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Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Part of: Multivariate analysis Geometric probability and stochastic geometry Global differential geometry

Published online by Cambridge University Press:  01 July 2016

Stephan Huckemann  and
Herbert Ziezold
Stephan Huckemann*
Affiliation:
University of Kassel
Herbert Ziezold*
Affiliation:
University of Kassel
*
∗ Postal address: Department of Mathematics, University of Kassel, D-34109 Kassel, Germany.
∗ Postal address: Department of Mathematics, University of Kassel, D-34109 Kassel, Germany.
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Abstract

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Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

Keywords

Shape analysisprincipal component analysisRiemannian manifoldgeodesic

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62H11: Directional data; spatial statistics 53C22: Geodesics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 299 - 319
Copyright
Copyright © Applied Probability Trust 2006 

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