Phase-II CUSUM Control Chart for Functional Data

Functional data monitoring is an emerging field that applies the principles of functional data analysis to improve the tracking and evaluation of processes characterized by continuous measurements over time or space. Modern measuring tools, equipped with computerized data collecting systems, have made it possible to collect functional data. This study aims to develop the cumulative sum (CUSUM) control chart for phase II functional data monitoring and provide a robust framework for detecting small, sustainable shifts in the process mean. Unlike traditional CUSUM charts, which focus on scalar data, this innovative approach captures the continuous nature of functional data, enabling the analysis of entire curves and shapes. To this end, this study discusses the construction of the functional CUSUM (FCUSUM) control chart, including the selection of suitable basis functions for functional representation and the empirical derivation of control limits tailored to functional data. Detailed run-length properties are studied under different scenarios to evaluate the effectiveness of the proposed methodology in identifying small shifts compared to existing methods. The real-time data of the vertical density profile of particleboard, produced by a real manufacturing process, is used for the FCUSUM chart which further illustrates its practical utility in industrial quality control settings.