Real-Time Monitoring of Dynamic Tensor Data with Longitudinal Patterns: A Tensor Graphical LASSO Approach

Dynamic tensor data with longitudinal patterns are increasingly prevalent across various applications, and their real-time monitoring is crucial for identifying irregular longitudinal behavior and preventing terrible consequences. However, existing online monitoring methods often fail to account for the dynamic nature of tensor data or are designed solely for vectorized data, making them inapplicable to general-order tensor processes. In this article, we propose a novel real-time monitoring method tailored for cases where longitudinal behavior manifests in tensor form. Our approach leverages the inherent advantages of tensor structures by stacking observed tensor data in an individual process, forming a dynamic tensor where time naturally serves as one of the tensor modes. Subsequently, we introduce an efficient tensor graphical LASSO procedure to estimate the in-control parameters, taking into consideration sparsity and addressing high-dimensionality challenges. In Phase II, we apply a decorrelation procedure based on Cholesky decomposition to detrend tensor process and eliminate temporal correlation. Finally, we design an exponentially weighted moving average control chart for the standardized process to identify irregular longitudinal behavior. We validate the effectiveness of our method through simulations and apply it to real-time passenger flow surveillance in the Hong Kong Mass Transit Railway transportation network.