Data-rich multivariable detection and diagnosis using eigenspace analysis

Abstract

With the rapid growth of data-acquisition technology and computing resources, a plethora of data can now be collected at high frequency. Because a large number of characteristics or variables are collected, interdependency among variables is expected and hence the variables are correlated. As a result, multivariate statistical process control is receiving increased attention. This thesis addresses multivariate quality control techniques that are capable of detecting covariance structure change as well as providing information about the real nature of the change occurring in the process. Eigenspace analysis is especially advantageous in data rich manufacturing processes because of its capability of reducing the data dimension. The eigenspace and Cholesky matrices are decompositions of the sample covariance matrix obtained from multiple samples. Detection strategies using the eigenspace and Cholesky matrices compute second order statistics and use this information to detect subtle changes in the process. Probability distributions of these matrices are discussed. In particular, the precise distribution of the Cholesky matrix is derived using Bartlett's decomposition result for a Wishart distribution matrix. Asymptotic properties regarding the distribution of these matrices are studied in the context of consistency of an estimator. The eigenfactor, a column vector of the eigenspace matrix, can then be treated as a random vector and confidence intervals can be established from the given distribution. In data rich environments, when high correlation exists among measurements, dominant eigenfactors start emerging from the data. Therefore, a process monitoring strategy using only the dominant eigenfactors is desirable and practical. The applications of eigenfactor analysis in semiconductor manufacturing and the automotive industry are demonstrated.