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Statistical Methods Generalizing Principal Component Analysis to Non-Euclidean Spaces

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Handbook of Variational Methods for Nonlinear Geometric Data

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.

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Acknowledgements

The authors acknowledge support from the Niedersachen Vorab of the Volkswagen Foundation and support from DFG HU 1575-7.

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Authors and Affiliations

  1. Felix-Bernstein-Institute for Mathematical Statistics in the Biosciences, University of Göttingen, Göttingen, Germany

    Stephan Huckemann & Benjamin Eltzner

Authors
  1. Stephan Huckemann
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  2. Benjamin Eltzner
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Correspondence to Stephan Huckemann .

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Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Huckemann, S., Eltzner, B. (2020). Statistical Methods Generalizing Principal Component Analysis to Non-Euclidean Spaces. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_10

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