Multivariate statistical process monitoring and diagnostic analysis of compositional data

Abstract

Compositional data are found in many types of data. Some examples of compositional data are particle size distributions, alloy composition and chemical milling bath composition, colour compositions of paintings, chemical compositions of basalt specimens, as well as household expenditure. In industry, process monitoring of compositional data are of interest for feed and product compositions, and to detect and diagnose potential deviations from expected performance. However, compositional data are subject to certain properties and constraints that complicate the analysis thereof, such as a large number of variables and a unit-sum constraint. In this study, the multivariate statistical analysis of compositional data is reviewed and discussed for purposes of data interpretation, and for application to multivariate statistical process monitoring. Log-transformations are applied to the compositional data, followed by a reduction in the number of variables using principal components analysis (PCA). PCA biplots are used as a visual inspection of the data, providing a way to estimate certain properties of the compositional data through geometric features of the biplots. Specifically, it is shown how correlations and relationships between the components are quantified from the biplot properties. The log-transformed and PCA-reduced data are used to perform multivariate statistical process monitoring. The T2, SPE and combined statistics are used to illustrate multivariate process monitoring of compositional data. In addition, variable contributions are calculated based on the various monitoring statistics for faulty data. Using simulated compositional data, and the well-known Tennessee Eastman Process, it is illustrated that faults can be accurately detected, together with the correct variable contributions, for compositional data.