Signal Interpretation in Hotelling's T2 Control Chart for Compositional Data

Abstract

Nowadays, control of concentrations of elements is of crucial importance in industry. Concentrations are expressed in terms of proportions or percentages which means that they are compositional data (CoDa). CoDa are defined as vectors of positive elements that represent parts of a whole and usually add to a constant sum. Classical T2 control chart is not appropriate for CoDa, for which is better to use a compositional T2 control chart (T2C CC). This paper generalizes the interpretation of the out-of-control signals of the individual T2C CC for more than three components. We propose two methods for identifying the ratio of components that mainly contribute to the signal. The first one is suitable for low dimensional problems and consists on finding the log ratio of components that maximizes the univariate T2 statistic. The second one is an optimized method for large dimensional problems that simplifies the calculus by transforming the coordinates into the sphere. We illustrate the T2C CC signal interpretation with a practical example from the chemical and pharmaceutical industry