Exploratory tools for outlier detection in compositional data with structural zeros

ABSTRACT

The analysis of compositional data using the log-ratio approach is based on ratios between the compositional parts. Zeros in the parts thus cause serious difficulties for the analysis. This is a particular problem in case of structural zeros, which cannot be simply replaced by a non-zero value as it is done, e.g. for values below detection limit or missing values. Instead, zeros to be incorporated into further statistical processing. The focus is on exploratory tools for identifying outliers in compositional data sets with structural zeros. For this purpose, Mahalanobis distances are estimated, computed either directly for subcompositions determined by their zero patterns, or by using imputation to improve the efficiency of the estimates, and then proceed to the subcompositional and subgroup level. For this approach, new theory is formulated that allows to estimate covariances for imputed compositional data and to apply estimations on subgroups using parts of this covariance matrix. Moreover, the zero pattern structure is analyzed using principal component analysis for binary data to achieve a comprehensive view of the overall multivariate data structure. The proposed tools are applied to larger compositional data sets from official statistics, where the need for an appropriate treatment of zeros is obvious.

KEYWORDS:

• Structural zeros
• Aitchison geometry on the simplex
• covariance estimation
• Mahalanobis distance
• principal component analysis

AMS SUBJECT CLASSIFICATION:

• 97K80
• 97K70
• 97K40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the COST Action CRoNoS IC1408, the Internal Grant Agency of the Palacky University in Olomouc [IGA_PrF_2015_013, IGA_PrF_2016_025], the Austrian Science Fund (FWF) [I 1910-N26], and by the K-project DEXHELPP through COMET - Competence Centers for Excellent Technologies supported by BMVIT, BMWFI and the province Vienna, administrated by the Austrian Research Promotion Agency (FFG).