Stationary Point Processes Research Articles

Colony size affects breeding density, but not spatial distribution type, in a seabird

Abstract The spatial distribution of individuals within populations can result in fine-scale density dependence and affect the social environment that is encountered. As such, it is important to quantify within-population spatial structuring and understand the factors that shape it. In this study, we make use of point process statistics to test whether colony size affects the statistical type of spatial nest distribution produced by common terns (Sterna hirundo) breeding at identical manmade rectangular and homogeneous islands of fixed physical size. Comparing subcolonies of variable density both within and across years, we find that inter-nest distances are smaller at higher local and overall breeding density, but that the spatial distribution type does not vary across the observed densities. This suggests that the birds’ main settlement rules do not depend on density. In our case, analyses of fine-scale density dependence or potential social effects therefore do not need to account for between-individual heterogeneity in settlement decision rules or acceptance of these rules. We urge, however, other studies to similarly test for density dependence of the spatial distribution of individuals before undertaking such “downstream” analyses.

Hyperuniform and rigid stable matchings

We study a stable partial matching τ of the d‐dimensional lattice with a stationary determinantal point process Ψ on Rd with intensity α>1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψτ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψτ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process Ψ, whose so‐called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behavior. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Furthermore, if Ψ is a Poisson process then Ψτ has an exponentially decreasing truncated pair correlation function.

Inhomogeneous higher-order summary statistics for point processes on linear networks

As a workaround for the lack of transitive transformations on linear network structures, which are required to consider different notions of distributional invariance, including stationarity, we introduce the notions of pseudostationarity and intensity reweighted moment pseudostationarity for point processes on linear networks. Moreover, using arbitrary so-called regular linear network distances, e.g. the Euclidean and the shortest-path distance, we further propose geometrically corrected versions of different higher-order summary statistics, including the inhomogeneous empty space function, the inhomogeneous nearest neighbour distance distribution function and the inhomogeneous J-function. Such summary statistics detect interactions of order higher than two. We also discuss their nonparametric estimators and through a simulation study, considering models with different types of spatial interaction and different networks, we study the performance of our proposed summary statistics by means of envelopes. Our summary statistic estimators manage to capture clustering, regularity as well as Poisson process independence. Finally, we make use of our new summary statistics to analyse two different datasets: motor vehicle traffic accidents and spiderwebs.

Large deviations for the Kardar– Parisi–Zhang equation from the Kadomtsev–Petviashvili equation

Recently, Quastel and Remenik (2019 (arXiv:1908.10353)) found a remarkable relation between some solutions of the finite time Kardar–Parisi–Zhang (KPZ) equation and the Kadomtsev–Petviashvili (KP) equation. Using this relation we obtain the large deviations at large time and at short time for the KPZ equation with droplet initial conditions, and at a short time with half-Brownian initial conditions. It is consistent with previous results and allows us to obtain sub-leading corrections, as well as results at intermediate time. In addition, we find that the appropriate generating function associated to the full Brownian initial condition also satisfies the KP equation. Finally, generating functions for some linear statistics of the Airy point process are also found to satisfy the KP property, and consequences are discussed.

Exponential Moments for Planar Tessellations

In this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical tessellations such as the Poisson–Voronoi, Poisson–Delaunay and Poisson line tessellation, we also treat the Johnson–Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

Distances and large deviations in the spatial preferential attachment model

This paper considers two asymptotic properties of a spatial preferential-attachment model introduced by E. Jacob and P. Mörters (In Algorithms and Models for the Web Graph (2013) 14–25 Springer). First, in a regime of strong linear reinforcement, we show that typical distances are at most of doubly-logarithmic order. Second, we derive a large deviation principle for the empirical neighbourhood structure and express the rate function as solution to an entropy minimisation problem in the space of stationary marked point processes.

Analysis of Extreme Peak Loads Using Point Processes: An Application Using South African Data

Extreme value modelling of peak load process is critical to the reliable specification of power generation, distribution and maintenance purposes during both peak and off-peak periods. In this study, a frequency assessment of extreme peak electricity demand for the four seasons of the year using South African data for the period, January 1997 to December 2013 is carried out. A point process approach from extreme value theory is proposed as an ingenious extreme value theory approach. The data are made stationary by using a time-varying threshold which has a positive shift factor. The non-linear detrended datasets are then grouped into summer, spring, winter and autumn according to the calendar dates in the Southern Hemisphere. The datasets were declustered to keep the series relatively independent using Ferro and Segers automatic declustering method. A stationary point process model is then fitted to each of the cluster maxima. The modelling framework, which is easily extensible to other peak load parameters, assumes that peak power follows a Poisson point process. The parameters of the developed model are estimated using the maximum likelihood method. Empirical results show that daily peak electricity demand could be experienced approximately 27, 16, 7 and 15 days per year in winter, spring, summer and autumn, respectively. The modelling approach could assist system operators of utility companies in scheduling maintenance of generating units including long term planning.

Using non-homogeneous point process statistics to find multi-species event clusters in an implanted semiconductor

The Poisson distribution of event-to--nearest-event radial distances is well known for homogeneous processes that do not depend on location or time. Here we investigate the case of a non-homogeneous point process where the event probability (and hence the neighbour configuration) depends on location within the event space. The particular non-homogeneous scenario of interest to us is ion implantation into a semiconductor for the purposes of studying interactions between the implanted impurities. We calculate the probability of a simple cluster based on nearest neighbour distances, and specialise to a particular two-species cluster of interest for qubit gates. We show that if the two species are implanted at different depths there is a maximum in the cluster probability and an optimum density profile.

Estimating the Parameters of Fitzhugh-Nagumo Neurons from Neural Spiking Data.

A theoretical and computational study on the estimation of the parameters of a single Fitzhugh–Nagumo model is presented. The difference of this work from a conventional system identification is that the measured data only consist of discrete and noisy neural spiking (spike times) data, which contain no amplitude information. The goal can be achieved by applying a maximum likelihood estimation approach where the likelihood function is derived from point process statistics. The firing rate of the neuron was assumed as a nonlinear map (logistic sigmoid) relating it to the membrane potential variable. The stimulus data were generated by a phased cosine Fourier series having fixed amplitude and frequency but a randomly shot phase (shot at each repeated trial). Various values of amplitude, stimulus component size, and sample size were applied to examine the effect of stimulus to the identification process. Results are presented in tabular and graphical forms, which also include statistical analysis (mean and standard deviation of the estimates). We also tested our model using realistic data from a previous research (H1 neurons of blowflies) and found that the estimates have a tendency to converge.

Downlink Analysis for the Typical Cell in Poisson Cellular Networks

Owing to its unparalleled tractability, the Poisson point process (PPP) has emerged as a popular model for the analysis of cellular networks. Considering a stationary point process of users, which is independent of the base station (BS) point process, it is well known that the typical user does not lie in the typical cell and thus may not truly represent the typical cell performance. Inspired by this observation, we present a construction that allows a direct characterization of the downlink performance of the typical cell. For this, we present an exact downlink analysis for the 1-D case and a remarkably accurate approximation for the 2-D case. Several useful insights about the differences and similarities in the two viewpoints (typical user vs. typical cell) are also provided.

Some properties of local weighted second-order statistics for spatio-temporal point processes

Diagnostics of goodness-of-fit in the theory of point processes are often considered through the transformation of data into residuals as a result of a thinning or a rescaling procedure. We alternatively consider here second-order statistics coming from weighted measures. Motivated by Adelfio and Schoenberg (Ann Inst Stat Math 61(4):929–948, 2009) for the temporal and spatial cases, we consider an extension to the spatio-temporal context in addition to focussing on local characteristics. In particular, our proposed method assesses goodness-of-fit of spatio-temporal models by using local weighted second-order statistics, computed after weighting the contribution of each observed point by the inverse of the conditional intensity function that identifies the process. Weighted second-order statistics directly apply to data without assuming homogeneity nor transforming the data into residuals, eliminating thus the sampling variability due to the use of a transforming procedure. We provide some characterisations and show a number of simulation studies.

Local Entropy Statistics for Point Processes

Point processes are often described with functionals, such as the probability generating functional, the Laplace functional, and the factorial cumulant generating functional. These are used to facilitate modelling of different processes and to determine important statistics via functional differentiation. In information theory, generating functions have also been defined for probability densities to determine information quantities such as the Shannon information and Kullback-Leibler divergence, though as yet there are no such analogues for point processes. The purpose of this article is to exploit the advantages of both types of generating function to facilitate the derivation of information statistics for point processes. In particular, a generating functional for point processes is introduced for determining statistics related to entropy and relative entropy based on Golomb’s information function and Moyal’s probability generating functional. It is shown that the information generating functional permits the derivation of a suite of statistics, including localised Shannon entropy and Kullback-Leibler divergence calculations.

A Location-Dependent Base Station Cooperation Scheme for Cellular Networks

The link quality in cellular networks strongly depends on the location of the users relative to the serving and interfering base stations (BSs). This paper proposes a location-dependent BS cooperation scheme for general cellular networks, where BSs are modeled using a stationary point process and the Voronoi diagram forms the cell structure. The cooperation scheme is based on the relative average received signal strength from the three strongest BSs. For the channel model where Rayleigh fading and power-law path loss are considered, each cell is partitioned into three regions based on the relative distance to the three nearest BSs: the cell center region, cell edge region, and cell corner region. The area fraction of each region is tuned by the so-called cooperation level $\gamma \in [ {0},{1}]$ . We study the scheme where users in the above regions receive the non-coherent joint transmission from one, two, and three nearest BSs, respectively. As such, the scheme primarily helps users vulnerable to interference. We analyze the signal-to-interference ratio (SIR) in Poisson networks and show that a moderate $\gamma $ jointly improves the average SIR performance and the network fairness.

Simple Approximations of the SIR Meta Distribution in General Cellular Networks

Compared to the standard success (coverage) probability, the meta distribution of the signal-to-interference ratio (SIR) provides much more fine-grained information about the network performance. We consider general heterogeneous cellular networks (HCNs) with base station tiers modeled by arbitrary stationary and ergodic non-Poisson point processes. The exact analysis of non-Poisson network models is notoriously difficult, even in terms of the standard success probability, let alone the meta distribution. Hence, we propose a simple approach to approximate the SIR meta distribution for non-Poisson networks based on the ASAPPP (“approximate SIR analysis based on the Poisson point process”) method. We prove that the asymptotic horizontal gap $G_{0}$ between its standard success probability and that for the Poisson point process exactly characterizes the gap between the $b$ th moment of the conditional success probability, as the SIR threshold goes to 0. The gap $G_{0}$ allows two simple approximations of the meta distribution for general HCNs: 1) the per-tier approximation by applying the shift $G_{0}$ to each tier and 2) the effective gain approximation by directly shifting the meta distribution for the homogeneous independent Poisson network. Given the generality of the model considered and the fine-grained nature of the meta distribution, these approximations work surprisingly well.

Renewal processes, population dynamics, and unimodular trees

AbstractBased on a simple object, an i.i.d. sequence of positive integer-valued random variables {an}n∊ℤ, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time n and dies at time n + an. This dynamics is that of a D/GI/∞ queue, with arrivals at integer times and service times given by {an}n∊ℤ. Second, the directed random graph Tf on ℤ generated by the random map f(n) = n + an. Assuming only that E [a0] < ∞ and P [a0 = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regeneration epoch. We identify a unimodular structure in this dynamics. More precisely, Tf is a unimodular directed tree, in which f(n) is the parent of n. This tree has a unique bi-infinite path. Moreover, Tf splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regeneration epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regeneration integers form stationary and mixing point processes on ℤ.

The Fractal Features of the Background Activity of Neurons in the Auditory Center of the Frog Midbrain

Abstract—Statistics were obtained for background firing activity recorded from several neurons of the auditory center of the midbrain (semicircular torus) in the common frog. The methods were those used to analyze the temporal flow of fractal point processes. Most neurons showed a power-law increase in the variance-to-mean ratio (Fano factor) of pulse number with the time window increasing from hundreds of milliseconds to several tens of seconds, although cells whose background firing activity was close to random were also observed. A method based on the registration of local features (the Allan factor as a function of the same time window) confirmed the fractal properties for many cells. A power-law growth of the Allan factor usually began at large time windows. The third method of fractal analysis (the Hurst index) also argued in favor of the randomness and fractality of background firing activity. The relationships with other statistics of random point processes were analyzed for the fractal characteristics. A comparison with data obtained for neurons of the medulla oblongata indicated that the variability of firing activity slight increases along the auditory pathway.

Saltcedar (Tamarix mascatensis) inhibits growth and spatial distribution of eshnan (Seidlitzia rosmarinus) by enrichment of soil salinity in a semi-arid desert

BACKGROUND AND AIMS: Understanding the species associations in dryland sparse and patchy vegetation requires information on species intra- and interspecific interactions and their response to environmental heterogeneity. To better understand the coexistence mechanisms of saltcedar (Tamarix mascatensis) and eshnan (Seidlitzia rosmarinus) in central deserts of Iran, we analysed the spatial distributions and soil chemical properties of their mixed stands. METHODS: We applied uni- and bivariate point process statistics together with mark correlation functions and soil information under the species canopy and in the interspaces to evaluate plant-to-plant interactions. RESULTS: Tamarix and Seidlitzia were both highly aggregated in conspecific clusters, and the species were significantly segregated at small spatial scales. Interestingly, we found that the density of conspecific neighbours in Tamarix clusters was higher than Seidlitzia (approximately 9 times more), although the mean density of Tamarix (29.4 shrubs ha⁻¹) was lower than Seidlitzia (357.3 shrubs ha⁻¹). For both species, intraspecific interactions had negative effects on their size, which was more evident in Tamarix. Our results showed that the size of Seidlitzia was strongly and negatively depended on the proximity of nearby Tamarix. Additionally, the two species improved soil chemical properties and moisture under their canopy compared to bare soil; however, soil salinity under Tamarix (24.29 dS m⁻¹) was higher than Seidlitzia (1.68 dS m⁻¹). CONCLUSIONS: According to the segregated spatial pattern of the species and negative effects of Tamarix neighbourhood on Seidlitzia growth, we suggest that high soil salinity beneath Tamarix canopy likely reduces interspecific interactions between the two species and affects Seidlitzia growth. Furthermore, our results indicate no spatial mixture of Tamarix and Seidlitzia, contrary to our expectations, although we still encourage further research on the co-occurrence of the species in the region.

Marked point processes in discrete time

We develop a general framework for stationary marked point processes in discrete time. We start with a careful analysis of the sample paths. Our initial representation is a sequence $$\{(t_j,k_j): j\in {\mathbb {Z}}\}$$ of times $$t_j\in {\mathbb {Z}}$$ and marks $$k_j\in {\mathbb {K}}$$ , with batch arrivals (i.e., $$t_j=t_{j+1}$$ ) allowed. We also define alternative interarrival time and sequence representations and show that the three different representations are topologically equivalent. Then, we develop discrete analogs of the familiar stationary stochastic constructs in continuous time: time-stationary and point-stationary random marked point processes, Palm distributions, inversion formulas and Campbell’s theorem with an application to the derivation of a periodic-stationary Little’s law. Along the way, we provide examples to illustrate interesting features of the discrete-time theory.

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

AbstractWe establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.

Mark-mark scatterplots improve pattern analysis in spatial plant ecology

Point process statistics provides valuable tools for many ecological studies, where ‘points’ are commonly determined to represent the locations of plants or animals and ‘marks’ are additional items such as species or size. In the statistical analysis of marked point patterns, various correlation functions are used such as the mark variogram or the mark correlation function. Often the interpretation of these functions is not easy and the non-spatial ecologist is in need of support. In order to make the analysis of spatial point patterns more accessible to ecologists, we introduced and tested a new graphical method, the mark-mark scatterplot. This plot visualises the marks of point pairs of inter-point distances r smaller than some small distance rmax. We tested the application of the mark-mark scatterplot by reconsidering three quite different tree patterns: a pattern of longleaf pine trees from the southern US which was strongly influenced by fires, a tropical tree pattern of the species Shorea congestiflora from Sri Lanka and a Scots pine pattern from Siberia (Russia). The new method yielded previously undetected cause-effect information on mark behaviour at short inter-point distances and thus improved the analysis with mark correlation functions as well as complemented the information they provided. We discovered important new correlations in clusters of trees at close proximity. The application of the mark-mark scatterplot will facilitate the interpretation of point process summary statistics and will make point process analysis more accessible to ecologists not specialized in point process statistics.