2,809 research outputs found
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
Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds
We have introduce a new vision of stochastic processes through the geometry
induced by the dilation. The dilation matrices of a given processes are
obtained by a composition of rotations matrices, contain the measure
information in a condensed way. Particularly interesting is the fact that the
obtention of dilation matrices is regardless of the stationarity of the
underlying process. When the process is stationary, it coincides with the
Naimark Dilation and only one rotation matrix is computed, when the process is
non-stationary, a set of rotation matrices are computed. In particular, the
periodicity of the correlation function that may appear in some classes of
signal is transmitted to the set of dilation matrices. These rotation matrices,
which can be arbitrarily close to each other depending on the sampling or the
rescaling of the signal are seen as a distinctive feature of the signal. In
order to study this sequence of matrices, and guided by the possibility to
rescale the signal, the correct geometrical framework to use with the
dilation's theoretic results is the space of curves on manifolds, that is the
set of all curve that lies on a base manifold. To give a complete sight about
the space of curve, a metric and the derived geodesic equation are provided.
The general results are adapted to the more specific case where the base
manifold is the Lie group of rotation matrices. The notion of the shape of a
curve can be formalized as the set of equivalence classes of curves given by
the quotient space of the space of curves and the increasing diffeomorphisms.
The metric in the space of curve naturally extent to the space of shapes and
enable comparison between shapes.Comment: 19 pages, draft pape
On the consistency of Fr\'echet means in deformable models for curve and image analysis
A new class of statistical deformable models is introduced to study
high-dimensional curves or images. In addition to the standard measurement
error term, these deformable models include an extra error term modeling the
individual variations in intensity around a mean pattern. It is shown that an
appropriate tool for statistical inference in such models is the notion of
sample Fr\'echet means, which leads to estimators of the deformation parameters
and the mean pattern. The main contribution of this paper is to study how the
behavior of these estimators depends on the number n of design points and the
number J of observed curves (or images). Numerical experiments are given to
illustrate the finite sample performances of the procedure
- …