Controle estatístico de processos multivariados baseado em funções cópula.

Abstract

The Statistical Process Control (SPC) is a powerful set of quality tools used to reduce variability and obtain stability of products, services or processes (Montgomery, 2016). The Control Chart is a widely used process monitoring technique, whose main goal is to detect the occurrence of special causes that lead to the change of process as soon as it occurs. A major challenge in statistical quality control is the monitoring and detection of changes in quality characteristics evaluated simultaneously. The multivariate process control chart based on the Hotelling’s T 2 statistic is the most popular for monitoring the mean vector. However, it assumes that the data follow a multivariate normal distribu tion (which in practice rarely occurs) and are uncorrelated. Ba´ıllo and Cuevas (2006) proposed the use of tolerance regions obtained from density level set estimates as a de tection tool. Recently, Verdier (2013) suggested the use of copula-based models, which are simple and flexible tools for multivariate modeling, for obtaining such estimates. In this work, we present an extension of the non-normal approach based on copula functions introduced by Verdier (2013), that is, we explore the trivariate copulas case in addition to the bivariate one. For both situations, we consider the parametric and semi-parametric approaches (the latter one, with the use of Kernel margins, as in Verdier, 2013), and we also present a totally non-parametric approach. Thus, we first compared the bivariate tolerance region derived from the copula modeling with the usual one based on the Ho telling’s T 2 statistic, both constructed under the approach of density level set estimation. Particularly for the multivariate case (d > 2), we propose a new control chart for visual detection of changes in the process mean vector, called cn-chart, which is analogous to the traditional tolerance region. The simulations performed here allowed the variation of: (i) the original data distribution, where we considered the parametric (parametric copula and marginal distributions), semi-parametric (parametric copula and Kernel mar gins) and non-parametric (non-parametric copula) cases; (ii) the degree of association between the variables (weak, moderate and strong); (iii) and the magnitude of changes in the mean vector. Finally, we applied the proposed methodology to a bivariate data set related to water quality measured by potential of hydrogen (pH) and phosphates, as well as to a trivariate data set on measurements of the deflection, curvature and resistivity from brass and steel bimetal thermostats. Both data sets are available in the MSQC package (Santos-Fernandez, 2012) of the R software.