629 research outputs found
Central limit theorems for Poisson hyperplane tessellations
We derive a central limit theorem for the number of vertices of convex
polytopes induced by stationary Poisson hyperplane processes in .
This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab.
30 (1998) 640--656] for intersection points of motion-invariant Poisson line
processes in . Our proof is based on Hoeffding's decomposition of
-statistics which seems to be more efficient and adequate to tackle the
higher-dimensional case than the ``method of moments'' used in [Adv. in Appl.
Probab. 30 (1998) 640--656] to treat the case . Moreover, we extend our
central limit theorem in several directions. First we consider -flat
processes induced by Poisson hyperplane processes in for . Second we derive (asymptotic) confidence intervals for the
intensities of these -flat processes and, third, we prove multivariate
central limit theorems for the -dimensional joint vectors of numbers of
-flats and their -volumes, respectively, in an increasing spherical
region.Comment: Published at http://dx.doi.org/10.1214/105051606000000033 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Cell shape analysis of random tessellations based on Minkowski tensors
To which degree are shape indices of individual cells of a tessellation
characteristic for the stochastic process that generates them? Within the
context of stochastic geometry and the physics of disordered materials, this
corresponds to the question of relationships between different stochastic
models. In the context of image analysis of synthetic and biological materials,
this question is central to the problem of inferring information about
formation processes from spatial measurements of resulting random structures.
We address this question by a theory-based simulation study of shape indices
derived from Minkowski tensors for a variety of tessellation models. We focus
on the relationship between two indices: an isoperimetric ratio of the
empirical averages of cell volume and area and the cell elongation quantified
by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for
these quantities, as well as for distributions thereof and for correlations of
cell shape and volume, are presented for Voronoi mosaics of the Poisson point
process, determinantal and permanental point processes, and Gibbs hard-core and
random sequential absorption processes as well as for Laguerre tessellations of
polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data
are complemented by mechanically stable crystalline sphere and disordered
ellipsoid packings and area-minimising foam models. We find that shape indices
of individual cells are not sufficient to unambiguously identify the generating
process even amongst this limited set of processes. However, we identify
significant differences of the shape indices between many of these tessellation
models. Given a realization of a tessellation, these shape indices can narrow
the choice of possible generating processes, providing a powerful tool which
can be further strengthened by density-resolved volume-shape correlations.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 Jense
Fault Tolerance in Cellular Automata at High Fault Rates
A commonly used model for fault-tolerant computation is that of cellular
automata. The essential difficulty of fault-tolerant computation is present in
the special case of simply remembering a bit in the presence of faults, and
that is the case we treat in this paper. We are concerned with the degree (the
number of neighboring cells on which the state transition function depends)
needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We
consider both the traditional transient fault model (where faults occur
independently in time and space) and a recently introduced combined fault model
which also includes manufacturing faults (which occur independently in space,
but which affect cells for all time). We also consider both a purely
probabilistic fault model (in which the states of cells are perturbed at
exactly the fault rate) and an adversarial model (in which the occurrence of a
fault gives control of the state to an omniscient adversary). We show that
there are cellular automata that can tolerate a fault rate (with
) with degree , even with adversarial combined
faults. The simplest such automata are based on infinite regular trees, but our
results also apply to other structures (such as hyperbolic tessellations) that
contain infinite regular trees. We also obtain a lower bound of
, even with purely probabilistic transient faults only
Asymptotic goodness-of-fit tests for the Palm mark distribution of stationary point processes with correlated marks
We consider spatially homogeneous marked point patterns in an unboundedly
expanding convex sampling window. Our main objective is to identify the
distribution of the typical mark by constructing an asymptotic
-goodness-of-fit test. The corresponding test statistic is based on a
natural empirical version of the Palm mark distribution and a smoothed
covariance estimator which turns out to be mean square consistent. Our approach
does not require independent marks and allows dependences between the mark
field and the point pattern. Instead we impose a suitable -mixing
condition on the underlying stationary marked point process which can be
checked for a number of Poisson-based models and, in particular, in the case of
geostatistical marking. In order to study test performance, our test approach
is applied to detect anisotropy of specific Boolean models.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ523 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin
note: substantial text overlap with arXiv:1205.504
Towards generalized measures grasping CA dynamics
In this paper we conceive Lyapunov exponents, measuring the rate of separation between two initially close configurations, and Jacobians, expressing the sensitivity of a CA's transition function to its inputs, for cellular automata (CA) based upon irregular tessellations of the n-dimensional Euclidean space. Further, we establish a relationship between both that enables us to derive a mean-field approximation of the upper bound of an irregular CA's maximum Lyapunov exponent. The soundness and usability of these measures is illustrated for a family of 2-state irregular totalistic CA
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Stochastic Geometry, Spatial Statistics and Statistical Physics
[no abstract available
Typical Geometry, Second-Order Properties and Central Limit Theory for Iteration Stable Tessellations
Since the seminal work of Mecke, Nagel and Weiss, the iteration stable (STIT)
tessellations have attracted considerable interest in stochastic geometry as a
natural and flexible yet analytically tractable model for hierarchical spatial
cell-splitting and crack-formation processes. The purpose of this paper is to
describe large scale asymptotic geometry of STIT tessellations in
and more generally that of non-stationary iteration infinitely
divisible tessellations. We study several aspects of the typical first-order
geometry of such tessellations resorting to martingale techniques as providing
a direct link between the typical characteristics of STIT tessellations and
those of suitable mixtures of Poisson hyperplane tessellations. Further, we
also consider second-order properties of STIT and iteration infinitely
divisible tessellations, such as the variance of the total surface area of cell
boundaries inside a convex observation window. Our techniques, relying on
martingale theory and tools from integral geometry, allow us to give explicit
and asymptotic formulae. Based on these results, we establish a functional
central limit theorem for the length/surface increment processes induced by
STIT tessellations. We conclude a central limit theorem for total edge
length/facet surface, with normal limit distribution in the planar case and
non-normal ones in all higher dimensions.Comment: 51 page
Traffic Analysis in Random Delaunay Tessellations and Other Graphs
In this work we study the degree distribution, the maximum vertex and edge
flow in non-uniform random Delaunay triangulations when geodesic routing is
used. We also investigate the vertex and edge flow in Erd\"os-Renyi random
graphs, geometric random graphs, expanders and random -regular graphs.
Moreover we show that adding a random matching to the original graph can
considerably reduced the maximum vertex flow.Comment: Submitted to the Journal of Discrete Computational Geometr
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