Stationary Point Processes Research Articles

Distribution analysis and modeling of air voids in concrete as spatial point processes

The air voids present in concrete were evaluated as a 2D spatial point process. The characteristic distance between the points was defined using the nearest neighbor distance function and resulted to be very similar to the spacing factor, which has been used for long. This similarity was discussed by comparing the cubic lattice structure of the air voids associated to the point process model and the spacing factor. Due to the random air-void distribution within the cement paste matrix, a method to estimate the characteristic distance by simulation was also proposed. The characteristics of the random point patterns were consistent with the air-void structures assumed for the spacing factor. Hence, the characteristic distance could be used as a quality parameter instead of the spacing factor and the point process statistics was a useful evaluating method for air-void distribution in concrete.

Limit theorems for persistence diagrams

The persistent homology of a stationary point process on $\mathbf{R}^{N}$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

Systematic time expansion for the Kardar–Parisi–Zhang equation, linear statistics of the GUE at the edge and trapped fermions

We present a systematic short time expansion for the generating function of the one point height probability distribution for the KPZ equation with droplet initial condition, which goes much beyond previous studies. The expansion is checked against a numerical evaluation of the known exact Fredholm determinant expression. We also obtain the next order term for the Brownian initial condition. Although initially devised for short time, a resummation of the series allows to obtain also the long time large deviation function, found to agree with previous works using completely different techniques. Unexpected similarities with stationary large deviations of TASEP with periodic and open boundaries are discussed. Two additional applications are given. (i) Our method is generalized to study the linear statistics of the Airy point process, i.e. of the GUE edge eigenvalues. We obtain the generating function of the cumulants of the empirical measure to a high order. The second cumulant is found to match the result in the bulk obtained from the Gaussian free field by Borodin and Ferrari [1,2], but we obtain systematic corrections to the Gaussian free field (higher cumulants, expansion towards the edge). This also extends a result of Basor and Widom [3] to a much higher order. We obtain large deviation functions for the Airy point process for a variety of linear statistics test functions. (ii) We obtain results for the counting statistics of trapped fermions at the edge of the Fermi gas in both the high and the low temperature limits.

Reach of repulsion for determinantal point processes in high dimensions

Abstract Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.

Some properties of stationary determinantal point processes on Z

We study properties of stationary determinantal point processes $\X$ on $\Z$ from different points of views. It is proved that $\X\cap \N$ is almost surely Bohr-dense and good universal for almost everywhere convergence in $L^1$, and that $\X$ is not syndetic but $\X +\X = \mathbb{Z}$. For the associated centered random field, we obtain a sub-Gaussian property, a Salem-Littlewood inequality and a Khintchine-Kahane inequality. Results can be generalized to $\Z^d$.

Analyzing linear spatial features in ecology.

The spatial analysis of dimensionless points (e.g., tree locations on a plot map) is common in ecology, for instance using point-process statistics to detect and compare patterns. However, the treatment of one-dimensional linear features (fiber processes) is rarely attempted. Here we appropriate the methods of vector sums and dot products, used regularly in fields like astrophysics, to analyze a data set of mapped linear features (logs) measured in 12×1-ha forest plots. For this demonstrative case study, we ask two deceptively simple questions: do trees tend to fall downhill, and if so, does slope gradient matter? Despite noisy data and many potential confounders, we show clearly that topography (slope direction and steepness) of forest plots does matter to treefall. More generally, these results underscore the value of mathematical methods of physics to problems in the spatial analysis of linear features, and the opportunities that interdisciplinary collaboration provides. This work provides scope for a variety of future ecological analyzes of fiber processes in space.

Tail asymptotics of signal-to-interference ratio dis­tribution in spatial cellular network models

TAIL ASYMPTOTICS OF SIGNAL-TO-INTERFERENCE RATI ODISTRIBUTION IN SPATIAL CELLULAR NETWORK MODELSWe consider a spatial stochastic model of wireless cellular networks, where the base stations BSs are deployed according to a simple and stationary point process on Rd, d > 2. In this model, we investigate tail asymptotics of the distribution of signal-to-interference ratio SIR, which is a key quantity in wireless communications. In the case where the pathloss function representing signal attenuation is unbounded at the origin, we derive the exact tail asymptotics of the SIR distribution under an appropriate sufficient condition. While we show that widely-used models based on a Poisson point process and on a determinantal point process meet the sufficient condition, we also give a counterexample violating it. In the case of bounded path-loss functions, we derive a logarithmically asymptotic upper bound on the SIR tail distribution for the Poisson-based and -Ginibrebased models. A logarithmically asymptotic lower bound with the same order as the upper bound is also obtained for the Poisson-based model.

A second-order test to detect spatio-temporal anisotropic effects in point patterns

ABSTRACTSecond-order spatio-temporal orientation methods provide a natural tool for the analysis of anisotropic spatio-temporal point process data. In this paper, we generalize a method based on the spatial point pair orientation distribution function to test for evidence of spatio-temporal anisotropy, by exploring the fact that the space–time orientation function is the distribution function of a uniform random variable on for any second-order intensity reweighted stationary and isotropic spatio-temporal point process. We present a numerical procedure based on this result to test for anisotropic effects, illustrated through a simulation study considering several space–time structures including Poisson and cluster configurations. The resulting testing procedure is applied to analyse the spatio-temporal distribution of earthquakes in Southern California for the period 1984–2004. Our results confirm that our approach is able to detect directional components at distinct spatio-temporal scales.

Geometry and Topology of the Boolean Model on a Stationary Point Processes : A Brief Survey

Geometry and Topology of the Boolean Model on a Stationary Point Processes : A Brief Survey

Asymptotic goodness-of-fit tests for point processes based on scaled empirical K-functions

ABSTRACTWe study sequences of scaled edge-corrected empirical (generalized) K-functions (modifying Ripley's K-function) each of them constructed from a single observation of a d-dimensional fourth-order stationary point process in a sampling window which grows together with some scaling rate unboundedly as . Under some natural assumptions it is shown that the normalized difference between scaled empirical and scaled theoretical K-function converges weakly to a mean zero Gaussian process with simple covariance function. This result suggests discrepancy measures between empirical and theoretical K-function with known limit distribution which allow to perform goodness-of-fit tests for checking a hypothesized point process based only on its intensity and (generalized) K-function. Similar test statistics are derived for testing the hypothesis that two independent point processes in have the same distribution without explicit knowledge of their intensities and K-functions.

Detection of intensity bursts using Hawkes processes: An application to high-frequency financial data.

Given a stationary point process, an intensity burst is defined as a short time period during which the number of counts is larger than the typical count rate. It might signal a local nonstationarity or the presence of an external perturbation to the system. In this paper we propose a procedure for the detection of intensity bursts within the Hawkes process framework. By using a model selection scheme we show that our procedure can be used to detect intensity bursts when both their occurrence time and their total number is unknown. Moreover, the initial time of the burst can be determined with a precision given by the typical interevent time. We apply our methodology to the midprice change in foreign exchange (FX) markets showing that these bursts are frequent and that only a relatively small fraction is associated with news arrival. We show lead-lag relations in intensity burst occurrence across different FX rates and we discuss their relation with price jumps.

Brillinger-mixing point processes need not to be ergodic

Recently, it has been proved that a stationary Brillinger-mixing point process is mixing (of any order) if its moment measures determine the distribution uniquely. In this paper we construct a family of non-ergodic stationary point processes as mixture of two distinct Brillinger-mixing Neyman–Scott processes having the same moment measures.

An Analytical Framework for Modeling a Spatially Repulsive Cellular Network

We propose a new cellular network model that captures both deterministic and random aspects of base station (BS) deployments. Namely, the BS locations are modeled as the superposition of two independent stationary point processes: a random shifted grid with intensity $ \lambda _{g} $ and a Poisson point process (PPP) with intensity $ \lambda _{p} $ . Grid and PPP deployments are special cases with $\lambda _{p} \to 0$ and $\lambda _{g} \to 0$ , with actual deployments in between these two extremes, as we demonstrate with deployment data. Assuming that each user is associated with the BS that provides the strongest average received signal power, we obtain the probability that a typical user is associated with either a grid or PPP BS. Assuming Rayleigh fading channels, we derive the expression for the coverage probability of the typical user, resulting in the following observations. First, the association and the coverage probability of the typical user are fully characterized as functions of intensity ratio $ \rho _\lambda =\lambda _{p}/\lambda _{g}$ . Second, the user association is biased toward the BSs located on a grid. Finally, the proposed model predicts the coverage probability of the actual deployment with great accuracy.

Point-shift foliation of a point process

A point-shift $F$ maps each point of a point process $\Phi $ to some point of $\Phi $. For all translation invariant point-shifts $F$, the $F$-foliation of $\Phi $ is a partition of the support of $\Phi $ which is the discrete analogue of the stable manifold of $F$ on $\Phi $. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts $F$, there exists a point-shift $F_\bot $, the orbits of which are the $F$-foils of $\Phi $, and which is measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.

Modeling of Cellular Networks Using Stationary and Nonstationary Point Processes

The spatial topology of the base stations in wireless networks has a profound impact on their performance evaluation and analysis. It is important to identify a proper and accurate point process model before applying any theoretical stochastic geometry analysis. In this paper, we present and describe a network-data-supported technique for fitting stationary and nonstationary point process models to real-life cellular networks using maximum likelihood/pseudolikelihood and minimum contrast methods. Nonstationary processes are of particular interest, since real-life wireless networks most often do not have a homogeneous spatial distribution. When fitting with nonstationary models, both spatial inhomogeneity and covariate effects are considered. We introduce covariates into the point process models as a potential (or secondary) effect that further influences the distribution of wireless nodes, in order to bridge the gaps between the results and measures of stationary models and simulations of real-life cellular networks. The covariates considered account for population densities in urban areas and distance from the base stations to their closest main roads in rural areas. Simulated envelope tests are used for the evaluation of goodness-of-fit. However, such envelope tests are insufficient to conclusively distinguish between the fitted models. Thus, we apply other metrics such as the Akaike information criterion and root mean square deviation to differentiate among different fitted models. Additionally, the probability of coverage is also considered as the supplementary criterion for the goodness-of-fit and model selection.

Probability approximation of point processes with Papangelou conditional intensity

We give general bounds in the Gaussian and Poisson approximations of innovations (or Skorohod integrals) defined on the space of point processes with Papangelou conditional intensity. We apply the general results to Gibbs point processes with pair potential and determinantal point processes. In particular, we provide explicit error bounds and quantitative limit theorems for stationary, inhibitory and finite range Gibbs point processes with pair potential and $\beta$-Ginibre point processes.

Stochastic earthquake interevent time modeling from exponentiated Weibull distributions

In view of the growing importance of stochastic earthquake modeling in disaster preparation, the present study introduces a new family of exponentiated Weibull distribution and examines its performance in earthquake interevent time analysis in a stationary point process. This three-parameter (one scale and two shapes) distribution not only covers the Weibull distribution, exponentiated exponential distribution, Burr-type X distribution, Rayleigh distribution, and exponential distribution as special sub-families, but also offers monotone and non-monotone hazard shapes. Here we first describe some of the exponentiated Weibull distribution properties, such as the survival rate, mode, median, and hazard rate. We then provide statistical inference and goodness-of-fit measures to examine the suitability of exponentiated Weibull model in comparison with other popular models, like exponential, gamma, lognormal, Weibull, and exponentiated exponential. Finally, we conduct real data analysis to assess the usefulness and flexibility of exponentiated Weibull distribution in the context of seismic interevent time modeling and associated applications. Results suggest that the exponentiated Weibull distribution has a comparable performance with other popular distributions of its nature. However, further investigations are necessary to confirm the importance and flexibility of exponentiated Weibull distribution in statistical seismology.

Crown plasticity in Scots pine (Pinus sylvestris L.) as a strategy of adaptation to competition and environmental factors

Competition for canopy space is a major ecological process in forest dynamics and some tree species are able to shift their crowns away from competition pressure to improve light capture. As a result canopy structure is an expression of the spatial pattern of resource acquisition. A novel combination of spatial analysis, tessellations and computer simulations of crown movements allowed us to disentangle the main processes that govern crown movements in Pinus sylvestris L. forests. By applying spatial point process statistics to three sets of coordinates (stem, crown and tessellation centre points), we analysed and modelled the interaction between individuals along different tree-crown development stages in Valsain forest, Sierra of Guadarrama (Spain). Overall we found that Scots pine trees were able to respond to the presence of neighbours in monospecific stands. The trees involved tended to avoid competition by shifting their crowns towards empty spaces. Mature trees showed more crown displacement than smaller trees and made an effective use of canopy space. The efficient use of space in older stands was indicated by more regular crown patterns compared to those of younger stands, which usually were more clustered. We also found that crown asymmetry was often a combination of some minor abiotic and biotic ecological processes. In our study, wind and slope did not have a big effect on crown displacement, while biotic interactions showed a strong effect on canopy structure causing trees to be regularly distributed and thus using canopy space more efficiently.

Technical Note: The MAPt/Pht/∞ Queueing System and Multiclass [MAPt/Pht/∞]K Queueing Network

In this paper we demonstrate how a key adjustment to known numerically exact methods for evaluating time-dependent moments of the number of entities in the Pht/Pht/∞ queueing system and [Pht/Pht/∞]K queueing network may be implemented to capture the effect of autocorrelation that may be present in arrivals to the more general MAPt/Pht/∞ queueing system and multiclass [MAPt/Pht/∞]K queueing network. The MAPt is more general than the Pht arrival process in that it allows for stationary nonrenewal point processes, as well as the time-dependent generalization of nonrenewal point processes. Modeling real-world systems with bursty arrival processes such as those in telecommunications and transportation, for example, necessitate the use of nonrenewal processes. Finally, we show that the covariance of the number of entities at different nodes and times may be described by a single closed differential equation. The online supplement is available at https://doi.org/10.1287/ijoc.2016.0736 .

General Point Sampling with Adaptive Density and Correlations

Analyzing and generating sampling patterns are fundamental problems for many applications in computer graphics. Ideally, point patterns should conform to the problem at hand with spatially adaptive density and correlations. Although there exist excellent algorithms that can generate point distributions with spatially adaptive density or anisotropy, the pair-wise correlation model, blue noise being the most common, is assumed to be constant throughout the space. Analogously, by relying on possibly modulated pair-wise difference vectors, the analysis methods are designed to study only such spatially constant correlations. In this paper, we present the first techniques to analyze and synthesize point patterns with adaptive density and correlations. This provides a comprehensive framework for understanding and utilizing general point sampling. Starting from fundamental measures from stochastic point processes, we propose an analysis framework for general distributions, and a novel synthesis algorithm that can generate point distributions with spatio-temporally adaptive density and correlations based on a locally stationary point process model. Our techniques also extend to general metric spaces. We illustrate the utility of the new techniques on the analysis and synthesis of real-world distributions, image reconstruction, spatio-temporal stippling, and geometry sampling.