9,769 research outputs found
Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions
Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin
Uniform line fillings
Deterministic fabrication of random metamaterials requires filling of a space
with randomly oriented and randomly positioned chords with an on-average
homogenous density and orientation, which is a nontrivial task. We describe a
method to generate fillings with such chords, lines that run from edge to edge
of the space, in any dimension. We prove that the method leads to random but
on-average homogeneous and rotationally invariant fillings of circles, balls
and arbitrary-dimensional hyperballs from which other shapes such as rectangles
and cuboids can be cut. We briefly sketch the historic context of Bertrand's
paradox and Jaynes' solution by the principle of maximum ignorance. We analyse
the statistical properties of the produced fillings, mapping out the density
profile and the line-length distribution and comparing them to analytic
expressions. We study the characteristic dimensions of the space in between the
chords by determining the largest enclosed circles and balls in this pore
space, finding a lognormal distribution of the pore sizes. We apply the
algorithm to the direct-laser-writing fabrication design of optical
multiple-scattering samples as three-dimensional cubes of random but
homogeneously positioned and oriented chords.Comment: 10 pages, 12 figures; v3: restructured paper, more references, more
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The collocation and meshless methods for differential equations in R(2)
In recent years, meshless methods have become popular ones to solve differential equations. In this thesis, we aim at solving differential equations by using Radial Basis Functions, collocation methods and fundamental solutions (MFS). These methods are meshless, easy to understand, and even easier to implement
Sampling From A Manifold
We develop algorithms for sampling from a probability distribution on a
submanifold embedded in Rn. Applications are given to the evaluation of
algorithms in 'Topological Statistics'; to goodness of fit tests in exponential
families and to Neyman's smooth test. This article is partially expository,
giving an introduction to the tools of geometric measure theory
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