ABSTRACT
The methods of statistical inference relate mainly to studying a Euclidean cloud, that is, a family of statistical observations conceptualized as points in a multidimensional Euclidean space. Ready-made Euclidean clouds occur whenever observations are points recorded in the ambient physical space. For instance, the impacts of bullets on a target or the positions of bees in a swarm define Euclidean clouds. In general, Euclidean clouds are built using numerical data sets. In Geometric Data Analysis, clouds are constructed from contingency tables. In this chapter, the authors study the covariance structure of a Euclidean cloud and introduce the problem of determining the principal axes of a cloud. They present two notions that are both linked to the covariance structure: firstly the Mahalanobis distance, which will be used as a test statistic of the combinatorial test for the mean point of a cloud, and secondly the principal (or inertia) ellipsoids of a cloud, which provide fundamental elements to define compatibility regions.
