28,734 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
Constructing Reference Metrics on Multicube Representations of Arbitrary Manifolds
Reference metrics are used to define the differential structure on multicube
representations of manifolds, i.e., they provide a simple and practical way to
define what it means globally for tensor fields and their derivatives to be
continuous. This paper introduces a general procedure for constructing
reference metrics automatically on multicube representations of manifolds with
arbitrary topologies. The method is tested here by constructing reference
metrics for compact, orientable two-dimensional manifolds with genera between
zero and five. These metrics are shown to satisfy the Gauss-Bonnet identity
numerically to the level of truncation error (which converges toward zero as
the numerical resolution is increased). These reference metrics can be made
smoother and more uniform by evolving them with Ricci flow. This smoothing
procedure is tested on the two-dimensional reference metrics constructed here.
These smoothing evolutions (using volume-normalized Ricci flow with DeTurck
gauge fixing) are all shown to produce reference metrics with constant scalar
curvatures (at the level of numerical truncation error).Comment: 37 pages, 16 figures; additional introductory material added in
version accepted for publicatio
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
Three-dimensional Analytical Description of Magnetised Winds from Oblique Pulsars
Rotating neutron stars, or pulsars, are plausibly the source of power behind
many astrophysical systems, such as gamma-ray bursts, supernovae, pulsar wind
nebulae and supernova remnants. In the past several years, 3D numerical
simulations made it possible to compute pulsar spindown luminosity from first
principles and revealed that oblique pulsar winds are more powerful than
aligned ones. However, what causes this enhanced power output of oblique
pulsars is not understood. In this work, using time-dependent 3D
magnetohydrodynamic (MHD) and force-free simulations, we show that, contrary to
the standard paradigm, the open magnetic flux, which carries the energy away
from the pulsar, is laterally non-uniform. We argue that this non-uniformity is
the primary reason for the increased luminosity of oblique pulsars. To
demonstrate this, we construct simple analytic descriptions of aligned and
orthogonal pulsar winds and combine them to obtain an accurate 3D description
of the pulsar wind for any obliquity. Our approach describes both the warped
magnetospheric current sheet and the smooth variation of pulsar wind properties
outside of it. We find that generically the magnetospheric current sheet
separates plasmas that move at mildly relativistic velocities relative to each
other. This suggests that the magnetospheric reconnection is a type of driven,
rather than free, reconnection. The jump in magnetic field components across
the current sheet decreases with increasing obliquity, which could be a
mechanism that reduces dissipation in near-orthogonal pulsars. Our analytical
description of the pulsar wind can be used for constructing models of pulsar
gamma-ray emission, pulsar wind nebulae, and magnetar-powered core-collapse
gamma-ray bursts and supernovae.Comment: Submitted to MNRAS, comments welcome. 12 pages, 16 figures, uses
mn2e.cl
Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives
We consider the problem of testing uniformity on high-dimensional unit
spheres. We are primarily interested in non-null issues. We show that
rotationally symmetric alternatives lead to two Local Asymptotic Normality
(LAN) structures. The first one is for fixed modal location and allows
to derive locally asymptotically most powerful tests under specified .
The second one, that addresses the Fisher-von Mises-Langevin (FvML) case,
relates to the unspecified- problem and shows that the high-dimensional
Rayleigh test is locally asymptotically most powerful invariant. Under mild
assumptions, we derive the asymptotic non-null distribution of this test, which
allows to extend away from the FvML case the asymptotic powers obtained there
from Le Cam's third lemma. Throughout, we allow the dimension to go to
infinity in an arbitrary way as a function of the sample size . Some of our
results also strengthen the local optimality properties of the Rayleigh test in
low dimensions. We perform a Monte Carlo study to illustrate our asymptotic
results. Finally, we treat an application related to testing for sphericity in
high dimensions
Inference Under Convex Cone Alternatives for Correlated Data
In this research, inferential theory for hypothesis testing under general
convex cone alternatives for correlated data is developed. While there exists
extensive theory for hypothesis testing under smooth cone alternatives with
independent observations, extension to correlated data under general convex
cone alternatives remains an open problem. This long-pending problem is
addressed by (1) establishing that a "generalized quasi-score" statistic is
asymptotically equivalent to the squared length of the projection of the
standard Gaussian vector onto the convex cone and (2) showing that the
asymptotic null distribution of the test statistic is a weighted chi-squared
distribution, where the weights are "mixed volumes" of the convex cone and its
polar cone. Explicit expressions for these weights are derived using the
volume-of-tube formula around a convex manifold in the unit sphere.
Furthermore, an asymptotic lower bound is constructed for the power of the
generalized quasi-score test under a sequence of local alternatives in the
convex cone. Applications to testing under order restricted alternatives for
correlated data are illustrated.Comment: 31 page
Quantization of generic chaotic 3D billiard with smooth boundary I: energy level statistic
Numerical calculation and analysis of extremely high-lying energy spectra,
containing thousands of levels with sequential quantum number up to 62,000 per
symmetry class, of a generic chaotic 3D quantum billiard is reported. The shape
of the billiard is given by a simple and smooth de formation of a unit sphere
which gives rise to (almost) fully chaotic classical dynamics. We present an
analysis of (i) quantum length spectrum whose smooth part agrees with the 3D
Weyl formula and whose oscillatory part is peaked around the periods of
classical periodic orbits, (ii) nearest neighbor level spacing distribution and
(iii) number variance.
Although the chaotic classical dynamics quickly and uniformly explores almost
entire energy shell, while the measure of the regular part of phase space is
insignificantly small, we find small but significant deviations from GOE
statistics which are explained in terms of localization of eigenfunctions onto
lower dimensional classically invariant manifolds.Comment: 10 pages in plain Latex (6 figures in PCL format available upon
request) submitted to Phys. Lett.
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