17,181 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
From conformal to probabilistic prediction
This paper proposes a new method of probabilistic prediction, which is based
on conformal prediction. The method is applied to the standard USPS data set
and gives encouraging results.Comment: 12 pages, 2 table
Local proper scoring rules of order two
Scoring rules assess the quality of probabilistic forecasts, by assigning a
numerical score based on the predictive distribution and on the event or value
that materializes. A scoring rule is proper if it encourages truthful
reporting. It is local of order if the score depends on the predictive
density only through its value and the values of its derivatives of order up to
at the realizing event. Complementing fundamental recent work by Parry,
Dawid and Lauritzen, we characterize the local proper scoring rules of order 2
relative to a broad class of Lebesgue densities on the real line, using a
different approach. In a data example, we use local and nonlocal proper scoring
rules to assess statistically postprocessed ensemble weather forecasts.Comment: Published in at http://dx.doi.org/10.1214/12-AOS973 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On the multiplicity of arrangements of congruent zones on the sphere
Consider an arrangement of congruent zones on the -dimensional unit
sphere , where a zone is the intersection of an origin symmetric
Euclidean plank with . We prove that, for sufficiently large , it
is possible to arrange congruent zones of suitable width on such
that no point belongs to more than a constant number of zones, where the
constant depends only on the dimension and the width of the zones. Furthermore,
we also show that it is possible to cover by congruent zones such
that each point of belongs to at most zones, where the
is a constant that depends only on . This extends the corresponding
-dimensional result of Frankl, Nagy and Nasz\'odi (2016). Moreover, we also
examine coverings of with congruent zones under the condition that
each point of the sphere belongs to the interior of at most zones
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