1,274 research outputs found
Sojourn measures of Student and Fisher-Snedecor random fields
Limit theorems for the volumes of excursion sets of weakly and strongly
dependent heavy-tailed random fields are proved. Some generalizations to
sojourn measures above moving levels and for cross-correlated scenarios are
presented. Special attention is paid to Student and Fisher-Snedecor random
fields. Some simulation results are also presented.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ529 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Morphological Image Analysis of Quantum Motion in Billiards
Morphological image analysis is applied to the time evolution of the
probability distribution of a quantum particle moving in two and
three-dimensional billiards. It is shown that the time-averaged Euler
characteristic of the probability density provides a well defined quantity to
distinguish between classically integrable and non-integrable billiards. In
three dimensions the time-averaged mean breadth of the probability density may
also be used for this purpose.Comment: Major revision. Changes include a more detailed discussion of the
theory and results for 3 dimensions. Now: 10 pages, 9 figures (some are
colored), 3 table
High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere
In this paper, we shall be concerned with geometric functionals and excursion
probabilities for some nonlinear transforms evaluated on Fourier components of
spherical random fields. In particular, we consider both random spherical
harmonics and their smoothed averages, which can be viewed as random wavelet
coefficients in the continuous case. For such fields, we consider smoothed
polynomial transforms; we focus on the geometry of their excursion sets, and we
study their asymptotic behaviour, in the high-frequency sense. We focus on the
analysis of Euler-Poincar\'{e} characteristics, which can be exploited to
derive extremely accurate estimates for excursion probabilities. The present
analysis is motivated by the investigation of asymmetries and anisotropies in
cosmological data. The statistics we focus on are also suitable to deal with
spherical random fields which can only be partially observed, the canonical
example being provided by the masking effect of the Milky Way on Cosmic
Microwave Background (CMB) radiation data.Comment: Published at http://dx.doi.org/10.1214/15-AAP1097 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Second order analysis of geometric functionals of Boolean models
This paper presents asymptotic covariance formulae and central limit theorems
for geometric functionals, including volume, surface area, and all Minkowski
functionals and translation invariant Minkowski tensors as prominent examples,
of stationary Boolean models. Special focus is put on the anisotropic case. In
the (anisotropic) example of aligned rectangles, we provide explicit analytic
formulae and compare them with simulation results. We discuss which information
about the grain distribution second moments add to the mean values.Comment: Chapter of the forthcoming book "Tensor Valuations and their
Applications in Stochastic Geometry and Imaging" in Lecture Notes in
Mathematics edited by Markus Kiderlen and Eva B. Vedel Jensen. (The second
version mainly resolves minor LaTeX problems.
Parametric Inference for Biological Sequence Analysis
One of the major successes in computational biology has been the unification,
using the graphical model formalism, of a multitude of algorithms for
annotating and comparing biological sequences. Graphical models that have been
applied towards these problems include hidden Markov models for annotation,
tree models for phylogenetics, and pair hidden Markov models for alignment. A
single algorithm, the sum-product algorithm, solves many of the inference
problems associated with different statistical models. This paper introduces
the \emph{polytope propagation algorithm} for computing the Newton polytope of
an observation from a graphical model. This algorithm is a geometric version of
the sum-product algorithm and is used to analyze the parametric behavior of
maximum a posteriori inference calculations for graphical models.Comment: 15 pages, 4 figures. See also companion paper "Tropical Geometry of
Statistical Models" (q-bio.QM/0311009
Explicit excluded volume of cylindrically symmetric convex bodies
We represent explicitly the excluded volume Ve{B1,B2} of two generic
cylindrically symmetric, convex rigid bodies, B1 and B2, in terms of a family
of shape functionals evaluated separately on B1 and B2. We show that Ve{B1,B2}
fails systematically to feature a dipolar component, thus making illusory the
assignment of any shape dipole to a tapered body in this class. The method
proposed here is applied to cones and validated by a shape-reconstruction
algorithm. It is further applied to spheroids (ellipsoids of revolution), for
which it shows how some analytic estimates already regarded as classics should
indeed be emended
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