Multivariate statistical process control for inspection data from coordinate measuring machines

Traditional multivariate statistical process control techniques are not applicable when sample size is less than the number of quality characteristics because the sample covariance matrix is not positive-definite, hence not full rank and not invertible. With the emergence of coordinate measurement machines (CMM) in a computer-aided manufacturing environment, a large number of measurements on a part usually exceed the number of sample parts. In this paper, we propose a multiplicative time series model to estimate the positive-definite variance-covariance matrix when sample size is less than the number of quality characteristics (measurements at coordinate points). To estimate the proposed separable covariance models, measurement sites are formed as a multidimensional lattice where each dimensional observation series is analyzed by a univariate time series model. By forming multiplicative models of these univariate time series, the variance-covariance structure of observations is derived. An example demonstrates how the proposed methodology can be used to estimate the sample covariance matrix for multivariate statistical process control. Performance of the proposed methodology is notable when compared with those of Boyles' (1996) covariance model and conventional sample covariance with a fewer number of parameters.