Spatial In-Profile Monitoring via Latent Tensor Gaussian Process with Mixed Effects

Advanced sensing technology enables real-time data collection in two or higher-dimensional coordinate systems, known as spatial profiles. These data have attracted significant efforts toward anomaly detection and quality control in manufacturing. However, most of the existing monitoring methods face detection delays as they require complete profiles before implementation. To accommodate both the spatial correlation and sequential nature of the data, Gaussian process (GP) modeling offers a promising approach. Yet, the high-dimensional covariance matrix associated with densely sampled spatial profiles poses challenges in estimation accuracy. Additionally, they often treat spatial locations as the GP inputs, neglecting numerous latent factors involved in the generation of spatial profiles. This article proposes an in-profile monitoring (INPOM) control chart for spatial profiles based on a latent tensor GP with mixed effects (LTGP-ME) model. The random effects component is constructed by an LTGP, which preserves the multiway structure of spatial profiles in the tensor domain while capturing nonstationary spatial correlation through latent factors. Fixed effects component can be further incorporated to capture the relationship between spatial profiles and additional covariates. We develop an expectation-maximization algorithm for parameter estimation, exploring model identifiability and convergence properties. Based on the prediction errors of LTGP-ME, a Hotelling T2 statistic is further constructed for INPOM. The effectiveness and applicability of the proposed approach are demonstrated through extensive simulation studies and a real case study in additive manufacturing.