Restricted Relevance Vector Machine for Missing Data and Application to Virtual Metrology

In semiconductor manufacturing, virtual metrology (VM) is a method of predicting physical measurements of wafer qualities using in-process information from sensors on production equipment. The relevance vector machine (RVM) is a sparse Bayesian kernel machine that has been widely used for VM modeling in semiconductor manufacturing. Missing values from equipment sensors, however, preclude training an RVM model due to missing kernels from incomplete instances. Moreover, imputation for such kernels can lead to a loss of model sparsity. In this work, we propose a restricted RVM (RRVM) that selects its basis functions from only complete instances to handle incomplete data for VM. We conduct the experiments using toy data and real-life data from an etching process for wafer fabrication. The results indicate the model’s competitive prediction accuracy with massive missing data while maintaining model sparsity. Note to Practitioners—In recent decades, virtual metrology (VM) has focused on wafer fabrication in semiconductor manufacturing due to its advantages for process monitoring and automation. Typically, signals from production process equipment can predict wafer qualities in VM, which often leads to high data dimensionality. The relevance vector machine (RVM) is an algorithm that can provide a sparse solution to a Bayesian kernel method for a prediction model. Missing components in incomplete data due to sensor failures in wafer fabrication processes, however, hinder model training, and the existing approaches to handling missing data using imputation may lead to a loss of model sparsity. This article proposes a new method for RVM with incomplete data to train a model built on fully available instances by incorporating the available components of incomplete instances into model training. Using the proposed method, one can predict wafer qualities building a model trained to maintain its sparsity. Experiments indicate that the proposed model achieves competitive prediction performance and maintains model sparsity when incomplete instances are used.