Laplacian Convolutional Representation for Traffic Time Series Imputation

TL;DR

The paper tackles missing data in spatiotemporal traffic time series by proposing the Laplacian Convolutional Representation (LCR), which couples a circulant-matrix nuclear-norm objective for global low-rank structure with a Laplacian kernel-based temporal regularization for local smoothness. By leveraging the circular convolution and the FFT, the authors derive efficient ADMM-based updates that operate in the frequency domain, yielding an O(T log T) per-iteration complexity. A univariate LCR is extended to the multivariate case (LCR-2D) via a circulant tensor-nuclear-norm framework and a separable spatiotemporal kernel, enabling joint imputation of high-dimensional traffic data. Empirical results on Portland speed/volume data, HighD/CitySim speed fields, and PeMS show that LCR and LCR-2D outperform CirccNNM, ConvNNM, and standard low-rank tensor/matrix methods, particularly at high missing rates, while remaining scalable to large datasets. The work provides a principled, FFT-friendly pathway to fuse global low-rank structure with local temporal regularization, with clear potential for broad impact in traffic analytics and other spatiotemporal domains.

Abstract

Spatiotemporal traffic data imputation is of great significance in intelligent transportation systems and data-driven decision-making processes. To perform efficient learning and accurate reconstruction from partially observed traffic data, we assert the importance of characterizing both global and local trends in time series. In the literature, substantial works have demonstrated the effectiveness of utilizing the low-rank property of traffic data by matrix/tensor completion models. In this study, we first introduce a Laplacian kernel to temporal regularization for characterizing local trends in traffic time series, which can be formulated as a circular convolution. Then, we develop a low-rank Laplacian convolutional representation (LCR) model by putting the circulant matrix nuclear norm and the Laplacian kernelized temporal regularization together, which is proved to meet a unified framework that has a fast Fourier transform (FFT) solution in log-linear time complexity. Through extensive experiments on several traffic datasets, we demonstrate the superiority of LCR over several baseline models for imputing traffic time series of various time series behaviors (e.g., data noises and strong/weak periodicity) and reconstructing sparse speed fields of vehicular traffic flow. The proposed LCR model is also an efficient solution to large-scale traffic data imputation over the existing imputation models.

Figures (9)

• Figure 1: Undirected and circulant graphs on the relational data samples $\{{x}_{1},{x}_{2},\ldots,{x}_{5}\}$ with certain degrees.
• Figure 2: Illustration of the proposed LCR model.
• Figure 3: Empirical time complexity. The model is tested 50 times on each generated data.
• Figure 4: Univariate traffic time series imputation on the freeway traffic speed time series. The blue curve represents the ground truth time series, while the red curve refers to the reconstructed time series produced by LCR. Here, partial observations are illustrated as blue circles.
• Figure 5: Univariate traffic time series imputation on the freeway traffic speed time series. In this case, we mask 95% observations as missing values and only have 14 speed observations for training the model.

Theorems & Definitions (13)

• Definition 1: Laplacian Kernel
• Remark 1
• Remark 2
• Theorem 1: Convolution Theorem brunton2022data
• Remark 3
• Remark 4
• Lemma 1
• proof
• Lemma 2
• proof