3 research outputs found
Stochastic Development Regression on Non-Linear Manifolds
We introduce a regression model for data on non-linear manifolds. The model
describes the relation between a set of manifold valued observations, such as
shapes of anatomical objects, and Euclidean explanatory variables. The approach
is based on stochastic development of Euclidean diffusion processes to the
manifold. Defining the data distribution as the transition distribution of the
mapped stochastic process, parameters of the model, the non-linear analogue of
design matrix and intercept, are found via maximum likelihood. The model is
intrinsically related to the geometry encoded in the connection of the
manifold. We propose an estimation procedure which applies the Laplace
approximation of the likelihood function. A simulation study of the performance
of the model is performed and the model is applied to a real dataset of Corpus
Callosum shapes
An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data
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
estimation procedures for inference of parameters and prediction of principal
components, and we give examples of properties of the model on embedded
surfaces