ABSTRACT
This idea of analyzing data on non-Euclidean sample spaces gained traction
as computational power increased, allowing for this new direction to be im-
plemented practically. Non-Euclidean data analysis includes directional data
(Watson (1983) [333]), direct similarity shape (Kendall (1984) [177]), tectonic
plates (Chang (1988) [63]), certain Lie groups (Kim (2000) [188]), Stiefel ma-
nifolds (Hendricks and Landsman (1998) [154]), projective shape manifolds
(Patrangenaru (2001) [268]), and affine shape manifolds (Patrangenaru and
Mardia (2003) [274]). A common features of all these sample spaces is that
they are all homogeneous spaces. Given a homogeneous space, it is a statisti-
cian’s choice to select an appropriate homogeneous Riemannian structure on
the sample space, that in her or his view would best address the data analy-
sis in question. While the analysis of directional and shape data that domi-
nated object data analysis in its initial phase are on compact, more recently, in
brain imaging , proteomics, and astronomy, homogeneous spaces of noncom-
pact type arose as well (see Dryden et al. (2009) [89] and Bandulasiri et al.
(2009) [10]).
