1,777 research outputs found
Quasi-likelihood for Spatial Point Processes
Fitting regression models for intensity functions of spatial point processes
is of great interest in ecological and epidemiological studies of association
between spatially referenced events and geographical or environmental
covariates. When Cox or cluster process models are used to accommodate
clustering not accounted for by the available covariates, likelihood based
inference becomes computationally cumbersome due to the complicated nature of
the likelihood function and the associated score function. It is therefore of
interest to consider alternative more easily computable estimating functions.
We derive the optimal estimating function in a class of first-order estimating
functions. The optimal estimating function depends on the solution of a certain
Fredholm integral equation which in practice is solved numerically. The
approximate solution is equivalent to a quasi-likelihood for binary spatial
data and we therefore use the term quasi-likelihood for our optimal estimating
function approach. We demonstrate in a simulation study and a data example that
our quasi-likelihood method for spatial point processes is both statistically
and computationally efficient
Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard
A study of the quantum triangular billiard requires consideration of a
boundary value problem for the Green's function of the Laplacian on a trianglar
domain. Our main result is a reformulation of this problem in terms of coupled
non--singular integral equations. A non--singular formulation, via Fredholm's
theory, guarantees uniqueness and provides a mathematically firm foundation for
both numerical and analytic studies. We compare and contrast our reformulation,
based on the exact solution for the wedge, with the standard singular integral
equations using numerical discretisation techniques. We consider in detail the
(integrable) equilateral triangle and the Pythagorean 3-4-5 triangle. Our
non--singular formulation produces results which are well behaved
mathematically. In contrast, while resolving the eigenvalues very well, the
standard approach displays various behaviours demonstrating the need for some
sort of ``renormalisation''. The non-singular formulation provides a
mathematically firm basis for the generation and analysis of periodic orbit
expansions. We discuss their convergence paying particular emphasis to the
computational effort required in comparision with Einstein--Brillouin--Keller
quantisation and the standard discretisation, which is analogous to the method
of Bogomolny. We also discuss the generalisation of our technique to smooth,
chaotic billiards.Comment: 50 pages LaTeX2e. Uses graphicx, amsmath, amsfonts, psfrag and
subfigure. 17 figures. To appear Annals of Physics, southern sprin
Application of Fredholm integral equations inverse theory to the radial basis function approximation problem
This paper reveals and examines the relationship between the solution and stability of Fredholm integral equations and radial basis function approximation or interpolation. The underlying system (kernel) matrices are shown to have a smoothing property which is dependent on the choice of kernel. Instead of using the condition number to describe the ill-conditioning, hence only looking at the largest and smallest singular values of the matrix, techniques from inverse theory, particularly the Picard condition, show that it is understanding the exponential decay of the singular values which is critical for interpreting and mitigating instability. Results on the spectra of certain classes of kernel matrices are reviewed, verifying the exponential decay of the singular values. Numerical results illustrating the application of integral equation inverse theory are also provided and demonstrate that interpolation weights may be regarded as samplings of a weighted solution of an integral equation. This is then relevant for mapping from one set of radial basis function centers to another set. Techniques for the solution of integral equations can be further exploited in future studies to find stable solutions and to reduce the impact of errors in the data
Inverse Density as an Inverse Problem: The Fredholm Equation Approach
In this paper we address the problem of estimating the ratio
where is a density function and is another density, or, more generally
an arbitrary function. Knowing or approximating this ratio is needed in various
problems of inference and integration, in particular, when one needs to average
a function with respect to one probability distribution, given a sample from
another. It is often referred as {\it importance sampling} in statistical
inference and is also closely related to the problem of {\it covariate shift}
in transfer learning as well as to various MCMC methods. It may also be useful
for separating the underlying geometry of a space, say a manifold, from the
density function defined on it.
Our approach is based on reformulating the problem of estimating
as an inverse problem in terms of an integral operator
corresponding to a kernel, and thus reducing it to an integral equation, known
as the Fredholm problem of the first kind. This formulation, combined with the
techniques of regularization and kernel methods, leads to a principled
kernel-based framework for constructing algorithms and for analyzing them
theoretically.
The resulting family of algorithms (FIRE, for Fredholm Inverse Regularized
Estimator) is flexible, simple and easy to implement.
We provide detailed theoretical analysis including concentration bounds and
convergence rates for the Gaussian kernel in the case of densities defined on
, compact domains in and smooth -dimensional sub-manifolds of
the Euclidean space.
We also show experimental results including applications to classification
and semi-supervised learning within the covariate shift framework and
demonstrate some encouraging experimental comparisons. We also show how the
parameters of our algorithms can be chosen in a completely unsupervised manner.Comment: Fixing a few typos in last versio
Spatiospectral concentration on a sphere
We pose and solve the analogue of Slepian's time-frequency concentration
problem on the surface of the unit sphere to determine an orthogonal family of
strictly bandlimited functions that are optimally concentrated within a closed
region of the sphere, or, alternatively, of strictly spacelimited functions
that are optimally concentrated within the spherical harmonic domain. Such a
basis of simultaneously spatially and spectrally concentrated functions should
be a useful data analysis and representation tool in a variety of geophysical
and planetary applications, as well as in medical imaging, computer science,
cosmology and numerical analysis. The spherical Slepian functions can be found
either by solving an algebraic eigenvalue problem in the spectral domain or by
solving a Fredholm integral equation in the spatial domain. The associated
eigenvalues are a measure of the spatiospectral concentration. When the
concentration region is an axisymmetric polar cap the spatiospectral projection
operator commutes with a Sturm-Liouville operator; this enables the
eigenfunctions to be computed extremely accurately and efficiently, even when
their area-bandwidth product, or Shannon number, is large. In the asymptotic
limit of a small concentration region and a large spherical harmonic bandwidth
the spherical concentration problem approaches its planar equivalent, which
exhibits self-similarity when the Shannon number is kept invariant.Comment: 48 pages, 17 figures. Submitted to SIAM Review, August 24th, 200
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