Statistical Methods Generalizing Principal Component Analysis to Non-Euclidean Spaces

Stephan Huckemann () and Benjamin Eltzner ()
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Stephan Huckemann: University of Göttingen, Felix-Bernstein-Institute for Mathematical Statistics in the Biosciences
Benjamin Eltzner: University of Göttingen, Felix-Bernstein-Institute for Mathematical Statistics in the Biosciences

Chapter Chapter 10 in Handbook of Variational Methods for Nonlinear Geometric Data, 2020, pp 317-338 from Springer

Abstract: Abstract Very generally speaking, statistical data analysis builds on descriptors reflecting data distributions. In a linear context, well studied nonparametric descriptors are means and PCs (principal components, the eigenorientations of covariance matrices). In 1963, T.W. Anderson derived his celebrated result of joint asymptotic normality of PCs under very general conditions. As means and PCs can also be defined geometrically, there have been various generalizations of PC analysis (PCA) proposed for manifolds and manifold stratified spaces. These generalizations play an increasingly important role in statistical dimension reduction of non-Euclidean data. We review their beginnings from Procrustes analysis (GPA), over principal geodesic analysis (PGA) and geodesic PCA (GPCA) to principal nested spheres (PNS), horizontal PCA, barycentric subspace analysis (BSA) and backward nested descriptors analysis (BNDA). Along with this, we review the current state of the art of their asymptotic statistical theory and applications for statistical testing, including open challenges, e.g. new insights into scenarios of nonstandard rates and asymptotic nonnormality.

Date: 2020
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DOI: 10.1007/978-3-030-31351-7_10

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