Nonparametric passenger flow monitoring using a minimum distance criterion

Monitoring real-time passenger flow in urban rapid transit systems is very important to maintain social stability and prevent unexpected group events and system failure. To monitor passenger flow, data are collected by sensors deployed in important stations and many existing control charts can be applied. However, because of unknown complex distributions and the requirement to detect shifts of all ranges effectively, conventional methods may perform poorly. Nevertheless, while there are certain charting schemes that truncate the Log-Likelihood Ratio (LLR) function to detect large shifts more quickly, they can cause massive loss of information by truncation, and can only handle particular distributions, leading to unstable online monitoring. In this article, we propose a nonparametric CUSUM charting scheme to monitor passenger flow dynamically. We propose a novel minimum distance criterion to minimize the functional distance between the objective function and the original LLR function while maintaining its monotonically increasing property. By integrating this concept with kernel density estimation, our proposed chart does not require any parametric process distribution, it can be constructed easily in any situation, and it is sensitive to shifts of all sizes. Theoretical analysis, simulations and a real application to monitoring passenger flow in the Mass Transit Railway in Hong Kong show that our method performs well in various cases.