Non-homogeneous Poisson Point Process Research Articles

Stochastic model of seed dispersal with homogeneous and non-homogeneous Poisson processes under habitat reduction conditions.

This study presents a stochastic model of seed dispersal based on a branching random walk (BRW) framework, incorporating both homogeneous and non-homogeneous Poisson point processes (PPP). Building on the model introduced by Coletti et al. (2023), we examine the effects of habitat reduction on seed dispersal dynamics. We analyze the phase transition behavior of the BRW model under varying conditions of habitat fragmentation, focusing on how these conditions influence the critical dispersal rate. Specifically, we study a BRW on the real line with a non-homogeneous PPP driven by a log-normal density, constrained between spatial barriers. Our simulations localize the critical dispersal rate with respect to barrier positions and compare this dependence between homogeneous and non-homogeneous models.

Nonhomogeneous Stochastic Geometry Analysis of Massive LEO Communication Constellations

Providing truly ubiquitous connectivity requires development of low Earth orbit (LEO) satellite Internet, whose theoretical study is lagging behind network-specific simulations. In this paper, we derive analytical expressions for the downlink coverage probability and average data rate of a massive inclined LEO constellation in terms of total interference power’s Laplace transform in the presence of fading and shadowing, ergo presenting a stochastic geometry-based analysis. We assume the desired link to experience Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> fading, which serves to represent different fading scenarios by varying integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> , while the interfering channels can follow any fading model without an effect on analytical tractability. To take into account the inherent non-uniform distribution of satellites across different latitudes, we model the LEO network as a nonhomogeneous Poisson point process with its intensity being a function of satellites’ actual distribution in terms of constellation size, the altitude of the constellation, and the inclination of orbital planes. From the numerical results, we observe optimum points for both the constellation altitude and the number of orthogonal frequency channels; interestingly, the optimum user’s latitude is greater than the inclination angle due to the presence of fewer interfering satellites. Overall, the presented study facilitates general stochastic evaluation and planning of satellite Internet constellations without specific orbital simulations or tracking data on satellites’ exact positions in space.

Nonhomogeneous Euclidean first-passage percolation and distance learning

Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We define such a sample distance and prove the convergence, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean first-passage percolation for nonhomogeneous Poisson point processes.

On one method for modeling an nonhomogeneous Poisson point process

On one method for modeling an nonhomogeneous Poisson point process

Bayesian Mixture Poisson Regression for Modeling Spatial Point Pattern of Primary Health Centers in Surabaya

Primary Health Centres (PHC) are the first referral health facilities for Indonesian people to seek treatment. The varying distribution of PHC location in central Surabaya, therefore, causes its process to follow the Non-homogeneous Poisson Point Process (NHPP). We use Bayesian analysis coupled with Markov Chain Monte Carlo (MCMC) to model the mixture Poisson regression on NHPP intensity estimation. The result shows that two mixture components are significantly involved in the model along with four variables; i.e., the total population, the number of clean households, Accessibility Index, and the length of road; that produce the smallest Deviance Information Criteria (DIC). Keywords: Bayesian Analysis; Mixture Poisson Regression; NHPP Intensity; Primary Health Centres

Bayesian modeling and decision theory for non-homogeneous Poisson point processes

We present a flexible hierarchical Bayesian model and develop a comprehensive Bayesian decision theoretic framework for point process theory. We closely investigate the commonly used point process model for independent events, the Poisson process, using a finite mixture of exponential family components to model the intensity function. We employ a Bayesian hierarchical framework for parameter estimation and illustrate the Bayesian computations involved. We demonstrate the effectiveness of the Bayes rule under the Kullback–Leibler and Hellinger loss functions and compare them with the usual estimator, the posterior mean under squared error loss. The methodology is exemplified through simulations and a motivating application involving estimation of the intensity surface of homicide incidents in Chicago during 2015.

Coverage and rate analysis in two‐tier heterogeneous networks under suburban and urban scenarios

AbstractMassive boost in data traffic demand and inconsistent user's behavior have necessitated modern cellular networks to evolve toward heterogeneous architectural framework consisting of macro and small cells to accommodate ever‐increasing user's density. Literature survey reveals that the deployment of additional small cells can encounter the booming coverage, capacity, and QoS constraints by maintaining the overall operational cost of the network. In this paper, at first, a suitable model based on nonhomogeneous Poisson point process (NHPPP) is designed for heterogeneous wireless network (HetNet) consisting of two‐tiers eNodeBs (Macro and Small Cell), where each of the tiers is differentiated in terms of transmitting power, eNodeB density, and supported data rate. Subsequently, analytical expressions are derived for coverage probability (CP) and average rate (AR) to assess the performance of the HetNet. The contribution of the paper further lies in integrating the K‐means clustering algorithm with NHPPP to find the optimal locations of the small cell eNodeBs for extended coverage and rate improvement. The proposed model is investigated under differently dense scenarios like urban and suburban areas in India. It establishes the requisite of an optimal number of small cells along with the traditional infrastructure to maximize the performance in terms of CP and AR. Finally, the proposed integrated model is compared with the traditional homogeneous Poisson point process (HPPP) and NHPPP for coverage and rate analysis. It is observed that the K‐means clustering algorithm in integration with NHPPP overshadows both HPPP and NHPPP in terms of coverage and rate under both urban and suburban deployment scenarios.

Convergence of point processes associated with coupon collector’s and Dixie cup problems

We prove that, in the coupon collector's problem, the point processes given by the times of $r$-th arrivals for coupons of each type, centered and normalized in a proper way, converge toward a non-homogeneous Poisson point process. This result is then used to derive some generalizations and infinite-dimensional extensions of classical limit theorems on the topic.

Super-resolution of spatiotemporal event-stream image

Super-resolution (SR) is a useful technology to generate a high-resolution (HR) visual output from the low-resolution (LR) visual inputs overcoming the physical limitations of the cameras. However, SR has not been applied to enhance the spatial resolution of event-stream images captured by the frame-free dynamic vision sensor (DVS). SR of event-stream image aims to recover the same statistics of events which is fundamentally different from the existing frame-based schemes. In this work, a two-stage scheme is proposed to solve the spatial SR problem of the spatiotemporal event-stream image. We use a nonhomogeneous Poisson point process to model the event sequence, and sample the events of each pixel by simulating a nonhomogeneous Poisson process according to the specified event number and rate function. Firstly, the event number of each pixel of the HR DVS image is generated by obtaining the HR event-count map (ECM) from the LR DVS recording with a sparse representation based method. The rate function over time line of the point process of each HR pixel is computed using a spatiotemporal filter on the corresponding LR neighbor pixels. Secondly, the event sequence of each new pixel is obtained with a thinning based event sampling algorithm. A metric is proposed to assess the event-stream SR quality. The effectiveness of the proposed method is demonstrated through obtaining HR event-stream images from a series of DVS recordings. This work enables many potential.

Coverage and rate analysis in the uplink of millimeter wave cellular networks with fractional power control

In this paper, using the concept of stochastic geometry, we present an analytical framework to evaluate the signal-to-interference-and-noise-ratio (SINR) coverage in the uplink of millimeter wave cellular networks. By using a distance-dependent line-of-sight (LOS) probability function, the location of LOS and non-LOS users are modeled as two independent non-homogeneous Poisson point processes, with each having a different pathloss exponent. The analysis takes account of per-user fractional power control (FPC), which couples the transmission of users based on location-dependent channel inversion. We consider the following scenarios in our analysis: (1) pathloss-based FPC (PL-FPC) which is performed using the measured pathloss and (2) distance-based FPC (D-FPC) which is performed using the measured distance. Using the developed framework, we derive expressions for the area spectral efficiency. Results suggest that in terms of SINR coverage, D-FPC outperforms PL-FPC scheme at high SINR where the future networks are expected to operate. It achieves equal or better area spectral efficiency compared with the PL-FPC scheme. Contrary to the conventional ultra-high frequency cellular networks, in both FPC schemes, the SINR coverage decreases as the cell density becomes greater than a threshold, while the area spectral efficiency experiences a slow growth region.

Coverage, Capacity, and Energy Efficiency Analysis in the Uplink of mmWave Cellular Networks

In this paper, using the concept of stochastic geometry, we present an analytical framework to evaluate the signal-to-interference-and-noise-ratio (SINR) coverage in the uplink of millimeter wave cellular networks. By using a distance-dependent line-of-sight (LOS) probability function, the location of LOS and nonLOS users are modeled as two independent nonhomogeneous Poisson point processes, with each having a different pathloss exponent. The analysis takes account of per-user fractional power control (FPC), which couples the transmission of users based on location-dependent channel inversion. We consider the following scenarios in our analysis:1) pathloss-based FPC (PL-FPC) which is performed using the measured pathloss and 2) distance-based FPC (D-FPC) which is performed using the measured distance. Using the developed framework, we derive expressions for the area spectral efficiency and energy efficiency. Results suggest that in terms of SINR coverage, D-FPC outperforms PL-FPC scheme at high SINR where the future networks are expected to operate. It achieves equal or better area spectral efficiency and energy efficiency compared with the PL-FPC scheme. Contrary to the conventional ultra-high frequency cellular networks, in both FPC schemes, the SINR coverage decreases as the cell density becomes greater than a threshold, while the area spectral efficiency experiences a slow growth region.

The Meta Distribution of the SIR for Cellular Networks With Power Control

The meta distribution of the signal-to-interference ratio (SIR) provides fine-grained information about the performance of individual links in a wireless network. This paper focuses on the analysis of the meta distribution of the SIR for both the cellular network uplink and downlink with fractional power control. For the uplink scenario, an approximation of the interfering user point process with a non-homogeneous Poisson point process is used. The moments of the meta distribution for both scenarios are calculated. Some bounds, the analytical expression, the mean local delay, and the beta approximation of the meta distribution are provided. The results give interesting insights into the effect of the power control in both the uplink and downlink. Detailed simulations show that the approximations made in the analysis are well justified.

Multiple change points detection and clustering in dynamic networks

The increasing amount of data stored in the form of dynamic interactions between actors necessitates the use of methodologies to automatically extract relevant information. The interactions can be represented by dynamic networks in which most existing methods look for clusters of vertices to summarize the data. In this paper, a new framework is proposed in order to cluster the vertices while detecting change points in the intensities of the interactions. These change points are key in the understanding of the temporal interactions. The model used involves non-homogeneous Poisson point processes with cluster-dependent piecewise constant intensity functions and common discontinuity points. A variational expectation maximization algorithm is derived for inference. We show that the pruned exact linear time method, originally developed for change points detection in univariate time series, can be considered for the maximization step. This allows the detection of both the number of change points and their location. Experiments on artificial and real datasets are carried out, and the proposed approach is compared with related methods.

Optimized Random Deployment of Energy Harvesting Sensors for Field Reconstruction in Analog and Digital Forwarding Systems

This work examines the large-scale deployment of energy harvesting sensors for the purpose of sensing and reconstruction of a spatially correlated Gaussian random field. The sensors are powered solely by energy harvested from the environment and are deployed randomly according to a spatially nonhomogeneous Poisson point process whose density depends on the energy arrival statistics at different locations. Random deployment is suitable for applications that require deployment over a wide and/or hostile area. During an observation period, each sensor takes a local sample of the random field and reports the data to the closest data-gathering node if sufficient energy is available for transmission. The realization of the random field is then reconstructed at the fusion center based on the reported sensor measurements. For the purpose of field reconstruction, the sensors should, on the one hand, be more spread out over the field to gather more informative samples, but should, on the other hand, be more concentrated at locations with high energy arrival rates or large channel gains toward the closest data-gathering node. This tradeoff is exploited in the optimization of the random sensor deployment in both analog and digital forwarding systems. More specifically, given the statistics of the energy arrival at different locations and a constraint on the average number of sensors, the spatially-dependent sensor density and the energy-aware transmission policy at the sensors are determined for both cases by minimizing an upper bound on the average mean-square reconstruction error. The efficacy of the proposed schemes are demonstrated through numerical simulations.

Uplink Interference Analysis for Two-Tier Cellular Networks With Diverse Users Under Random Spatial Patterns

Multi-tier architecture improves the spatial reuse of radio spectrum in cellular networks, but it introduces complicated heterogeneity in the spatial distribution of transmitters, which brings new challenges in interference analysis. In this work, we present a stochastic geometric model for evaluating the uplink interference in a two-tier network considering multi-type users and base stations. Each type of tier-1 users and tier-2 base stations are modeled as independent homogeneous Poisson point processes, and tier-2 users are modeled as locally non-homogeneous clustered Poisson point processes centered at tier-2 base stations. By applying a superposition-aggregation-superposition approach, we quantify the interference at both tiers. Our model is also able to capture the impact of two types of exclusion regions, where either tier-2 base stations or tier-2 users are restricted to avoid cross-tier interference. As an important application of this analytical model, an intensity planning scenario is investigated, in which we aim to maximize the total income of the network operator with respect to the intensities of tier-2 cells, under constraints on the outage probabilities of tier-1 and tier-2 users. The result of our interference analysis suggests that this maximization can be converted to a standard convex optimization problem. Finally, numerical studies further demonstrate the correctness of our analysis.

Coverage and Rate Analysis for Millimeter-Wave Cellular Networks

Millimeter wave (mmWave) holds promise as a carrier frequency for fifth generation cellular networks. Because mmWave signals are sensitive to blockage, prior models for cellular networks operated in the ultra high frequency (UHF) band do not apply to analyze mmWave cellular networks directly. Leveraging concepts from stochastic geometry, this paper proposes a general framework to evaluate the coverage and rate performance in mmWave cellular networks. Using a distance-dependent line-of-site (LOS) probability function, the locations of the LOS and non-LOS base stations are modeled as two independent non-homogeneous Poisson point processes, to which different path loss laws are applied. Based on the proposed framework, expressions for the signal-to-noise-and-interference ratio (SINR) and rate coverage probability are derived. The mmWave coverage and rate performance are examined as a function of the antenna geometry and base station density. The case of dense networks is further analyzed by applying a simplified system model, in which the LOS region of a user is approximated as a fixed LOS ball. The results show that dense mmWave networks can achieve comparable coverage and much higher data rates than conventional UHF cellular systems, despite the presence of blockages. The results suggest that the cell size to achieve the optimal SINR scales with the average size of the area that is LOS to a user.

Online clutter estimation using a Gaussian kernel density estimator for multitarget tracking

In this study, the spatial distribution of false alarms is assumed to be a non-homogeneous Poisson point (NHPP) process. Then, a new method is developed under the kernel density estimation (KDE) framework to estimate the spatial intensity of false alarms for the multitarget tracking problem. In the proposed method, the false alarm spatial intensity estimation problem is decomposed into two subproblems: (i) estimating the number of false alarms in one scan and (ii) estimating the variation of the intensity function value in the measurement space. Under the NHPP assumption, the only parameter that needs to be estimated for the first subproblem is the mean of false alarm number, and the empirical mean is used here as the maximum likelihood estimate of that parameter. Then, for the second subproblem, an online multivariate local adaptive Gaussian kernel density estimator is proposed. Furthermore, the proposed estimation method is seamlessly integrated with widely used multitarget trackers, like the joint integrated probabilistic data association algorithm and the multiple hypotheses tracking algorithm. Simulation results show that the proposed KDE-based method can provide a better estimate of the false alarm spatial intensity and help the multitarget trackers yield superior performance in scenarios with spatially non-homogeneous false alarms.

Downlink Performance Analysis for a Generalized Shotgun Cellular System

In this paper, we analyze the signal-to-interference-plus-noise ratio (SINR) performance at a mobile station (MS) in a random cellular network. The cellular network is formed by base stations (BSs) placed in a one-, two-, or three-dimensional space according to a possibly non-homogeneous Poisson point process, which is a generalization of the so-called shotgun cellular system. We develop a sequence of equivalence relations for the SCSs and use them to derive semi-analytical expressions for the coverage probability at the MS when the transmissions from each BS may be affected by random fading with arbitrary distributions as well as attenuation following arbitrary path-loss models. For homogeneous Poisson point processes in the interference-limited case with power-law path-loss model, we show that the SINR distribution is the same for all fading distributions and is not a function of the base station density. In addition, the influence of random transmission power, power control, and multiple channel reuse groups on the downlink performance is also discussed. The techniques developed for the analysis of SINR have applications beyond cellular networks and can be used in similar studies for cognitive radio networks, femtocell networks, and other heterogeneous and multi-tier networks.

A general study of extremes of stationary tessellations with examples

Let m be a random tessellation in Rd, d≥1, observed in a bounded Borel subset W and f(⋅) be a measurable function defined on the set of convex bodies. A point z(C), called the nucleus of C, is associated with each cell C of m. Applying f(⋅) to all the cells of m, we investigate the order statistics of f(C) over all cells C∈m with nucleus in Wρ=ρ1/dW when ρ goes to infinity. Under a strong mixing property and a local condition on m and f(⋅), we show a general theorem which reduces the study of the order statistics to the random variable f(C), where C is the typical cell of m. The proof is deduced from a Poisson approximation on a dependency graph via the Chen–Stein method. We obtain that the point process {(ρ−1/dz(C),aρ−1(f(C)−bρ)),C∈m,z(C)∈Wρ}, where aρ>0 and bρ are two suitable functions depending on ρ, converges to a non-homogeneous Poisson point process. Several applications of the general theorem are derived in the particular setting of Poisson–Voronoi and Poisson–Delaunay tessellations and for different functions f(⋅) such as the inradius, the circumradius, the area, the volume of the Voronoi flower and the distance to the farthest neighbor.

Simulating Nonhomogeneous Poisson Point Process Based On Multi Criteria Intensity Function And Comparison With Its Simple Form

In this paper we first study the general nonhomogeneous Poisson point process based on strict form of an intensity function and its algorithm for generating it. Then, we employ multi criteria intensity function instead of simple form and establish a new algorithm, then we compare the efficiency of our new algorithm based on this modified intensity function.