Multivariate Process Analysis with Lattice Data

Modern dimensional inspection techniques often produce more measurements per part than parts measured within reasonable timeframes. This poses a problem for multivariate process monitoring and capability analysis: The sample covariance matrix is not positive definite, hence not full rank and not invertible. When the measurement sites form a multidimensional lattice, spatially stationary covariance models provide positive definite estimates regardless of the number of measurements per part. I show that these estimates may be used in place of the sample covariance matrix to extend, and in some cases improve, standard multivariate methods. I describe a general class of lattices for which positive definite estimates are obtained via simple averaging or a closed-form EM algorithm. The proposed estimation and analysis procedures are illustrated in three case studies.