Monitoring autocorrelated compositional data vectors using an enhanced residuals Hotelling [formula omitted] control chart

Abstract

The most prevalent use of multivariate statistical process monitoring (SPM) is in the production industries. In SPM, data are usually assumed to be uncorrelated in both univariate and multivariate control charts. But, in the real industrial process, the data are often autocorrelated. Specifically, while using compositional data (CoDa), autocorrelation (AC) cannot be neglected, as the CoDa vectors have values close to one another. Hence the presence of AC in CoDa vectors is a very common phenomenon. The AC affects the control chart’s signal interpretation and decreases the detection capability. This paper uses a time series autoregressive moving average (ARMA) model on constrained CoDa based on an algebraic system in the simplex sample space (SD) without transformation. For the theoretical modeling foundation, the mathematical concepts in SD are presented. The AC and the partial AC functions of CoDa are defined to fit the CoDa ARMA model. The ordinary least square estimation is used to estimate the parameters of the CoDa ARMA model. After fitting the best CoDa ARMA model, a residual-based Hotelling T2 control chart for the CoDa ARMA model is used. The average run length of the proposed chart has been evaluated using the Monte Carlo simulations. Different levels of AC and sizes of the shift are considered. Two illustrative examples of sources of energy consumption in China and Japan are presented. The Hotelling T2 based on residuals of the CoDa ARMA model is better than the Hotelling T2 based on isometric log-ratio transformed original data to detect the shift in the mean vector.