342 research outputs found
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
A Note on the Kullback-Leibler Divergence for the von Mises-Fisher distribution
We present a derivation of the Kullback Leibler (KL)-Divergence (also known
as Relative Entropy) for the von Mises Fisher (VMF) Distribution in
-dimensions.Comment: 8 pages 1 figur
Isotropic Multiple Scattering Processes on Hyperspheres
This paper presents several results about isotropic random walks and multiple
scattering processes on hyperspheres . It allows one to
derive the Fourier expansions on of these processes. A
result of unimodality for the multiconvolution of symmetrical probability
density functions (pdf) on is also introduced. Such
processes are then studied in the case where the scattering distribution is von
Mises Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs
on are obtained. Both Fourier expansion and asymptotic
approximation allows us to compute estimation bounds for the parameters of
Compound Cox Processes (CCP) on .Comment: 16 pages, 4 figure
High-dimensional tests for spherical location and spiked covariance
Rotationally symmetric distributions on the p-dimensional unit hypersphere,
extremely popular in directional statistics, involve a location parameter theta
that indicates the direction of the symmetry axis. The most classical way of
addressing the spherical location problem H_0:theta=theta_0, with theta_0 a
fixed location, is the so-called Watson test, which is based on the sample mean
of the observations. This test enjoys many desirable properties, but its
implementation requires the sample size n to be large compared to the dimension
p. This is a severe limitation, since more and more problems nowadays involve
high-dimensional directional data (e.g., in genetics or text mining). In this
work, we therefore introduce a modified Watson statistic that can cope with
high-dimensionality. We derive its asymptotic null distribution as both n and p
go to infinity. This is achieved in a universal asymptotic framework that
allows p to go to infinity arbitrarily fast (or slowly) as a function of n. We
further show that our results also provide high-dimensional tests for a problem
that has recently attracted much attention, namely that of testing that the
covariance matrix of a multinormal distribution has a "theta_0-spiked"
structure. Finally, a Monte Carlo simulation study corroborates our asymptotic
results
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