Detecting outliers in compositional data using Invariant coordinate selection

Invariant coordinate (or component) selection (ICS) is a multivariate statistical method introduced by Tyler et al. (J R Stat Soc Ser B (Stat Methodol) 71(3):549–592, 2009) and based on the simultaneous diagonalization of two scatter matrices. A model-based approach of ICS, called invariant coordinate analysis, has already been adapted for compositional data in Muehlmann et al. (Independent component analysis for compositional data. In Daouia, A, Ruiz-Gazen A (eds) Advances in Contemporary Statistics and Econometrics: Festschrift in Honor of Christine Thomas-Agnan. Springer, New York, pp. 525–545, 2021). In a model-free context, ICS is also helpful at identifying outliers Nordhausen and Ruiz-Gazen (J Multivar Anal 188:104844, 2022). We propose to develop a version of ICS for outlier detection in compositional data. This version is first introduced in coordinate space for a specific choice of isometric log-ratio coordinate system associated to a contrast matrix and follows the outlier detection procedure proposed by Archimbaud et al. (Comput Stat Data Anal 128:184–199, 2018a). We then show that the procedure is independent of the choice of contrast matrix and can be defined directly in the simplex. To do so, we establish some properties of the set of matrices satisfying the zero-sum property and introduce a simplex definition of the Mahalanobis distance and the one-step M-estimators class of scatter matrices. We also need to define the family of elliptical distributions in the simplex. We then show how to interpret the results directly in the simplex using two artificial datasets and a real dataset of market shares in the automobile industry.