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.

先进传感技术可在二维及以上坐标系中实现实时数据采集，这类数据统称为空间轮廓（spatial profiles）。此类数据已在制造领域的异常检测与质量控制方向受到广泛关注。然而，现有多数监控方法需先获取完整轮廓方能实施检测，因而存在检测延迟问题。为兼顾数据的空间相关性与序列特性，高斯过程（Gaussian Process, GP）建模是一种极具潜力的解决方案。但针对高密度采样的空间轮廓，其对应的高维协方差矩阵会给参数估计精度带来挑战；此外，现有方法常将空间位置直接作为GP的输入，忽略了空间轮廓生成过程中涉及的诸多潜在因子。

本文提出一种基于带混合效应的潜在张量高斯过程（Latent Tensor Gaussian Process with Mixed Effects, LTGP-ME）模型的轮廓内监控（in-profile monitoring, INPOM）控制图。该模型的随机效应分量由LTGP构建，可在张量域保留空间轮廓的多维度结构，并通过潜在因子捕捉非平稳空间相关性；固定效应分量则可进一步融入，以刻画空间轮廓与额外协变量间的关联。本文开发了用于参数估计的期望-最大化（Expectation-Maximization, EM）算法，并探讨了模型的可辨识性与收敛特性。基于LTGP-ME的预测误差，本文进一步构建了用于INPOM的霍特林T²（Hotelling T²）统计量。通过大量仿真实验与一项增材制造领域的真实案例研究，验证了所提方法的有效性与适用性。