71 research outputs found
Consistency of likelihood estimation for Gibbs point processes
Strong consistency of the maximum likelihood estimator (MLE) for parametric
Gibbs point process models is established. The setting is very general. It
includes pairwise pair potentials, finite and infinite multibody interactions
and geometrical interactions, where the range can be finite or infinite. The
Gibbs interaction may depend linearly or non-linearly on the parameters, a
particular case being hardcore parameters and interaction range parameters. As
important examples, we deduce the consistency of the MLE for all parameters of
the Strauss model, the hardcore Strauss model, the Lennard-Jones model and the
area-interaction model
Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction
General models of Gibbs Delaunay-Voronoi tessellations, which can be viewed
as extensions of Ord's process, are considered. The interaction may occur on
each cell of the tessellation and between neighbour cells. The tessellation may
also be subjected to a geometric hardcore interaction, forcing the cells not to
be too large, too small, or too flat. This setting, natural for applications,
introduces some theoretical difficulties since the interaction is not
necessarily hereditary. Mathematical results available for studying these
models are reviewed and further outcomes are provided. They concern the
existence, the simulation and the estimation of such tessellations. Based on
these results, tools to handle these objects in practice are presented: how to
simulate them, estimate their parameters and validate the fitted model. Some
examples of simulated tessellations are studied in details
Some convergence results on quadratic forms for random fields and application to empirical covariances
Limit theorems are proved for quadratic forms of Gaussian random fields in
presence of long memory. We obtain a non central limit theorem under a minimal
integrability condition, which allows isotropic and anisotropic models. We
apply our limit theorems and those of Ginovian (99) to obtain the asymptotic
behavior of the empirical covariances of Gaussian fields, which is a particular
example of quadratic forms. We show that it is possible to obtain a Gaussian
limit when the spectral density is not in . Therefore the dichotomy
observed in dimension between central and non central limit theorems
cannot be stated so easily due to possible anisotropic strong dependence in
A tutorial on estimator averaging in spatial point process models
Assume that several competing methods are available to estimate a parameter
in a given statistical model. The aim of estimator averaging is to provide a
new estimator, built as a linear combination of the initial estimators, that
achieves better properties, under the quadratic loss, than each individual
initial estimator. This contribution provides an accessible and clear overview
of the method, and investigates its performances on standard spatial point
process models. It is demonstrated that the average estimator clearly improves
on standard procedures for the considered models. For each example, the code to
implement the method with the R software (which only consists of few lines) is
provided
Aggregation of isotropic autoregressive fields
This note constitutes a corrigendum to the article of Azomahou, JSPI,
139:2581-2597. The aggregation of isotropic four nearest neighbors
autoregressive models on the lattice, with random coefficient, is investigated.
The spectral density of the resulting random field is studied in details for a
large class of law of the AR coefficient. Depending on this law, the aggregated
field may exhibit short memory or isotropic long memory
A two-sample test for comparison of long memory parameters
We construct a two-sample test for comparison of long memory parameters based
on ratios of two rescaled variance (V/S) statistics studied in [Giraitis L.,
Leipus, R., Philippe, A., 2006. A test for stationarity versus trends and unit
roots for a wide class of dependent errors. Econometric Theory 21, 989--1029].
The two samples have the same length and can be mutually independent or
dependent. In the latter case, the test statistic is modified to make it
asymptotically free of the long-run correlation coefficient between the
samples. To diminish the sensitivity of the test on the choice of the bandwidth
parameter, an adaptive formula for the bandwidth parameter is derived using the
asymptotic expansion in [Abadir, K., Distaso, W., Giraitis, L., 2009. Two
estimators of the long-run variance: Beyond short memory. Journal of
Econometrics 150, 56--70]. A simulation study shows that the above choice of
bandwidth leads to a good size of our comparison test for most values of
fractional and ARMA parameters of the simulated series
Brillinger mixing of determinantal point processes and statistical applications
Stationary determinantal point processes are proved to be Brillinger mixing.
This property is an important step towards asymptotic statistics for these
processes. As an important example, a central limit theorem for a wide class of
functionals of determinantal point processes is established. This result yields
in particular the asymptotic normality of the estimator of the intensity of a
stationary determinantal point process and of the kernel estimator of its pair
correlation
Self-similar prior and wavelet bases for hidden incompressible turbulent motion
This work is concerned with the ill-posed inverse problem of estimating
turbulent flows from the observation of an image sequence. From a Bayesian
perspective, a divergence-free isotropic fractional Brownian motion (fBm) is
chosen as a prior model for instantaneous turbulent velocity fields. This
self-similar prior characterizes accurately second-order statistics of velocity
fields in incompressible isotropic turbulence. Nevertheless, the associated
maximum a posteriori involves a fractional Laplacian operator which is delicate
to implement in practice. To deal with this issue, we propose to decompose the
divergent-free fBm on well-chosen wavelet bases. As a first alternative, we
propose to design wavelets as whitening filters. We show that these filters are
fractional Laplacian wavelets composed with the Leray projector. As a second
alternative, we use a divergence-free wavelet basis, which takes implicitly
into account the incompressibility constraint arising from physics. Although
the latter decomposition involves correlated wavelet coefficients, we are able
to handle this dependence in practice. Based on these two wavelet
decompositions, we finally provide effective and efficient algorithms to
approach the maximum a posteriori. An intensive numerical evaluation proves the
relevance of the proposed wavelet-based self-similar priors.Comment: SIAM Journal on Imaging Sciences, 201
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