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

arXiv: https://arxiv.org/abs/2110.04352 · HTML: https://mlsanity.com/paper/2110.04352/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 31 references that are in the map.
- First posted on arXiv 8 Oct 2021.

## Builds on (most cited first, 8 of 8)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Unsupervised Deep Anomaly Detection for Multi-Sensor Time-Series Signals](https://mlsanity.com/paper/2107.12626/) (2021)
- [Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery](https://mlsanity.com/paper/1307.5870/) (2013)
- [Low-Rank Autoregressive Tensor Completion for Spatiotemporal Traffic Data Imputation](https://mlsanity.com/paper/2104.14936/) (2021)
- [Low-Rank Hankel Tensor Completion for Traffic Speed Estimation](https://mlsanity.com/paper/2105.11335/) (2021)
- [Missing Slice Recovery for Tensors Using a Low-rank Model in Embedded Space](https://mlsanity.com/paper/1804.01736/) (2018)
- [Static and Dynamic Robust PCA and Matrix Completion: A Review](https://mlsanity.com/paper/1803.00651/) (2018)
- [Efficient Optimization Algorithms for Robust Principal Component Analysis and Its Variants](https://mlsanity.com/paper/1806.03430/) (2018)

## Built on by (most cited first, 4 of 4)

- [Correlating sparse sensing for large-scale traffic speed estimation: A Laplacian-enhanced low-rank tensor kriging approach](https://mlsanity.com/paper/2210.11780/) (2022)
- [Spatiotemporal Implicit Neural Representation as a Generalized Traffic Data Learner](https://mlsanity.com/paper/2405.03185/) (2024)
- [Anomaly Detection for Fraud in Cryptocurrency Time Series](https://mlsanity.com/paper/2207.11466/) (2022)
- [Adaptive Anomaly Detection in Network Flows with Low-Rank Tensor Decompositions and Deep Unrolling](https://mlsanity.com/paper/2409.11529/) (2024)

## Abstract (arXiv)

Spatiotemporal traffic data (e.g., link speed/flow) collected from sensor networks can be organized as multivariate time series with additional spatial attributes. A crucial task in analyzing such data is to identify and detect anomalous observations and events from the data with complex spatial and temporal dependencies. Robust Principal Component Analysis (RPCA) is a widely used tool for anomaly detection. However, the traditional RPCA purely relies on the global low-rank assumption while ignoring the local temporal correlations. In light of this, this study proposes a Hankel-structured tensor version of RPCA for anomaly detection in spatiotemporal data. We treat the raw data with anomalies as a multivariate time series matrix (location × time) and assume the denoised matrix has a low-rank structure. Then we transform the low-rank matrix to a third-order tensor by applying temporal Hankelization. In the end, we decompose the corrupted matrix into a low-rank Hankel tensor and a sparse matrix. With the Hankelization operation, the model can simultaneously capture the global and local spatiotemporal correlations and exhibit more robust performance. We formulate the problem as an optimization problem and use tensor nuclear norm (TNN) to approximate the tensor rank and l1 norm to approximate the sparsity. We develop an efficient solution algorithm based on the Alternating Direction Method of Multipliers (ADMM). Despite having three hyper-parameters, the model is easy to set in practice. We evaluate the proposed method by synthetic data and metro passenger flow time series and the results demonstrate the accuracy of anomaly detection.