Correlated Bayesian Co-Training for Virtual Metrology

A rising challenge in manufacturing data analysis is training robust regression models using limited labeled data. In this work, we investigate a semi-supervised regression scenario, where a manufacturing process operates on multiple mutually correlated states. We exploit this inter-state correlation to improve regression accuracy by developing a novel co-training method, namely Correlated Bayesian Co-training (CBCT). CBCT adopts a block Sparse Bayesian Learning framework to enhance multiple individual regression models which share the same support. Additionally, CBCT casts a unified prior distribution on both the coefficient magnitude and the inter-state correlation. The model parameters are estimated using maximum-a-posteriori estimation (MAP), while hyper-parameters are estimated using the expectation-maximization (EM) algorithm. Experimental results from two industrial examples shows that CBCT successfully leverages inter-state correlation to reduce the modeling error by up to 79.40%, compared to other conventional approaches. This suggests that CBCT is of great value to multi-state manufacturing applications.