Gauge imprecision and multivariate process capability analysis

In manufacturing industry process capability analysis often entails assessing processes or products based on more than one engineering specification. In these processes the quality is measured by the joint level of several correlated quality characteristics and the capability analysis should be based on a multivariate statistical technique. Some authors have recently proposed alternative definitions of Multivariate Process Capability Indices (MPCIs) based on different approaches. In general, MPCIs are constructed using: the ratio of the area or volume of a tolerance region to the area or volume of a process region; the proportion of nonconforming items; the principal component analysis (PCA). This paper considers the use of PCA in measuring the capability of a multivariate process. The use of this multivariate methodology for assessing the capability of a multivariate process has been initially proposed by Wang and Chen [Quality Engineering (1998-99): 21-27]. These authors proposed to assess the multivariate capability considering a subset of principal components (PCs). They have defined MPCIs using the univariate process capability indices computed for each PC and computing the MPCIs as the geometric mean of the capability indices of the PCs. This definition of capability index has the drawback that all the PCs are equally weighted. However, as it is well known, the first PC will be more relevant than the last ones. To solve this problem new indices have been proposed which allow for potential differences in the portions of the variance explained by the PCs considered. These differences are taken account by assigning unequal weights to the univariate index values corresponding to the principal components used and in particular proportional to the portions of the variance explained by them as determined by their respective eigenvalues. The research works related to MCPIs assume no gauge measurement errors. However, in industrial applications despite the use of highly advanced measuring instruments, account needs to be taken of measurement gauge imprecision. In this current work, we study the effects of multivariate measurement errors on MPCIs computed using PCA. We assume a Gaussian and additive multivariate measurement error model. Therefore the observable quality characteristics, which is usually obtained from some physical measurements, is a multivariate normally distributed random variable with covariance matrix given by the sum of the true quality characteristics covariance matrix and the measurement error covariance matrix. The measurement error covariance matrix can be regarded as a small perturbation of the true quality characteristics covariance matrix and the effects of multivariate measurement error on MPCIs can be investigated by studying the spectral properties of the perturbed matrix. We examine several cases for the covariance error matrix in which analytical results on the eigenvalues and eigenvectors of the perturbed matrix can be obtained. To study the effects of multivariate measurement error on MPCIs we use an example from the literature in the context of multivariate process. The analysis shows that multivariate measurement errors result in important effects on all the considered indices and that changes are more evident for the un-weighted indices