A model-based monitoring framework for tensor count data in passenger flow surveillance

Tensor count data are increasingly prevalent across numerous applications, and passenger flow data in urban rail transit systems serve as a typical example. Real-time monitoring of passenger flow tensors is essential for identifying irregular behaviors and preventing severe consequences. However, existing online monitoring methods often fail to accommodate the unique characteristics of count data or are designed specifically for vectorized data, rendering them unsuitable for general-order tensor count processes. In this paper, we introduce a novel monitoring method, specifically designed for scenarios where count data appear in tensor form. The proposed method is based on a new Tensor Poisson Log-Normal Model. To address the estimation difficulties arising from the multi-dimensional latent variables in the model, we develop an efficient variational Gaussian approximation approach for Phase I modeling. In Phase II surveillance, an online parameter estimation algorithm is formulated based on the Laplace approximation method for the real-time computation requirements in online monitoring. Subsequently, we design an exponentially weighted likelihood based monitoring statistic to identify anomalies in online monitoring. Finally, we validate the effectiveness and superiority of our method through comprehensive simulations and apply it to real-time passenger flow surveillance in the Hong Kong Mass Transit Railway.