix L can be transferred to a third-order tensor L ∈ R N ×(T −τ +1)×τ . By doing so, the Hankel tensor can preserve the global pattern of the low-rank data and introduce a higher-order dependency/correlation structure within a local temporal domain [16]. Figure 1 illustrates the vanilla RPCA model and the proposed Hankel-structured tensor RPCA (HT-RPCA) model, respectively.…”

Traffic State Estimation with Anisotropic Gaussian Processes from Vehicle Trajectories

“…However, modelbased TSE may not always be accurate because it may not fully capture the complexity of real-world traffic. Conversely, with massive traffic data and machine learning techniques available, TSE can be achieved in a purely data-driven manner, as demonstrated by some recent works [6,7]. However, the training of a data-driven approach typically requires a large external training dataset and a validation dataset with full information.…”