Signal interpretation in Hotelling's T2 control chart for compositional data

Nowadays, the control of the 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. The classical T-2 Control Chart is not appropriate for CoDa; rather, it is better to use a compositional T-2 Control Chart (T-C(2) CC). This article generalizes the interpretation of the out-of-control signals of the individual T-C(2) 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 in finding the log ratio of the components that maximizes the univariate T-2 statistic. The second approach is an optimized method for large-dimensional problems that simplifies the calculation by transforming the coordinates into an sphere. We illustrate the T-C(2) CC signal interpretation with a practical example from the chemical and pharmaceutical industry.