5,215 research outputs found
Pair correlation functions and limiting distributions of iterated cluster point processes
We consider a Markov chain of point processes such that each state is a super
position of an independent cluster process with the previous state as its
centre process together with some independent noise process. The model extends
earlier work by Felsenstein and Shimatani describing a reproducing population.
We discuss when closed term expressions of the first and second order moments
are available for a given state. In a special case it is known that the pair
correlation function for these type of point processes converges as the Markov
chain progresses, but it has not been shown whether the Markov chain has an
equilibrium distribution with this, particular, pair correlation function and
how it may be constructed. Assuming the same reproducing system, we construct
an equilibrium distribution by a coupling argument
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
Averaging approach to phase coherence of uncoupled limit-cycle oscillators receiving common random impulses
Populations of uncoupled limit-cycle oscillators receiving common random
impulses show various types of phase-coherent states, which are characterized
by the distribution of phase differences between pairs of oscillators. We
develop a theory to predict the stationary distribution of pairwise phase
difference from the phase response curve, which quantitatively encapsulates the
oscillator dynamics, via averaging of the Frobenius-Perron equation describing
the impulse-driven oscillators. The validity of our theory is confirmed by
direct numerical simulations using the FitzHugh-Nagumo neural oscillator
receiving common Poisson impulses as an example
Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media
The continuous-time random walk (CTRW) model is useful for alleviating the
computational burden of simulating diffusion in actual media. In principle,
isotropic CTRW only requires knowledge of the step-size, , and
waiting-time, , distributions of the random walk in the medium and it then
generates presumably equivalent walks in free space, which are much faster.
Here we test the usefulness of CTRW to modelling diffusion of finite-size
particles in porous medium generated by loose granular packs. This is done by
first simulating the diffusion process in a model porous medium of mean
coordination number, which corresponds to marginal rigidity (the loosest
possible structure), computing the resulting distributions and as
functions of the particle size, and then using these as input for a free space
CTRW. The CTRW walks are then compared to the ones simulated in the actual
media.
In particular, we study the normal-to-anomalous transition of the diffusion
as a function of increasing particle size. We find that, given the same
and for the simulation and the CTRW, the latter predicts incorrectly the
size at which the transition occurs. We show that the discrepancy is related to
the dependence of the effective connectivity of the porous media on the
diffusing particle size, which is not captured simply by these distributions.
We propose a correcting modification to the CTRW model -- adding anisotropy
-- and show that it yields good agreement with the simulated diffusion process.
We also present a method to obtain and directly from the porous
sample, without having to simulate an actual diffusion process. This extends
the use of CTRW, with all its advantages, to modelling diffusion processes of
finite-size particles in such confined geometries.Comment: 9 pages, 7 figure
Semi-Markov representations of some stochastic point processes
Imperial Users onl
The random geometry of equilibrium phases
This is a (long) survey about applications of percolation theory in
equilibrium statistical mechanics. The chapters are as follows:
1. Introduction
2. Equilibrium phases
3. Some models
4. Coupling and stochastic domination
5. Percolation
6. Random-cluster representations
7. Uniqueness and exponential mixing from non-percolation
8. Phase transition and percolation
9. Random interactions
10. Continuum modelsComment: 118 pages. Addresses: [email protected]
http://www.mathematik.uni-muenchen.de/~georgii.html [email protected]
http://www.math.chalmers.se/~olleh [email protected]
- …