Research On Some Topics In Multivariate Control Charts

In multivariate statistical process control (MSPC),when the quality characteristics are continuous, most control charts are restricted to the assumption of multivariate nor-mal distribution. However, in many applications, it is well recognized that the normal assumption may not hold. In this situation, the performance of these charts could po-tentially be highly affected. Thus some nonparametric or robust multivariate charts are needed. When the process involves multiple categorical quality characteristics, the distri-bution can be displayed by contingency table. Traditional methods to monitor them are usually developed for the "small cell number and large sample size". When the number of categorical variables increases, the number of cells in contingency table grows extremely fast, so that most of the cell entries are very small or zeros counts, this is so-called sparse contingency table. In this situation, the traditional methodologies are inadequate to use, it is necessary to develop a new method for monitoring the sparse contingency table.In Chapter 2, when the multivariate process distribution is unknown and only a set of in-control data is available, we extend the constant control limit of the multivariate cumulative sum (MCUSUM) control chart to a sequence of dynamic control limits which are determined by the conditional bootstrap distribution of the statistics given the sprint length. Simulation results show that the novel control chart with dynamic control limits offers a better ARL performance, compared with the traditional MCUSUM control chart. However, it is considerably computer-intensive. This leads to the development of a more flexible control chart which uses a continuous function of the sprint length as the control limit sequences.In Chapter 3, when process mean shifts occur in only a few number of components, we extend the classical multivariate LASSO control chart to a robust version. The new chart is distribution free under the family of elliptical direction distributions, indicating that the in-control run length distribution is the same for any continuous distribution in this family and the control limit can be acquired from multivariate standard normal distribution. Simulation results show that the proposed method is very efficient in detecting various sparse shifts under multivariate heavy-tailed and skewed distributions.In Chapter 4, when shifts only occur in a few of elements of the covariance matrix, we apply spatial-sign covariance matrix and maximum norm to the exponentially weighted moving average (EWMA) scheme to construct a robust multivariate control chart for monitoring the covariance matrix. The new chart is distribution-free under the elliptical directions distribution family. Comparison studies demonstrate that the novel method is very powerful in detecting various sparse shifts for heavy-tailed distributions and is more robust under skewed distributions.In Chapter 5, when contingency table is sparse, a two-stage group lasso method is firstly developed to perform model selection and parameter estimation in high-dimension log-linear models. Then, based on modified Pearson χ2 statistics, a new EWMA control chart is proposed. Compared with the the traditional control charts which based on Pear-son χ2 and likelihood ratio statistics, the new chart, is more superior in various coefficient shifts, especially in small and medium shifts.