1,725 research outputs found
Upper bounds for spatial point process approximations
We consider the behavior of spatial point processes when subjected to a class
of linear transformations indexed by a variable T. It was shown in Ellis [Adv.
in Appl. Probab. 18 (1986) 646-659] that, under mild assumptions, the
transformed processes behave approximately like Poisson processes for large T.
In this article, under very similar assumptions, explicit upper bounds are
given for the d_2-distance between the corresponding point process
distributions. A number of related results, and applications to kernel density
estimation and long range dependence testing are also presented. The main
results are proved by applying a generalized Stein-Chen method to discretized
versions of the point processes.Comment: Published at http://dx.doi.org/10.1214/105051604000000684 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Variational approach for spatial point process intensity estimation
We introduce a new variational estimator for the intensity function of an
inhomogeneous spatial point process with points in the -dimensional
Euclidean space and observed within a bounded region. The variational estimator
applies in a simple and general setting when the intensity function is assumed
to be of log-linear form where is a spatial
covariate function and the focus is on estimating . The variational
estimator is very simple to implement and quicker than alternative estimation
procedures. We establish its strong consistency and asymptotic normality. We
also discuss its finite-sample properties in comparison with the maximum first
order composite likelihood estimator when considering various inhomogeneous
spatial point process models and dimensions as well as settings were is
completely or only partially known.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ516 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Spatial point process theory
No abstract
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
Score, Pseudo-Score and Residual Diagnostics for Spatial Point Process Models
We develop new tools for formal inference and informal model validation in
the analysis of spatial point pattern data. The score test is generalized to a
"pseudo-score" test derived from Besag's pseudo-likelihood, and to a class of
diagnostics based on point process residuals. The results lend theoretical
support to the established practice of using functional summary statistics,
such as Ripley's -function, when testing for complete spatial randomness;
and they provide new tools such as the compensator of the -function for
testing other fitted models. The results also support localization methods such
as the scan statistic and smoothed residual plots. Software for computing the
diagnostics is provided.Comment: Published in at http://dx.doi.org/10.1214/11-STS367 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Orthogonal series estimation of the pair correlation function of a spatial point process
The pair correlation function is a fundamental spatial point process
characteristic that, given the intensity function, determines second order
moments of the point process. Non-parametric estimation of the pair correlation
function is a typical initial step of a statistical analysis of a spatial point
pattern. Kernel estimators are popular but especially for clustered point
patterns suffer from bias for small spatial lags. In this paper we introduce a
new orthogonal series estimator. The new estimator is consistent and
asymptotically normal according to our theoretical and simulation results. Our
simulations further show that the new estimator can outperform the kernel
estimators in particular for Poisson and clustered point processes
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