Cluster Point Process Research Articles

Spatial-temporal rainfall models based on poisson cluster processes

We fit stochastic spatial-temporal models to high-resolution rainfall radar data using Approximate Bayesian Computation (ABC). We consider models constructed from cluster point-processes, starting with the model of Cox, Isham and Northrop, which is the current state of the art. We then generalise this model to allow for more realistic rainfall intensity gradients and for a richer covariance structure that can capture negative correlation between the intensity and size of localised rain cells. The use of ABC is of central importance, as it is not possible to fit models of this complexity using previous approaches. We also introduce the use of Simulated Method of Moments (SMM) to initialise the ABC fit.

Information criteria for inhomogeneous spatial point processes

SummaryThe theoretical foundation for a number of model selection criteria is established in the context of inhomogeneous point processes and under various asymptotic settings: infill, increasing domain and combinations of these. For inhomogeneous Poisson processes we consider Akaike's information criterion and the Bayesian information criterion, and in particular we identify the point process analogue of ‘sample size’ needed for the Bayesian information criterion. Considering general inhomogeneous point processes we derive new composite likelihood and composite Bayesian information criteria for selecting a regression model for the intensity function. The proposed model selection criteria are evaluated using simulations of Poisson processes and cluster point processes.

NScluster: An R Package for Maximum Palm Likelihood Estimation for Cluster Point Process Models Using OpenMP

NScluster is an R package used for simulation and parameter estimation for NeymanScott cluster point process models and their extensions. For parameter estimation, NScluster uses the maximum Palm likelihood estimation procedure. As some estimation procedures proposed herein require heavy calculation, NScluster can use parallel computation via OpenMP and achieve significant speedup in some cases. In this paper, we discuss results obtained using a laptop PC and a shared memory supercomputer. In addition, we examine the performance characteristics of parallel computation via OpenMP.

Factors affecting the spatial density of longleaf pine plantations under the Conservation Reserve Program in Georgia, United States

Several initiatives are promoting longleaf pine (Pinus palustris) restoration in the Southern United States for conserving biodiversity and mitigating climate change. Under the Conservation Reserve Program (CRP), landowners plant longleaf pine on farmlands and pasturelands in exchange for annual payments and other financial incentives. While the introduction of this program was an important step towards longleaf pine restoration, there exists a necessity to understand how the decisions of landowners to enroll in CRP are manifested over space. An understanding about the role of a range of factors on the spatial density of longleaf pine plantations enrolled under CRP would help in prioritizing locations for longleaf pine plantations, thereby increasing the efficacy of conservation-related resources. We evaluate the effects of socioeconomic, topographic, and distance variables on the spatial density of longleaf plantations enrolled under CRP using clustered point process model in Georgia, a state with 47% of the total acreage under CRP-supported longleaf plantations. Only 30% of the current longleaf pine locations under CRP were within the Significant Geographic Area. About 86% of current longleaf plantations under CRP are present on former croplands and pasturelands. The spatial density of longleaf pine plantations is negatively associated with the distance from cropland and pastureland and positively associated with land capability classes, and distance from the sawmills. Longleaf pine restoration efforts should focus on those croplands or pasturelands which are closer to existing CRP-supported plantations, located on lower soil capability classes, and are far from wood consuming mills. Our study would support current programs that are supporting longleaf pine restoration in the Southern United States.

Inference for cluster point processes with over- or under-dispersed cluster sizes

Cluster point processes comprise a class of models that have been used for a wide range of applications. While several models have been studied for the probability density function of the offspring displacements and the parent point process, there are few examples of non-Poisson distributed cluster sizes. In this paper, we introduce a generalization of the Thomas process, which allows for the cluster sizes to have a variance that is greater or less than the expected value. We refer to this as the cluster sizes being over- and under-dispersed, respectively. To fit the model, we introduce minimum contrast methods and a Bayesian MCMC algorithm. These are evaluated in a simulation study. It is found that using the Bayesian MCMC method, we are in most cases able to detect over- and under-dispersion in the cluster sizes. We use the MCMC method to fit the model to nerve fiber data, and contrast the results to those of a fitted Thomas process.

Performance Analysis of C-RAN with Different Deployment Distributions

In cloud radio access networks (C-RAN) for 5G and beyond, the user-centric system model has gained popularity during downlink performance analysis for various reasons like tractability of stochastic geometric approach and distributed beamforming methods. In this paper, the coverage analysis of C-RAN with different distributions of BSs, e.g. uniform, PPP, Matern hard-core point process (MHCPP) of types I, II, and III are studied. We present the coverage analysis with the uniform distribution of BSs and compare it with other deployment distributions of BSs. The effect of threshold distance on the performance of MHCPP distributions is explored. The Monte Carlo simulation of heterogeneous C-RAN with different cluster point processes is observed and compared with the PPP model. The effect of the range of child clusters on the performance analysis is observed in the cluster point processes. To avoid the complexity in the study, interference is not considered in the analysis, and its counterpart simulations are also not considered while performing the comparison.

Poles of pair correlation functions: When they are real?

The most common standard estimator of the pair correlation function (PCF) of a point process has a pole at zero, which is in most cases a statistical artifact. However, sometimes it makes sense to assume that a pole really exists. We propose two independent approaches for the proof of existence of a PCF’s pole and for the determination of its order. In the first, we use a summary characteristic F that transforms the PCF’s pole order to the location of F’s pole, while the other one uses a natural estimation method based on Ripley’s K-function. These methods are applied to simulated samples of two classical point process models and two cluster point process models with special geometries. Finally, we use the approach in the statistical analysis of a classical point pattern of pine trees and a highly clustered pattern of nonmetallic inclusions in steel.

Interacting cluster point process model for epidermal nerve fibers

We propose an interacting cluster model for the spatial distribution of epidermal nerve fibers (ENF). The model consists of a spatial process of parent points modeling the base points of nerve fiber bundles. To each base point there is an offspring point process of fibers end points associated. The parent process is, possibly, inhibited by fiber endings that belong to different base bundles. The fibers themselves have random length and spatial orientation. We consider a non-orphan process where we can connect each offspring to a parent. We examine the processes that can be described with our model, how coefficient estimation can be performed under the Bayesian paradigm via Markov chain Monte Carlo methods, and detail approaches for the implementation of efficient MCMC sampling schemes. We study the performance of the estimation procedure through a simulation study. An application to data from skin blister biopsy images of ENFs is presented.

Fitting the Bartlett–Lewis rainfall model using Approximate Bayesian Computation

The Bartlett–Lewis (BL) rainfall model is a stochastic model for the rainfall at a single point in space, constructed using a cluster point process. The cluster process is constructed by taking a primary/parent process, called the storm arrival process in our context, and then attaching to each storm point a finite secondary/daughter point process, called a cell arrival process. To each cell arrival point we then attach a rain cell, with an associated rainfall duration and intensity. The total rainfall at time t is then the sum of the intensities from all active cells at that time. Because it has an intractable likelihood function, in the past the BL model has been fitted using the Generalised Method of Moments (GMM). The purpose of this paper is to show that Approximate Bayesian Computation (ABC) can also be used to fit this model, and moreover that it gives a better fit than GMM. GMM fitting matches theoretical and observed moments of the process, and thus is restricted to moments for which you have an analytic expression. ABC fitting compares the observed process to simulations, and thus places no restrictions on the statistics used to compare them. The penalty we pay for this increased flexibility is an increase in computational time.

Architectures of exoplanetary systems – I. A clustered forward model for exoplanetary systems around Kepler’s FGK stars

ABSTRACTObservations of exoplanetary systems provide clues about the intrinsic distribution of planetary systems, their architectures, and how they formed. We develop a forward modelling framework for generating populations of planetary systems and ‘observed’ catalogues by simulating the Kepler detection pipeline (SysSim). We compare our simulated catalogues to the Kepler DR25 catalogue of planet candidates, updated to include revised stellar radii from Gaia DR2. We constrain our models based on the observed 1D marginal distributions of orbital periods, period ratios, transit depths, transit depth ratios, transit durations, transit duration ratios, and transit multiplicities. Models assuming planets with independent periods and sizes do not adequately account for the properties of the multiplanet systems. Instead, a clustered point process model for exoplanet periods and sizes provides a significantly better description of the Kepler population, particularly the observed multiplicity and period ratio distributions. We find that $0.56^{+0.18}_{-0.15}$ of FGK stars have at least one planet larger than 0.5R⊕ between 3 and 300 d. Most of these planetary systems ($\sim 98{{\ \rm per\ cent}}$) consist of one or two clusters with a median of three planets per cluster. We find that the Kepler dichotomy is evidence for a population of highly inclined planetary systems and is unlikely to be solely due to a population of intrinsically single planet systems. We provide a large ensemble of simulated physical and observed catalogues of planetary systems from our models, as well as publicly available code for generating similar catalogues given user-defined parameters.

Drivers of forest regeneration patterns in drought prone mixed-species forests in the Northern Calcareous Alps

In temperate mountain forests of Northern Calcareous Alps, shallow soils, in combination with steep slopes and high radiation input on sun-exposed slopes, imply severe conditions for tree regeneration. Considering the remarkable trend of temperature increase and related climatic extremes in the Greater Alpine Area, it is likely that conditions for successful regeneration further deteriorate at these sites. Hence, investigating the relation of regeneration patterns to site conditions is crucial for recent and future forest management. Spatial point process models are appropriate tools to test the effect of environmental covariates in plant pattern generation. By applying these methods, we aim at identifying driving factors of forest regeneration dynamics of Fagus sylvatica, Abies alba and Picea abies. We used tree regeneration data from two sample sites along transects from closed canopy to forest gaps of natural origin. A set of spatially continuous explanatory variables (light environment, rock outcrops, microrelief, distance to mature trees, estimated seed density and presence of already established saplings) were generated. We fitted nonstationary parametric intensity functions with Poisson point process models and Thomas cluster point process models. The first and second order processes were analysed with pair correlation summary statistics. Interactions between regeneration patterns of different species and height classes were assessed with cross-pair correlation summary statistics. Successful germination and growth, at least at some microsites, is possible for any of the investigated species when ungulate browsing is prevented by fencing. Light is the most important environmental factor for describing the regeneration patterns. As higher shading has a positive influence, it is apparently not a limiting resource but rather a proxy for other, latent unfavourable conditions like drying of seedbeds. We thus propose to limit canopy opening sizes to a maximum of 10–12 m. No indications of seed limitation as a cause for regeneration failure of any of the main tree species were detected. Conditions for A. alba seedlings and partly also for P. abies seedlings are more favourable under the canopy of F. sylvatica than under the canopies of conspecifics or the other conifer species. Therefore, we conclude that at our study sites with the current species mixture, at least for the two conifer species, safe microsites are more important than seed availability. With increasing height of P. abies, the spatial segregation of small A. alba and F. sylvatica seedlings became stronger, indicating that competition for resources becomes more dominant and determinant for species coexistence.

Macroscopic analysis of shot-noise Cox random balls

In this paper, we consider a cluster model of weighted Euclidean random balls generated by a shot-noise Cox process. It is an example of cluster point process. We perform a scaling on the model by shrinking the radii of the balls and compensate this effect by increasing the (mean) number of balls in each cluster, or/and increasing the (mean) number of clusters. We consider two different scenarios, say a local and a global scenario. Heuristically, in the first scenario, we focus on the mean number of large balls in a cluster while in the second one, we focus on the global mean number of large balls in the model. According to the different scenarios, the cluster structure can persist at the limit or disappear.

ORTHOGONAL SERIES ESTIMATION OF THE PAIR CORRELATION FUNCTION OF A SPATIAL POINT PROCESS

The pair correlation function is a fundamental spatial point process characteristic that, given the intensity function, determines second order moments of the point process. Non-parametric estimation of the pair correlation function is a typical initial step of a statistical analysis of a spatial point pattern. Kernel estimators are popular but especially for clustered point patterns suffer from bias for small spatial lags. In this paper we introduce a new orthogonal series estimator. The new estimator is consistent and asymptotically normal according to our theoretical and simulation results. Our simulations further show that the new estimator can outperform the kernel estimators in particular for Poisson and clustered point processes.

On the Performance of Downlink NOMA in Multi-Cell mmWave Networks

This letter investigates the performance of non-orthogonal multiple access (NOMA) in a multi-cell downlink millimeter-wave (mmWave) network. A Poisson cluster point process is utilized to model the considered system. Key features of mmWave networks, such as directional beamforming and distance-dependent line of sight/non-line of sight, are taken into consideration. Nakagami fading is considered for small-scale fading. Closed-form expressions for outage probabilities achieved by the NOMA users are obtained. Simulation results are provided to demonstrate the accuracy of the analytical results and also show that NOMA can outperform OMA in a multi-cell mmWave network.

Cluster capture-recapture to account for identification uncertainty on aerial surveys of animal populations.

Capture-recapture methods for estimating wildlife population sizes almost always require their users to identify every detected animal. Many modern-day wildlife surveys detect animals without physical capture-visual detection by cameras is one such example. However, for every pair of detections, the surveyor faces a decision that is often fraught with uncertainty: are they linked to the same individual? An inability to resolve every such decision to a high degree of certainty prevents the use of standard capture-recapture methods, impeding the estimation of animal density. Here, we develop an estimator for aerial surveys, on which two planes or unmanned vehicles (drones) fly a transect over the survey region, detecting individuals via high-definition cameras. Identities remain unknown, so one cannot discern if two detections match to the same animal; however, detections in close proximity are more likely to match. By modeling detection locations as a clustered point process, we extend recently developed methodology and propose a precise and computationally efficient estimator of animal density that does not require individual identification. We illustrate the method with an aerial survey of porpoise, on which cameras detect individuals at the surface of the sea, and we need to take account of the fact that they are not always at the surface.

Pair correlation functions and limiting distributions of iterated cluster point processes

Abstract We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein (1975) and Shimatani (2010) describing a reproducing population. We discuss when closed-form expressions of the first- and second-order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

PERFORMANCE OF MIXED EXPONENTIAL AND EXPONENTIAL DISTRIBUTION REPRESENTING RAIN CELL INTENSITY IN NEYMANSCOTT RECTANGULAR PULSE (NSRP) MODEL

Sub-daily timescale data such as hourly data are needed for modeling urban systems. However such series are not readily available as compared to daily rainfall series. Stochastic rainfall models are useful in estimating input for design work. One of the models that applies the clustered point process theory is the Neyman-Scott Rectangular Pulses (NSRP) model. The model uses a flexible model fitting procedure which involves matching approximately a chosen set of historical statistics which exceeds in number of set of parameters to be estimated. An optimization technique called Shuffle Complex Evolution (SCE-UA) was used to estimate the parameters. The performance of NSRP model was evaluated using 10 years hourly data taken from a station in Wilayah Persekutuan. Two distributions, namely exponential (EXP) and mixed exponential (MEXP) were used to model the cell intensities in the model. The models were evaluated on a monthly basis regarding their ability to preserve the statistical properties as well as the physical properties of the rainfall time-series over timescales of 1 h, 6 h and 24 h. The performance of the models with the two different distributions was evaluated and compared. The model with the mixed exponential (MEXP) distribution perform better in preserving most of the statistical and physical properties of the observed data

Stochastic Wireless Powered Communication Networks With Truncated Cluster Point Process

Wireless powered communication network (WPCN) can prolong the lifetime of energy constrained wireless networks in a sustainable way without replacing or recharging the batteries. A popular approach for analyzing stochastic WPCN is to model the location of different nodes according to a homogeneous Poisson point process. However, it is more reasonable to place user equipments (UEs) around a power beacon considering the inefficiency and difficulty of wireless energy transfer. This motivates us to model the location of UEs as a cluster point process (CPP). With practical transmission range considered, the truncated Matern CPP and Thomas CPP are proposed. The performances of coverage probability and the average received signal-to-interference-plus-noise ratio are derived. Then, we optimize the time allocation fraction to maximize the average network throughput by pseudo-convexity optimization, and the closed-form optimal results are presented.

Cache Miss Estimation for Non-Stationary Request Processes

The goal of the paper is to evaluate the miss probability of a Least Recently Used (LRU) cache, when it is offered a non-stationary request process given by a Poisson cluster point process. First, we construct a probability space using Palm theory, describing how to consider a tagged document with respect to the rest of the request process. This framework allows us to derive a fundamental integral formula for the expected number of misses of the tagged document. Then, we consider the limit when the cache size and the arrival rate go to infinity in proportion, and use the integral formula to derive an asymptotic expansion of the miss probability in powers of the inverse of the cache size. This enables us to quantify and improve the accuracy of the so-called Che approximation.

Extended convergence of the extremal process of branching Brownian motion

We extend the results of Arguin et al. [Probab. Theory Related Fields 157 (2013) 535–574] and Aïdékon et al. [Probab. Theory Related Fields 157 (2013) 405–451] on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the “location” of the particle in the underlying Galton–Watson tree. We show that the limit is a cluster point process on $\mathbb{R}_{+}\times\mathbb{R}$ where each cluster is the atom of a Poisson point process on $\mathbb{R}_{+}\times\mathbb{R}$ with a random intensity measure $Z(dz)\times C\mathrm{e}^{-\sqrt{2}x}\,dx$, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous result for the Gaussian free field by Biskup and Louidor [Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field (2016)].