Inhomogeneous Poisson Point Process Research Articles

Inhomogeneous Double Thinning—Modeling and Analysis of Cellular Networks by Using Inhomogeneous Poisson Point Processes

In this paper, we introduce a new methodology for modeling and analyzing downlink cellular networks, where the base stations (BSs) constitute a motion-invariant point process (PP) that exhibits some degree of interactions among the points, i.e., spatial repulsion or spatial clustering. The proposed approach is based on the theory of inhomogeneous Poisson PPs (I-PPPs) and is referred to as inhomogeneous double thinning (IDT) approach. In a PP, the distribution of the distance from a randomly distributed (typical) user to its nearest BS depends on the degree of spatial repulsion or clustering exhibited by the PP. In addition, the average number of interfering BSs that lies within a given distance from the typical user is a function of the repulsion and clustering characteristics of the PP. The proposed approach consists of approximating the original motion-invariant PP with an equivalent PP that is made of the superposition of two conditionally independent I-PPPs. The inhomogeneities of both PPs are created from the point of view of the typical user (“user-centric”): the first one is based on the distribution of the user’s distance to its nearest BS and the second one is based on the distance-dependent average number of interfering BSs around the user. The inhomogeneities are mathematically modeled through two distance-dependent thinning functions and a tractable expression of the coverage probability is obtained. Sufficient conditions on the parameters of the thinning functions that guarantee better or worse coverage compared with the baseline homogeneous PPP model are identified. The accuracy of the IDT approach is substantiated with the aid of empirical data for the spatial distribution of the BSs.

A quadrat neighborhood estimator for intensity function of point processes

In nonparametric estimation of the intensity function of a point process, assigning the local event weight is particularly important. This paper describes a sequential quadrat partitioning of the study region to define a quadrat neighborhood of a point. Based on this idea, a quadrat neighborhood estimator of intensity function is introduced. We extend this method to estimate the product density. Meanwhile, we show that under infill asymptotics our proposed estimator is asymptotically unbiased for inhomogeneous Poisson point process. Simulations are also used to investigate the performance of our proposed estimator.

Sppmix: Poisson point process modeling using normal mixture models

This paper describes the package sppmix for the statistical environment R. The sppmix package implements classes and methods for modeling spatial point patterns using inhomogeneous Poisson point processes, where the intensity surface is assumed to be a multiple of a finite additive mixture of normal components and the number of components is a finite, fixed or random integer. Extensions to the marked inhomogeneous Poisson point processes case are also presented. We provide an extensive suite of R functions that can be used to simulate, visualize and model point patterns, estimate the parameters of the models, assess convergence of the algorithms and perform model selection and checking in the proposed modeling context. In addition, several approaches have been implemented in order to handle the standard label switching issue which arises in any modeling approach involving mixture models. We adapt a hierarchical Bayesian framework in order to model the intensity surfaces and have implemented two major algorithms in order to estimate the parameters of the mixture models involved: the data augmentation and the birth–death Markov chain Monte Carlo (DAMCMC and BDMCMC). We used C++ (via the Rcpp package) in order to implement the most computationally intensive algorithms.

Coverage Analysis of Millimeter Wave Decode-and-Forward Networks With Best Relay Selection

In this paper, we investigate the coverage probability improvement of a millimeter wave network due to the deployment of spatially random decode-and-forward (DF) relays. The source and receiver are located at a fixed distance and all the relay nodes are distributed as a 2-D homogeneous Poisson point process (PPP). We first derive the spatial distribution of the set of decoding relays whose received signal-to-noise ratio (SNR) are above the minimum SNR threshold. This set is a 2-D inhomogeneous PPP. From this set, we select a relay that has minimum path loss to the receiver and derive the achievable coverage due to this selection. The analysis is developed using tools from stochastic geometry and is verified using Monte-Carlo simulation. The coverage probabilities of the direct link without relaying, a randomly chosen relay link, and the selected relay link are compared to show the significant performance gain when relay selection is used. We also analyze the effects of beam misalignment and different power allocations at the source and relay on coverage probability. In addition, rate coverage and spectral efficiency are compared for direct and selected relay links to show impressive performance gains with relaying.

Exclusive Region Design for Spatial Grid-Based Spectrum Database: A Stochastic Geometry Approach

This paper presents the stochastic-geometry analysis and designs a primary exclusive region (PER) for a spatial grid-based spectrum database system. The purpose of the spatial grid is to utilize information, such as the primary receiver (PR) antenna pattern, and the secondary transmitter (ST) density and transmission power, in each divided region. This paper introduces polar and square grids. For these spatial grids, the cumulants of the aggregate interference at a PR, from the STs, are derived, where the probability generating functional for the Poisson point process (PPP) is used on the assumption that the distribution of the STs in each divided region follows an inhomogeneous PPP. In addition, by introducing the allowable transmission probability of STs in each divided region, the PER optimization problem can be formulated as a continuous optimization problem. Numerical results demonstrate that a PER, corresponding to the information, is successfully designed and that more detailed information, due to a reduction in the area of each divided region, leads to a smaller and complex-shaped PER.

Periodicity of Strong Seismicity in Italy: Schuster Spectrum Analysis Extended to the Destructive Earthquakes of 2016

The strong earthquakes that occurred in Italy between 2009 and 2016 represent an abrupt acceleration of seismicity in respect of the previous 30 years. Such behavior seems to agree with the periodic rate change I observed in a previous paper. The present work improves that study by extending the data set up to the end of 2016, adopting the latest version of the historical seismic catalog of Italy, and introducing Schuster spectrum analysis for the detection of the oscillatory period and the assessment of its statistical significance. Applied to the declustered catalog of Mw ≥ 6 earthquakes that occurred between 1600 and 2016, the analysis individuates a marked periodicity of 46 years, which is recognized above the 95% confidence level. Monte Carlo simulation shows that the oscillatory behavior is stable in respect of random errors on magnitude estimation. A parametric oscillatory model for the annual rate of seismicity is estimated by likelihood maximization under the hypothesis of inhomogeneous Poisson point process. According to the Akaike Information Criterion, such model outperforms the simpler homogeneous one with constant annual rate. A further element emerges form the analysis: so far, despite recent earthquakes, the Italian seismicity is still within a long-term decreasing trend established since the first half of the twentieth century.

Integrated species distribution models: combining presence‐background data and site‐occupancy data with imperfect detection

Summary Two main sources of data for species distribution models (SDMs) are site‐occupancy (SO) data from planned surveys, and presence‐background (PB) data from opportunistic surveys and other sources. SO surveys give high quality data about presences and absences of the species in a particular area. However, due to their high cost, they often cover a smaller area relative to PB data, and are usually not representative of the geographic range of a species. In contrast, PB data is plentiful, covers a larger area, but is less reliable due to the lack of information on species absences, and is usually characterised by biased sampling. Here we present a new approach for species distribution modelling that integrates these two data types. We have used an inhomogeneous Poisson point process as the basis for constructing an integrated SDM that fits both PB and SO data simultaneously. It is the first implementation of an Integrated SO–PB Model which uses repeated survey occupancy data and also incorporates detection probability. The Integrated Model's performance was evaluated, using simulated data and compared to approaches using PB or SO data alone. It was found to be superior, improving the predictions of species spatial distributions, even when SO data is sparse and collected in a limited area. The Integrated Model was also found effective when environmental covariates were significantly correlated. Our method was demonstrated with real SO and PB data for the Yellow‐bellied glider (Petaurus australis) in south‐eastern Australia, with the predictive performance of the Integrated Model again found to be superior. PB models are known to produce biased estimates of species occupancy or abundance. The small sample size of SO datasets often results in poor out‐of‐sample predictions. Integrated models combine data from these two sources, providing superior predictions of species abundance compared to using either data source alone. Unlike conventional SDMs which have restrictive scale‐dependence in their predictions, our Integrated Model is based on a point process model and has no such scale‐dependency. It may be used for predictions of abundance at any spatial‐scale while still maintaining the underlying relationship between abundance and area.

Functional central limit theorem for subgraph counting processes

The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.

Stronger Wireless Signals Appear More Poisson

Keeler, Ross and Xia (2016) recently derived approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modelled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with different locations and random propagation effects.The aim of this note is to apply some of the main results of Keeler, Ross and Xia (2016) in a less general but more easily applicable form to illustrate how the results can be applied in practice. New results are derived that show that it is the strongest signals, after being weakened by random propagation effects, that behave like a Poisson process, which supports recent experimental work.

The greedy walk on an inhomogeneous Poisson process

The greedy walk is a deterministic walk that always moves from its current position to the nearest not yet visited point. In this paper we consider the greedy walk on an inhomogeneous Poisson point process on the real line. We prove that the property of visiting all points of the point process satisfies a $0$–$1$ law and determine explicit sufficient and necessary conditions on the mean measure of the point process for this to happen. Moreover, we provide precise results on threshold functions for the property of visiting all points.

Resource selection by recolonizing American black bears in a fragmented forest landscape

Abstract Following functional extirpation in Missouri, American black bear (Ursus americanus) populations in this state have been increasing in recent years through recolonization from re-established populations in northern Arkansas. To increase our understanding of resource selection by recolonizing black bears in the Ozark Highlands of the United States, we attached Global Positioning System (GPS) transmitters to 54 black bears during May–August 2010–2013, and used location data based on biological seasons. We constructed models with anthropogenic (distance to nearest development, distance to nearest road), biological (sex, age class, season), and environmental (distance to nearest water, land cover) categories. We used infinitely weighted logistic regression to approximate the inhomogeneous Poisson point process model for presence-only (i.e., GPS locations) data to fit models. We used Bayesian Information Criterion and found that the best-performing model in the set of 81 models included all independen...

Wireless Networks Appear Poissonian Due to Strong Shadowing

Geographic locations of cellular base stations sometimes can be well fitted with spatial homogeneous Poisson point processes. In this paper, we make a complementary observation. In the presence of the log-normal shadowing of sufficiently high variance, the statistics of the propagation loss of a single user with respect to different network stations are invariant with respect to their geographic positioning, whether regular or not, for a wide class of empirically homogeneous networks. Even in a perfectly hexagonal case they appear as though they were realized in a Poisson network model, i.e., form an inhomogeneous Poisson point process on the positive half-line with a power-law density characterized by the path-loss exponent. At the same time, the conditional distances to the corresponding base stations, given their observed propagation losses, become independent and log-normally distributed, which can be seen as a decoupling between the real and model geometry. The result applies also to the Suzuki (Rayleigh-log-normal) propagation model. We use the Kolmogorov-Smirnov test to empirically study the quality of the Poisson approximation and use it to build a linear-regression method for the statistical estimation of the value of the path-loss exponent.

A positive response of mountain pine beetle to pine forest-clearcut edges at the landscape scale in British Columbia, Canada

In British Columbia, large-scale salvage harvesting has been underway to recover timber value from forest stands infested by mountain pine beetle during the current outbreak. Understanding the response of beetles to clearcut edges particularly at the landscape scale is crucial to understanding the impacts of increased habitat fragmentation due to salvage harvesting on the spread of the beetle infestations. A novel proximity analysis approach based on null models of complete spatial randomness with three different spatial extents was developed to examine the spatial patterns of infestations in relation to cutblocks. Inhomogeneous Poisson point process models were fitted to predict how intensities of infestations varied with distances to the nearest cutblocks. Marked Poisson point process models were also fitted to evaluate the effects of the variables associated with the nearest cutblocks and adjacent infested pine stands on the edge response of beetles. The results clearly illustrated a significant positive edge response of beetles at the landscape scale. The intensities of infestations decreased non-linearly with distances to the nearest cutblocks. The results also suggested that the quality and distribution of key habitat resources could not fully explain the fundamental mechanisms underlying the edge response. The behavioural change of beetle dispersal at edges may also be an important factor contributing to a positive edge response. The results from this study may be useful in improving the efficacy of mountain pine beetle management efforts.

Kernel Intensity Estimation of 2-Dimensional Spatial Poisson Point Processes From k-Tree Sampling

To estimate the spatial intensity (density) of plants and animals, ecologists often sample populations by prespecifing a spatial array of points, then measuring the distance from each point to the k nearest organisms, a so-called k-tree sampling method. A variety of ad hoc methods are available for estimating intensity from k-tree sampling data, but they assume that two distinct points of the array do not share nearest neighbors. However, nearest neighbors are likely to be shared when the population intensity is low, as it is in our application. The purpose of this paper is twofold: (a) to derive and use for estimation the likelihood function for a k-tree sample under an inhomogeneous Poisson point-process model and (b) to estimate spatial intensity when nearest neighbors are shared. We derive the likelihood function for an inhomogeneous Poisson point-process with intensity λ(x,y) and propose a likelihood-based, kernel-smoothed estimator $\hat{\lambda}(x,y)$ . Performance of the method for k=1 is tested on four types of simulated populations: two homogeneous populations with low and high intensity, a population simulated from a bivariate normal distribution of intensity, and a “cliff” population in which the region is divided into high- and low-intensity subregions. The method correctly detected spatial variation in intensity across different subregions of the simulated populations. Application to 1-tree samples of carnivorous pitcher plants populations in four New England peat bogs suggests that the method adequately captures empirical patterns of spatial intensity. However, our method suffers from two evident sources of bias. First, like other kernel smoothers, it underestimates peaks and overestimates valleys. Second, it has positive bias analogous to that of the MLE for the rate parameter of exponential random variables.

Identification and estimation of superposed Neyman–Scott spatial cluster processes

This paper proposes an estimation method for superposed spatial point patterns of Neyman–Scott cluster processes of different distance scales and cluster sizes. Unlike the ordinary single Neyman–Scott model, the superposed process of Neyman–Scott models is not identified solely by the second-order moment property of the process. To solve the identification problem, we use the nearest neighbor distance property in addition to the second-order moment property. In the present procedure, we combine an inhomogeneous Poisson likelihood based on the Palm intensity with another likelihood function based on the nearest neighbor property. The derivative of the nearest neighbor distance function is regarded as the intensity function of the rotation invariant inhomogeneous Poisson point process. The present estimation procedure is applied to two sets of ecological location data.

Nondetection sampling bias in marked presence‐only data

Species distribution models (SDM) are tools used to determine environmental features that influence the geographic distribution of species' abundance and have been used to analyze presence-only records. Analysis of presence-only records may require correction for nondetection sampling bias to yield reliable conclusions. In addition, individuals of some species of animals may be highly aggregated and standard SDMs ignore environmental features that may influence aggregation behavior.We contend that nondetection sampling bias can be treated as missing data. Statistical theory and corrective methods are well developed for missing data, but have been ignored in the literature on SDMs. We developed a marked inhomogeneous Poisson point process model that accounted for nondetection and aggregation behavior in animals and tested our methods on simulated data.Correcting for nondetection sampling bias requires estimates of the probability of detection which must be obtained from auxiliary data, as presence-only data do not contain information about the detection mechanism. Weighted likelihood methods can be used to correct for nondetection if estimates of the probability of detection are available. We used an inhomogeneous Poisson point process model to model group abundance, a zero-truncated generalized linear model to model group size, and combined these two models to describe the distribution of abundance. Our methods performed well on simulated data when nondetection was accounted for and poorly when detection was ignored.We recommend researchers consider the effects of nondetection sampling bias when modeling species distributions using presence-only data. If information about the detection process is available, we recommend researchers explore the effects of nondetection and, when warranted, correct the bias using our methods. We developed our methods to analyze opportunistic presence-only records of whooping cranes (Grus americana), but expect that our methods will be useful to ecologists analyzing opportunistic presence-only records of other species of animals.

Firing-rate models for neurons with a broad repertoire of spiking behaviors

Instantaneous firing rates are commonly used to describe either the compound spiking activity of neuron ensembles (population rate) or the trial-averaged response of individual neurons to multiple repetitions of the same stimulus. The dynamics of firing rates is often studied by means of population or firing-rate models. The main motivation for using such models rather than spiking neuron models is to reduce the dimensionality and complexity of the microscopic dynamics to allow analytical tractability, efficient simulation, and intuitive understanding. In most cases, population models have to be treated as purely phenomenological descriptions. Only under simplifying assumptions can they be derived or extracted from the single-neuron dynamics. Previous studies in this field were mainly restricted to homogeneous populations of simple integrate-and-fire neurons which can, to some extent, mimic the rather stereotypical behaviour of regularly firing neurons (e.g., pyramidal cells in the neocortex). Electrophysiological studies have revealed a large diversity of neuron types with different dynamical behaviours, ranging from regular spiking to bursting cells, from accommodating to non-accommodating cells, from integrators to resonators, from depolarization- to hyperpolarization-activated cells, etc. So far, the effect of this neuron-type diversity on the dynamics of neural populations is unclear. Here, we investigate the rate-response characteristics for a variety of neuron types in simulation experiments. To this end, we extend previous work on simple integrate-and-fire neurons [1,2] to the multi-timescale adaptive threshold (MAT) [3] and the Izhikevich model [4]. The model neurons are probed by spike trains modeled as realizations of inhomogeneous Poisson or Gamma point processes with sinusoidally modulated intensity. Average rates, modulation amplitudes and phases of the period-averaged spike responses are measured for a broad range of stimulus, neuron and synapse parameters. The stationary and nonstationary response properties are parameterized by means of nonlinear activation functions and linear filter kernels, respectively, serving as ingredients for simple linear-nonlinear firing rate models. We show that linear-nonlinear rate models can accurately predict the population responses to novel test stimuli for a broad range of parameters and neuron types.

Distances Between Poisson k -Flats

The distances between flats of a Poisson k-flat process in the d-dimensional Euclidean space with k < d/2 are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a given threshold and midpoint in a fixed compact and convex set is considered. For a family of increasing convex subsets, the asymptotic variance is computed and a central limit theorem with an explicit rate of convergence is proven. Moreover, the asymptotic distribution of the m-th smallest distance between two flats is investigated and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous Poisson point process on the positive real half-axis. A similar result with a homogeneous limiting process is derived for distances around a fixed, strictly positive value. Our proofs rely on recent findings based on the Wiener–Ito chaos decomposition and the Malliavin–Stein method.

Determining High-Risk Zones for Unexploded World War II Bombs by Using Point Process Methodology

SummaryTo prevent accidents caused by unexploded bombs from the Second World War, high-risk zones need to be determined. We introduce two statistical methods to determine such zones by considering patterns of exploded and unexploded bombs as realizations of inhomogeneous spatial Poisson point processes. The first method is based on the intensity of the point process; the second method on its nearest neighbour distance. Although the performance of the two methods is similar, the intensity-based method has the advantage of a more direct specification of risk and should hence be preferred. Risk that is associated with a zone is defined as the probability that not all unexploded bombs are in the zone. We propose and develop a procedure to calculate this probability. This procedure is then also used to investigate consequences of additional spatial clustering, in which case the intensity-based method is shown to be conservative. Our work is motivated and illustrated by World War II explosive ordnance disposal by German authorities on the basis of aerial pictures taken by the allies.

Analytical Expressions for Formal Kinetics

In recent papers Rios and Villa resorted to developments in stochastic geometry to revisit theclassical KJMA theory and generalize it for situations in which nuclei were located in space accordingto both homogeneous and inhomogeneous Poisson point processes as well as according to Materncluster process and surface and bulk nucleation in small specimens. Rigorous mathematical methodswere employed to ensure the reliability of the new expressions. These results are briefly described.Analytical expression for inhomogeneous Poisson point process nucleation gives very good agreementwith Cellular Automata simulations. Cellular Automata simulations complement the analyticalsolutions by showing the corresponding microstructural evolution. These new results considerablyexpand the range of situations for which analytical solutions are available.