A Multivariate Functional CUSUM Control Chart for High-Dimensional Process Monitoring: Application to Air Quality

Abstract

This study proposes a multivariate functional CUSUM (MFCUSUM) control chart for Phase II monitoring of high-dimensional multivariate functional data. The proposed framework combines B-spline smoothing, multivariate functional principal component analysis (MFPCA), whitening transformation, and CUSUM accumulation, with control limits calibrated using a Monte Carlo bisection algorithm targeting a nominal in-control Average Run Length (ARL0) of 370. Simulation results under the evaluated mean-shift scenarios show that MFCUSUM with k = 0.15 consistently yields lower estimated ARL1 values than MFEWMA and Hotelling’s functional T2 chart, achieving ARL1 values of 85.294 and 42.193 for a small shift ( $\gamma = 0.25$ ) when p = 5 and p = 7, respectively. The estimated ARL1 exhibits a U-shaped relationship with the reference value k, indicating that detection performance is sensitive to the choice of the reference parameter and supporting its selection according to the anticipated shift magnitude. The proposed framework is further illustrated using a real-world air-quality dataset comprising seven functional variables observed over 355 days. In this application, the Phase II signal proportions were 89.7% for MFCUSUM, 75.5% for MFEWMA, and 0.0% for Hotelling’s functional T2 chart. These real-data results are presented as descriptive monitoring outcomes and should be interpreted in the absence of validated process-state labels.