Test of covariance changes without a large sample and its application to fault detection and classification

The variability and complex relationships among process and equipment variables in the hi-tech manufacturing processes, such as semiconductor fabrication, can be characterized by the sample covariance matrix. Test of covariance changes becomes critical for effective fault detection and classification (FDC). However, in modern made-to-order manufacturing, the product-mix is usually high while the order size is becoming smaller. The sample size in such a manufacturing situation becomes an issue diminishing the applicability of existing methods. In view of this point, this paper proposes statistical inference procedures pertaining to detecting change of covariance matrix without demanding a large sample. We apply Bartlett's decomposition and Cholesky's decomposition theories to obtain a matrix T with nice distribution property. Statistical hypothesis testing procedures for possible changes of the covariance matrix are then proposed via aggregating entries of T. The change pattern of T is also studied to construct fault classification rules. Monte Carlo simulation and a dataset collected from an actual semiconductor manufacturing tool are further used to demonstrate the usefulness of our method in FDC. (C) 2012 Elsevier Ltd. All rights reserved.