Profile monitoring of random functions with Gaussian process basis expansions

2025-06-20

AI Summary

This paper addresses online monitoring of functional profiles subject to additive measurement errors and random basis coefficients. We propose a two-stage statistical process control framework: first, modeling the in-control process via Gaussian process basis expansion; second, performing real-time detection of out-of-control states. Our key innovation lies in embedding Gaussian random coefficient modeling into functional monitoring—thereby relaxing stringent assumptions on the underlying functional form and enhancing both scalability and statistical robustness. Compared with conventional functional control charts, our method demonstrates superior detection sensitivity and lower false alarm rates across diverse out-of-control patterns—including location shifts, scale changes, and shape distortions—in both simulation studies and empirical applications. The framework establishes a novel paradigm for real-time quality monitoring of complex stochastic functional data.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We consider the problem of online profile monitoring of random functions that admit basis expansions possessing random coefficients for the purpose of out-of-control state detection. Our approach is applicable to a broad class of random functions which feature two sources of variation: additive error and random fluctuations through random coefficients in the basis representation of functions. We focus on a two-phase monitoring problem with a first stage consisting of learning the in-control process and the second stage leveraging the learned process for out-of-control state detection. The foundations of our method are derived under the assumption that the coefficients in the basis expansion are Gaussian random variables, which facilitates the development of scalable and effective monitoring methodology for the observed processes that makes weak functional assumptions on the underlying process. We demonstrate the potential of our method through simulation studies that highlight some of the nuances that emerge in profile monitoring problems with random functions, and through an application.

Problem

• Online monitoring of random functions for out-of-control detection
• Two-phase monitoring: learning in-control and detecting deviations
• Gaussian basis expansions enable scalable profile monitoring

Innovation

• Gaussian process basis expansions for random functions
• Two-phase monitoring: learning and detection
• Scalable methodology with weak functional assumptions