High breakdown estimation methods for Phase I multivariate control charts

Abstract

A goal of Phase I analysis of multivariate data is to identify multivariate outliers and step changes so that the Phase II estimated control limits are sufficiently accurate. High breakdown estimation methods based on the minimum volume ellipsoid (MVE) or the minimum covariance determinant (MCD) are well suited for detecting multivariate outliers in data. As a result of the inherent difficulties in their computation, many algorithms have been proposed to detect multivariate outliers. Due to their availability in standard software packages, we consider the subsampling algorithm to obtain the MVE estimators and the FAST-MCD algorithm to obtain the MCD estimators. Previous studies have not clearly determined which of these two available estimation methods is best for control chart applications. The comprehensive simulation study presented in this paper gives guidance for the correct use of each estimator. Control limits are provided. High breakdown estimation methods based on the MCD and MVE approaches can be applied to a wide variety of multivariate quality control data.

An Alternative Hotelling T^2 Control Chart Based on Minimum Vector Variance (MVV)

“…The C-step is similar to the one in the Fast MCD algorithm, except that the computation of covariance determinant is replaced by the vector variance. The complete algorithm is described as below: However, this standard approach is only effective in eliminating very extreme outliers and detecting large shift in the mean vector in small sample sizes, but it fails to detect more moderate outliers especially when number of variables increased (Vargas, 2003;Jensen et al, 2007;Chenouri et al, 2009). To alleviate the problem of this procedure, we proposed using MVV estimator in Phase I data, x i .…”

“…This is due to the fact that the Hotelling T 2 control charts based on MCD performed well in terms of probability of outliers detection. Theoretically, if the percentage of outliers' detection increases, the chart should also able to control the overall false alarm rate, α (Jensen et al, 2007). However the finding in Alfaro and Ortega (2009) shows a conflict between the percentage of outliers detection and the ability of robust control chart in controlling the overall false alarm rate when using robust charts under certain conditions.…”

“…The study on the significant role of MVE, MCD and RMCD estimators in scientific application can be easily found in the literature specifically in the construction of robust Hotelling T 2 chart. Vargas (2003) and Jensen et al (2007) introduced robust control charts based on MVE and MCD estimators for multivariate individual observations. They identified and removed the outliers in Phase I analysis and then calculate the classical estimators using the remaining good data points for Phase II analysis.…”

“…They identified and removed the outliers in Phase I analysis and then calculate the classical estimators using the remaining good data points for Phase II analysis. Through this approach, the computability and breakdown point of the estimators become more important, but statistical efficiency is not as crucial because the highly robust estimators will eventually be replaced by classical estimators in Phase II analysis (Jensen et al, 2007). They noticed some drawbacks when MVE and MCD are used in Phase I.…”

“…Besides possessing the nice properties of MCD estimator, this estimator is not unduly influenced by outliers and is more efficient than MCD. However, their approach is different from Vargas (2003) and Jensen et al (2007) whereby they used RMCD estimators in place of classical estimators in constructing Hotelling T 2 chart for Phase II data. Using the same approach as Chenouri et al (2009), Alfaro & Ortega (2009 make comparison study for the performance of Hotelling T 2 control chart based on robust estimators of MCD, MVE, RMCD, and trimmed estimator.…”