13 research outputs found
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
Contrast estimation for parametric stationary determinantal point processes
International audienceWe study minimum contrast estimation for parametric stationary determi-nantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley's K-function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation)
Contrast estimation for parametric stationary determinantal point processes
International audienceWe study minimum contrast estimation for parametric stationary determi-nantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley's K-function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation)
Contrast estimation for parametric stationary determinantal point processes
We study minimum contrast estimation for parametric stationary determi-nantal
point processes. These processes form a useful class of models for repulsive
(or regular, or inhibitive) point patterns and are already applied in numerous
statistical applications. Our main focus is on minimum contrast methods based
on the Ripley's K-function or on the pair correlation function. Strong
consistency and asymptotic normality of theses procedures are proved under
general conditions that only concern the existence of the process and its
regularity with respect to the parameters. A key ingredient of the proofs is
the recently established Brillinger mixing property of stationary determinantal
point processes. This work may be viewed as a complement to the study of Y.
Guan and M. Sherman who establish the same kind of asymptotic properties for a
large class of Cox processes, which in turn are models for clustering (or
aggregation)
The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applications.
We start with a simple introduction to topological data analysis where the most popular tool is called a persistence diagram. Briefly, a persistence diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative functional summary statistics have been suggested, but either they do not contain the full information of the persistence diagram or they are two-dimensional functions. We suggest a new functional summary statistic that is one-dimensional and hence easier to handle, and which under mild conditions contains the full information of the persistence diagram. Its usefulness is illustrated in statistical settings concerned with point clouds and brain artery trees. The supplementary materials include additional methods and examples, technical details, and the R code used for all examples.</p
Quantifying repulsiveness of determinantal point processes
Determinantal point processes (DPPs) have recently proved to be a useful
class of models in several areas of statistics, including spatial statistics,
statistical learning and telecommunications networks. They are models for
repulsive (or regular, or inhibitive) point processes, in the sense that nearby
points of the process tend to repel each other. We consider two ways to
quantify the repulsiveness of a point process, both based on its second-order
properties, and we address the question of how repulsive a stationary DPP can
be. We determine the most repulsive stationary DPP, when the intensity is
fixed, and for a given we investigate repulsiveness in the subclass of
-dependent stationary DPPs, that is, stationary DPPs with -compactly
supported kernels. Finally, in both the general case and the -dependent
case, we present some new parametric families of stationary DPPs that can cover
a large range of DPPs, from the stationary Poisson process (the case of no
interaction) to the most repulsive DPP.Comment: Published at http://dx.doi.org/10.3150/15-BEJ718 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Standard and robust intensity parameter estimation for stationary determinantal point processes
This work is concerned with the estimation of the intensity parameter of a stationary determinantal point process. We consider the standard estimator, corresponding to the number of observed points per unit volume and a recently introduced median-based estimator more robust to outliers. The consistency and asymptotic normality of estimators are obtained under mild assumptions on the determinantal point process. We illustrate the efficiency of the procedures in a simulation study