abstract. we provide a probabilistic and infinitesimal view of how the principal
component analysis procedure (pca) can be generalized to analysis of nonlinear
manifold valued data. starting with the probabilistic pca interpretation of the
euclidean pca procedure, we show how pca can be generalized to manifolds in
an intrinsic way that does not resort to linearization of the data space. the underlying probability model is constructed by mapping a euclidean stochastic process
to the manifold using stochastic development of euclidean semimartingales. the
construction uses a connection and bundles of covariant tensors to allow global
transport of principal eigenvectors, and the model is thereby an example of how
principal fiber bundles can be used to handle the lack of global coordinate system and orientations that characterizes manifold valued statistics. we show how
curvature implies non-integrability of the equivalent of euclidean principal subspaces, and how the stochastic flows provide an alternative to explicit construction
of such subspaces. we describe two estimation procedures for inference of parameters and prediction of principal components, and we give examples of properties
of the model on surfaces embedded in r3 .
principal component analysis, manifold valued statistics, stochastic development, probabilistic pca, anisotropic normal distributions, frame bundle