Stationary Point Processes Research Articles

Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations

AbstractIn this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of $$\mathbb {R}^n$$ R n . Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite $$(n-q)$$ ( n - q ) -moment, where $$1<q<n$$ 1 < q < n is the growth-exponent of the associated energy functionals. This assumption on the one hand ensures that the expectation of the nonlinear q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.

Linear Statistics of Determinantal Point Processes and Norm Representations

Abstract We study the asymptotic behavior of the fluctuations of smooth and rough linear statistics for determinantal point processes on the sphere and on the Euclidean space. The main tool is the generalization of some norm representation results for functions in Sobolev spaces and in the space of functions of bounded variation.

Summary statistics for spatio-temporal point processes on linear networks

We propose novel second/higher-order summary statistics for inhomogeneous spatio-temporal point processes when the spatial locations are limited to a linear network. More specifically, letting the spatial distance between events be measured by a regular distance metric, appropriate forms of K- and J-functions are introduced, and their theoretical relationships are studied. The theoretical forms of our proposed summary statistics are investigated under homogeneity, Poissonness, and independent thinning. Moreover, non-parametric estimators are derived, facilitating the use of our proposed summary statistics to study the spatio-temporal dependence between events. Through simulation studies, we demonstrate that our proposed J-function effectively identifies spatio-temporal clustering, inhibition, and randomness. Finally, we examine spatio-temporal dependencies for street crimes in Valencia, Spain, and traffic accidents in New York, USA.

Normality of smooth statistics for planar determinantal point processes

Normality of smooth statistics for planar determinantal point processes

Stochastic Models of Rainfall

Rainfall is the main input to most hydrological systems. To assess flood risk for a catchment area, hydrologists use models that require long series of subdaily, perhaps even subhourly, rainfall data, ideally from locations that cover the area. If historical data are not sufficient for this purpose, an alternative is to simulate synthetic data from a suitably calibrated model. We review stochastic models that have a mechanistic structure, intended to mimic physical features of the rainfall processes, and are constructed using stationary point processes. We describe models for temporal and spatial-temporal rainfall and consider how they can be fitted to data. We provide an example application using a temporal model and an illustration of data simulated from a spatial-temporal model. We discuss how these models can contribute to the simulation of future rainfall that reflects our changing climate.

Normal Approximation of Kabanov–Skorohod Integrals on Poisson Spaces

AbstractWe consider the normal approximation of Kabanov–Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of the Kabanov–Skorohod integral. The proofs rely on the Malliavin–Stein method and, in particular, on multiple applications of integration by parts formulae. As examples, we study some linear statistics of point processes that can be constructed by Poisson embeddings and functionals related to Pareto optimal points of a Poisson process.

Local Inhomogeneous Weighted Summary Statistics for Marked Point Processes

We introduce a family of local inhomogeneous mark-weighted summary statistics, of order two and higher, for general marked point processes. Depending on how the involved weight function is specified, these summary statistics capture different kinds of local dependence structures. We first derive some basic properties and show how these new statistical tools can be used to construct most existing summary statistics for (marked) point processes. We then propose a local test of random labeling. This procedure allows us to identify points, and consequently regions, where the random labeling assumption does not hold, for example, when the (functional) marks are spatially dependent. Through a simulation study we show that the test is able to detect local deviations from random labeling. We also provide an application to an earthquake point pattern with functional marks given by seismic waveforms.

Mean surface and volume particle tensors under L-restricted isotropy and associated ellipsoids

Abstract The convex-geometric Minkowski tensors contain information about shape and orientation of the underlying convex body. They therefore yield valuable summary statistics for stationary marked point processes with marks in the family of convex bodies, or, slightly more specialised, for stationary particle processes. We show here that if the distribution of the typical particle is invariant under rotations about a fixed k-plane, then the average volume tensors of the typical particle can be derived from k + 1-dimensional sections. This finding extends the well-known three-dimensional special case to higher dimensions. A corresponding result for the surface tensors is also proven. In the last part of the paper we show how Minkowski tensors can be used to define three ellipsoidal set-valued summary statistics, discuss their estimation and illustrate their construction and use in a simulation example. Two of these, the so-called Miles ellipsoid and the inertia ellipsoid, are based on mean volume tensors of ranks up to 2. The third, based on the mean surface tensor of rank 2, will be called the Blaschke ellipsoid and is only defined when the typical particle has a rotationally symmetric distribution about an axis, as we then can use uniqueness and reconstruction results for centred ellipsoids of revolution from their rank-2 surface tensor. The latter are also established here.

On estimating the structure factor of a point process, with applications to hyperuniformity

Hyperuniformity is the study of stationary point processes with a sub-Poisson variance in a large window. In other words, counting the points of a hyperuniform point process that fall in a given large region yields a small-variance Monte Carlo estimation of the volume. Hyperuniform point processes have received a lot of attention in statistical physics, both for the investigation of natural organized structures and the synthesis of materials. Unfortunately, rigorously proving that a point process is hyperuniform is usually difficult. A common practice in statistical physics and chemistry is to use a few samples to estimate a spectral measure called the structure factor. Its decay around zero provides a diagnostic of hyperuniformity. Different applied fields use however different estimators, and important algorithmic choices proceed from each field’s lore. This paper provides a systematic survey and derivation of known or otherwise natural estimators of the structure factor. We also leverage the consistency of these estimators to contribute the first asymptotically valid statistical test of hyperuniformity. We benchmark all estimators and hyperuniformity diagnostics on a set of examples. In an effort to make investigations of the structure factor and hyperuniformity systematic and reproducible, we further provide the Python toolbox structure_factor, containing all the estimators and tools that we discuss.

Singular Distribution Functions for Random Variables with Stationary Digits

Let F be the cumulative distribution function (CDF) of the base-q expansion $$\sum _{n=1}^\infty X_n q^{-n}$$ , where $$q\ge 2$$ is an integer and $$\{X_n\}_{n\ge 1}$$ is a stationary stochastic process with state space $$\{0,\ldots ,q-1\}$$ . In a previous paper we characterized the absolutely continuous and the discrete components of F. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: F is then either a uniform or a singular CDF on [0, 1]. Moreover, we study mixtures of such models. In most cases expressions and plots of F are given.

Convergence rate for geometric statistics of point processes having fast decay of dependence

Convergence rate for geometric statistics of point processes having fast decay of dependence

Asymptotic approximation of the likelihood of stationary determinantal point processes

AbstractContinuous determinantal point processes (DPPs) are a class of repulsive point processes on with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behavior when the observation window grows toward . This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on and compare favorably to common alternative estimation methods for continuous DPPs.

Analysis of spatial arrangement of fractures in two dimensions using point process statistics

Here we present a methodology that quantifies fracture arrangement in space as a function of position (barycenters), size (trace length in 2D), and orientation (variation in strike). We use point process statistics to analyze distances between barycenters and estimate a confidence interval for randomness that is compared with input data. The methodology can discriminate at least four types of spatial arrangements and identifies preferential associations of distances with fracture length and orientation. We apply our method to two naturally fractured formations and show that it can determine fracture arrangements under a significant statistical confidence.

Point process statistics improves particle size analysis

This paper re-considers the foundations of particle size statistics. While traditional particle size statistics consider their data as samples of random variables and use methods of classical mathematical statistics, here a particle sample is treated as a point process sample, and a suitable form of statistics is recommended. The whole sequence of ordered particle sizes is considered as a random variable in a suitable sample space. Instead of distribution functions, point process intensity functions are used. The application of point process data analysis is demonstrated for samples of fragments from single-particle crushing of glass balls. Three cases of data handling with point processes are presented: statistics for oversize particles, pooling of independent particle samples and pooling of piecewise particle data. Finally, the problem of goodness-of-fit testing for particle samples is briefly discussed. The point process approach turns out to be an extension of the classical approach, is simpler and more elegant, but retains all valuable traditional ideas. It is particularly strong in the analysis of oversize particles.Graphical abstract

Asymptotics of determinants with a rotation-invariant weight and discontinuities along circles

We study the moment generating function of the disk counting statistics of a two-dimensional determinantal point process which generalizes the complex Ginibre point process. This moment generating function involves an n×n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has discontinuities along circles centered at 0. These discontinuities can be thought of as a two-dimensional analogue of jump-type Fisher-Hartwig singularities. In this paper, we obtain large n asymptotics for this determinant, up to and including the term of order n−12. We allow for any finite number of discontinuities in the bulk, one discontinuity at the edge, and any finite number of discontinuities bounded away from the bulk. As an application, we obtain the large n asymptotics of all the cumulants of the disk counting function up to and including the term of order n−12, both in the bulk and at the edge. This improves on the best known results for the complex Ginibre point process, and for general values of our parameters these results are completely new. Our proof makes a novel use of the uniform asymptotics of the incomplete gamma function.

Local inhomogeneous second-order characteristics for spatio-temporal point processes occurring on linear networks

Point processes on linear networks are increasingly being considered to analyse events occurring on particular network-based structures. In this paper, we extend Local Indicators of Spatio-Temporal Association (LISTA) functions to the non-Euclidean space of linear networks, allowing to obtain information on how events relate to nearby events. In particular, we propose the local version of two inhomogeneous second-order statistics for spatio-temporal point processes on linear networks, the K- and the pair correlation functions. We put particular emphasis on the local K-functions, deriving come theoretical results which enable us to show that these LISTA functions are useful for diagnostics of models specified on networks, and can be helpful to assess the goodness-of-fit of different spatio-temporal models fitted to point patterns occurring on linear networks. Our methods do not rely on any particular model assumption on the data, and thus they can be applied for whatever is the underlying model of the process. We finally present a real data analysis of traffic accidents in Medellin (Colombia).

Neveu’s Exchange Formula for Analysis of Wireless Networks With Hotspot Clusters

Theory of point processes, in particular Palm calculus within the stationary framework, plays a fundamental role in the analysis of spatial stochastic models of wireless communication networks. Neveu’s exchange formula, which connects the respective Palm distributions for two jointly stationary point processes, is known as one of the most important results in the Palm calculus. However, its use in the analysis of wireless networks seems to be limited so far and one reason for this may be that the formula in a well-known form is based upon the Voronoi tessellation. In this paper, we present an alternative form of Neveu’s exchange formula, which does not rely on the Voronoi tessellation but includes the one as a special case. We then demonstrate that our new form of the exchange formula is useful for the analysis of wireless networks with hotspot clusters modeled using cluster point processes.

Finitary isomorphisms of renewal point processes and continuous-time regenerative processes

We show that a large class of stationary continuous-time regenerative processes are finitarily isomorphic to one another. The key is showing that any stationary renewal point process whose jump distribution is absolutely continuous with exponential tails is finitarily isomorphic to a Poisson point process. We further give simple necessary and sufficient conditions for a renewal point process to be finitarily isomorphic to a Poisson point process. This improves results and answers several questions of Soo [33] and of Kosloff and Soo [19].

Particle gradient descent model for point process generation

This paper presents a statistical model for stationary ergodic point processes, estimated from a single realization observed in a square window. With existing approaches in stochastic geometry, it is very difficult to model processes with complex geometries formed by a large number of particles. Inspired by recent works on gradient descent algorithms for sampling maximum-entropy models, we describe a model that allows for fast sampling of new configurations reproducing the statistics of the given observation. Starting from an initial random configuration, its particles are moved according to the gradient of an energy, in order to match a set of prescribed moments (functionals). Our moments are defined via a phase harmonic operator on the wavelet transform of point patterns. They allow one to capture multi-scale interactions between the particles, while controlling explicitly the number of moments by the scales of the structures to model. We present numerical experiments on point processes with various geometric structures, and assess the quality of the model by spectral and topological data analysis.

SCATTERING STATISTICS OF GENERALIZED SPATIAL POISSON POINT PROCESSES.

We present a machine learning model for the analysis of randomly generated discrete signals, modeled as the points of an inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by Mallat, our construction is naturally invariant to translations and reflections, but it decouples the roles of scale and frequency, replacing wavelets with Gabor-type measurements. We show that, with suitable nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes.