Inhomogeneous Poisson Point Process Research Articles

An Inhomogeneous Weibull–Hawkes Process to Model Underdispersed Acoustic Cues

AbstractA Hawkes point process describes self-exciting behaviour where event arrivals are triggered by historic events. These models are increasingly becoming a popular choice in analysing event-type data. Like all other inhomogeneous Poisson point processes, the waiting time between events in a Hawkes process is derived from an exponential distribution with mean one. However, as with many ecological and environmental data, this is an unrealistic assumption. We, therefore, extend and generalise the Hawkes process to account for potential under- or overdispersion in the waiting times between events by assuming the Weibull distribution as the foundation of the waiting times. We apply this model to the acoustic cue production times of sperm whales and show that our Weibull–Hawkes model better captures the inherent underdispersion in the interarrival times of echolocation clicks emitted by these whales.

Inhomogeneous Poisson processes in the disk and interpolation

Abstract We investigate different geometrical properties, related to Carleson measures and pseudo-hyperbolic separation, of inhomogeneous Poisson point processes on the unit disk. In particular, we give conditions so that these random sequences are almost surely interpolating for the Hardy, Bloch or weighted Dirichlet spaces.

COVID-19 Death Risk in Surabaya: Modeling by Spatial Point Process

The total death rate or Case Fatality Rate (CFR) due to COVID-19 in Surabaya is high, that is almost twice of the global CFR (1.4%). Utilization of high-resolution data has the potential to explore COVID-19 cases, not only recording cases at the district or city level but also at the patient’s domicile level so that they can provide more detailed spatial information. Meanwhile, research exploring the risk of death from COVID-19, especially in Surabaya using spatial point process model, has not yet been carried out. In this study, an analysis of the risk of death from COVID-19 in Surabaya will be carried out using the inhomogeneous Poisson point process model with covariates or external factors used including the density of the COVID-19 referral hospital location and the proportion of confirmed COVID-19 population aged > 60 years per districts. Our model shows that referral hospitals (exp( ) = 1.03295) and places of worship (exp( ) = 1.03835) have a significant effect on death risk from COVID-19. So, there is a need for special handling for areas that have a population with a vulnerable age (> 60 years) where at this age the human immune system will decrease.
 Keywords: COVID-19, heath risk, spatial point process, surabaya

Integrating presence‐only and detection/non‐detection data to estimate distributions and expected abundance of difficult‐to‐monitor species on a landscape‐scale

Abstract Estimating species distribution and abundance is foundational to effective management and conservation. Using an integrated species distribution model that combines presence‐only data from various sources with detection/non‐detection data from structured surveys, we estimated the distribution and expected abundance of three difficult‐to‐monitor mammals of management concern across New York State, namely, coyotes (Canis latrans), bobcats (Lynx rufus) and black bears (Ursus americanus). Three distinct landscape‐scale camera trap surveys provided detection/non‐detection data over 9 years between 2013 and 2021, and we augmented those data with incidental records of our focal species from public repositories. We used an inhomogeneous Poisson point process to construct an integrated model that fit both data types simultaneously. We demonstrate a simple application of spatial point density of all species records in the accessed public databases to inform the thinning process to account for unknown spatial sampling in the presence‐only data, often referred to as the ‘magic covariate’. Using this approach, we examine habitat associations and provide spatially explicit estimates in expected abundance across the entirety of New York State for all three focal species. As expected, coyotes were the most widely distributed and abundant species, with a strong positive association with agricultural land uses. Bobcats exhibited low expected abundance throughout the state and showed positive associations with deciduous forest and forest edge, and a negative association with road density. Finally, we observed considerable spatial variation in abundance of black bears with expected abundance increasing in association with various forest cover and composition covariates and decreasing with crop cover. We present insights into habitat associations and spatial variation in abundance, and provide management implications for each of the species of interest. Synthesis and applications. Our integrated modelling method allows for managers to use citizen sightings combined with detection/non‐detection surveys to estimate robust indices of abundance for both high‐ and low‐density, and wide‐spread versus patchily distributed species. Through comparison with previous studies, we highlight how broad‐scale programmes, such as the statewide efforts to estimate species distributions undertaken here, can benefit substantively from integrated models that leverage additional data (here, incidental records) from a larger region of space, and thus capture more landscape heterogeneity than is plausible within formalized surveys alone.

Modelling spatial point patterns of Asian Giant Hornets occurrences

This study explores statistical methods for modelling the spatial patterns of Asian giant hornets using sighting records in the Republic of Korea from 2008 to 2013. The focus is on simulating the spatial distribution of hornets, with a key aspect being the statistical inference of their intensity. Gaussian kernel-smoothing estimation is utilized to model the hornets intensity based on sighting records, which also transforms the occurrences of honeybees, a primary prey of the hornets, into a covariate. Results show a significant dependency of hornets intensity on honeybees intensity, with a calculated probability that a hornet sighting location has higher honeybee intensity than a random location. By this finding, the parametric modelling of the hornets spatial intensity is applied with the covariate of honeybees in each year, with the basic inhomogeneous Poisson point process along with the log-linear model. The model is refined for each year using backward stepwise selection based on the Akaike Information Criteria. Model validation confirms the Poisson process assumption and shows promising results with raw residuals against honeybee intensity. The analysis demonstrates that the spatial pattern modelling method employed is both sensible and valid.

Stochastic analysis of structural reliability of complex engineering systems

Purpose: The assessment of risks and prediction of reliability of complex engineering systems, in particular onshore gas pipelines subjected to external corrosion. Two methods are proposed using the structural reliability analysis.Methodology: Two strategies are considered for inspection and maintenance service of hazardous production facilities.Research findings: The model of structural reliability is proposed to estimate the metal rupture from external corrosion of the pipe section. Models are presented for its mechanical failure with respect to stochastic processes of loads and resistance.Value: The inhomogeneous Poisson point process is used to simulate the formation of new defects, and the Poisson distribution is used to simulate their growth. The first method focuses on the analysis of external corrosion of gas pipelines with metal loss and predicts rupture at a reference pipe segment, constructed with respect to average pipe rupture characteristics from the PHMSA database. The second model predicts reliability for untreated sections subject to external corrosion with metal loss.

Resident Population Density-Inspired Deployment of K-Tier Aerial Cellular Network

Using Unmanned Aerial Vehicles (UAVs) to enhance network coverage has proven a variety of benefits compared to terrestrial counterparts. One of the commonly used mathematical tools to model the locations of the UAVs is stochastic geometry (SG). However, in the existing studies, both users and UAVs are often modeled as homogeneous point processes. In this paper, we consider an inhomogeneous Poisson point process (PPP)-based model for the locations of the users that captures the degradation in the density of active users as we move away from the town center. In addition, we propose the deployment of aerial vehicles following the same inhomogeneity of the users to maximize the performance. In addition, a multi-tier network model is also considered to make better use of the rich space resources. Then, the analytical expressions of the coverage probability for a typical user and the total coverage probability are derived. Finally, we optimize the coverage probability with limitations of the total number of UAVs and the minimum local coverage probability. Finally we give the optimal UAV distribution parameters when the maximum overall coverage probability is reached.

On a special class of gibbs hard-core point processes modeling random patterns of non-overlapping grains

Inspired by issues of formal kinetics in materials science, we consider a class of processes with density with respect to an inhomogeneous finite Poisson point process, which may be regarded as a generalization of the classical Strauss hard-core process. We prove expressions for the intensity measure and the void probabilities, together with upper and lower bounds. A discussion on some special cases of interest, links with literature and a comparison between Matérn I and Strauss hard-core process are also provided. We apply such a special class of point processes in modeling patterns of non-overlapping grains and in the study of the mean volume density of particular birth-and-growth processes.

Performance of inhomogeneous Poisson point process models under different scenarios of uncertainty in species presence-only data

Haphazard and opportunistic species occurrence (PO) data are widely used in species distribution models (SDMs) instead of high-quality species data gathered using appropriate and structured sampling methods, which is expensive and often spatially limited. Despite their widespread use in ecology, PO data are prone to errors and uncertainties, such as imperfect detectability, positional imprecision, and spatial niche truncation, which make their use analytically challenging for effective and adaptive biodiversity management and conservation. Using simulated data, this study investigates the effects of these uncertainties on the performance of spatial point process based presence-only and integrated SDMs. We investigated three SDMs in this study, one that ignores imperfect detectability: the presence-only model (PO model), and two that account for it: the thinned presence-only model (THINPO model) and the integrated model (PBPC model). The ability of these SDMs to produce accurate maximum likelihood estimates of intensity model coefficients and reliable predictions of species distributions under different data quality scenarios was investigated. The results show that SDMs that account for imperfect detectability (THINPO or PBPC models) are not applicable in situations of high detectability. In this situation, the PO model produces the most accurate maximum likelihood estimates of the models’ coefficients ({hat{beta }}_k), and consequently the most accurate predictions of species distributions ({hat{lambda }}(s)). The effects of positional uncertainty and spatial niche truncation on this SDM output are minimal. However, in situations of low detectability, it is preferable to use the PBPC model. Positional uncertainty and spatial niche truncation have negligible effects on the output of this SDM, except when positionally uncertain PO data are analyzed along with truncated PC data. These minimal effects of spatial niche truncation on SDM outputs demonstrate the transferability of SDMs. However, the effects of all these uncertainties may depend on the characteristics of the species. Prior to modeling species distributions, a multivariate environmental similarity surface analysis should be performed to test the similarity between data from the restricted region to be used for model calibration and data from the entire range. If this analysis reveals dissimilarities, larger spatial and ecological scales should be considered to address the issue of spatial niche truncation. Further efforts could address the effects of species characteristics on SDMs performance and assess the effects of species-specific uncertainties.

UAV-Aided Post-Disaster Cellular Networks: A Novel Stochastic Geometry Approach

Motivated by the need for ubiquitous and reliable communications in post-disaster emergency management systems (EMSs), we hereby present a novel and efficient stochastic geometry (SG) framework. This mathematical model is specifically designed to evaluate the quality of service (QoS) experienced by a typical ground user equipment (UE) residing either inside or outside a generic area affected by a calamity. In particular, we model the functioning terrestrial base stations (TBSs) as an inhomogeneous Poisson point process (IPPP), and assume that a given number of uniformly distributed unmanned aerial vehicles (UAVs) equipped with cellular transceivers is deployed in order to compensate for the damage suffered by some of the existing TBSs. The downlink (DL) coverage probability is then derived based on the maximum average received power association policy and the assumption of Nakagami- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> fading conditions for all wireless links. The proposed numerical results show insightful trends in terms of coverage probability, depending on: distance of the UE from the disaster epicenter, disaster radius, quality of resilience (QoR) of the terrestrial network, and fleet of deployed ad-hoc aerial base stations (ABSs). The aim of this paper is therefore to prove the effectiveness of vertical heterogeneous networks (VHetNets) in emergency scenarios, which can both stimulate the involved authorities for their implementation and inspire researchers to further investigate related problems.

COVID-19 transmission risk in Surabaya and Sidoarjo: an inhomogeneous marked Poisson point process approach.

Understanding the spatio-temporal dynamics of COVID-19 transmission is necessary to plan better strategies for controlling the spread of the disease. However, only a few studies explore the COVID-19 transmission risk over a fine spatial resolution while considering relevant spatial and temporal factors. To this aim, we consider an inhomogeneous marked Poisson point process model to assess COVID-19 transmission risk using data of home addresses of confirmed cases, in relation to locations of sources of crowd (enterprise, market, and place of worship) and population density in Surabaya and Sidoarjo, Indonesia. Our marked model is able to analyze how the spatial covariates are varying with time, helping authorities to evaluate the information of covariates depending on the period in which restrictions are taking place. Our results show that enterprise, place of worship, and population densities have significant impact to the transmission risk in Surabaya and Sidoarjo. We finally provide predicted risk maps which provide additional information based on the demographic-based risk analysis to help conduct more efficient testing, tracing, and vaccination programs.

Spatial frequency of randomly acquired characteristics on outsoles.

The highest levels of source association in forensic footwear comparison rely on the agreement between randomly acquired characteristics (RACs) identified on questioned and exemplar test impressions. These features are presumed to be randomly acquired and independent. However, independent acquisition does not necessarily mean these features will be uniformly distributed across an outsole. The aim of this research was to determine if the distribution of RACs in a research dataset could be described by an inhomogeneous Poisson point process based on tread contact and wear. To achieve this goal, RAC spatial frequency from an empirical dataset of shoes was compared against simulated and modeled data assuming a Poisson point process. Deviations in count between the empirical and simulated/modeled predictions were examined using a Poisson rate test and Moran's I. Results indicate that RAC frequency over 67%-79% of an outsole can be reasonably well explained as a Poisson point process or by a Poisson generalized linear regression model (non-spatial GLM) with tread contact as a predictor. Moreover, if the predictor is extended to include both tread contact and wear, RAC counts over 84% of the spatial locations on an outsole are well-explained (although autocorrelation persists). Overall, results indicate that RACs are not uniformly distributed in this dataset, most likely because the factors that dictate RAC development (friction, gait, etc.) are not uniformly distributed. Although this observation in no way negates the use of RACs in forming source associations, the value of a correspondence can differ depending on its spatial location.

Inhomogeneous spatio-temporal point processes on linear networks for visitors’ stops data

We analyse the spatio-temporal distribution of visitors’ stops by touristic attractions in Palermo (Italy), using theory of stochastic point processes living on linear networks. We first propose an inhomogeneous Poisson point process model with a separable parametric spatio-temporal first-order intensity. We account for the spatial interaction among points on the given network, fitting a Gibbs point process model with mixed effects for the purely spatial component. This allows us to study first-order and second-order properties of the point pattern, accounting both for the spatio-temporal clustering and interaction and for the spatio-temporal scale at which they operate. Due to the strong degree of clustering in the data, we then formulate a more complex model, fitting a spatio-temporal log-Gaussian Cox process to the point process on the linear network, addressing the problem of the choice of the most appropriate distance metric.

Inhomogeneous Poisson point process for species distribution modelling: relative performance of methods accounting for sampling bias and imperfect detection

Inhomogeneous Poisson point process for species distribution modelling: relative performance of methods accounting for sampling bias and imperfect detection

Understanding the correlation between destination distribution and urban built environment from taxi GPS data

AbstractExploring the relationship between travel destinations and urban built environments can help to conduct reasonable measurements in urban transportation planning. Destination distribution is affected by various factors, and quantitative description of the spatial effects is a challenge in factor correlation analysis. In this study, inhomogeneous Poisson point processes are proposed to model the correlation between destination distribution and related categories of variables, including land use, trip origin distribution, demographic information, and transport networks. First, five models based on inhomogeneous Poisson point processes are constructed, considering different combinations of variables. Second, diagnostic methods, including the G‐function, smoothed residuals, and parameter influence function, are applied to evaluate the models' performance. Finally, in experiments, the taxi GPS trajectory data collected in Shenzhen City, China, are used to validate the effectiveness of the proposed method. Moreover, in the discussion, the variations of variable coefficients in different time periods (workdays, weekends, and holidays) are analyzed and explained. The results of this study may enhance understanding of the correlation between travel patterns and urban structure.

Performance Analysis of Clustered Wireless-Powered Ad Hoc Networks via β-Ginibre Point Processes

We consider a wireless-powered ad hoc network with clustered Power Beacons (PBs). The wireless-powered transmitters (WP-Txs) only activate stochastically deployed PBs within the circles centered at their locations. The PBs transmit radio frequency (RF) signals, which are harvested by the WP-Txs to transmit information to their target receivers. The nodes are modeled as following <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> -Ginibre Point Processes ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> -GPP), which incorporate repulsion among PBs or active WP-Txs. Lacking analytical mathematical tools to describe the distribution of clustered PBs, we establish lower and upper bounds on the energy outage probability, as well as an approximated expression of it. We then derive the information outage probability, an approximate expression of it by approximating the reduced Palm distribution of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> -GPP as an inhomogeneous Poisson point process (IPPP), and an analytical approximate expression of it by further ignoring the small-scale fading of interfering WP-Tx links. Finally, the transmission capacity is obtained based on the overall outage probability. The simulation results show that the derived theoretical expressions accurately reflect the performance of the wireless-powered ad hoc networks and the performance gap introduced by approximations is negligible. Transmission capacity optimization based on the derived expressions is also possible and demonstrated.

Salt marsh vegetation and secured woody debris facilitate the recolonization of Avicennia germinans

AbstractQuestionsDoes the presence of salt marsh vegetation affect the long‐term regeneration of the pioneer mangrove species Avicennia germinans in a degraded dwarf forest? Does immobilized coarse woody debris (CWD) affect regeneration similarly? Do larger trees suppress or facilitate intraspecific saplings?LocationDwarf mangrove forest in the high intertidal zone on the Bragança peninsula in northern Brazil.MethodsThe spatial patterns of A. germinans, the herbaceous halophyte Sesuvium portulacastrum, and CWD were mapped in three sample plots (each 400 m2) during two consecutive vegetation surveys, conducted in 2011 and 2014. Inhomogeneous Poisson and Thomas point process models were used to assess the distribution of A. germinans life‐history stages (seedlings, saplings, and adult dwarf trees), conditioned on the presence of S. portulacastrum and CWD. In addition, intraspecific interactions between trees and regeneration were assessed based on crown projection mapping. Bivariate point pattern analyses were used to assess the dependence of advance regeneration on dwarf A. germinans trees and S. portulacastrum.ResultsAvicennia germinans saplings and trees were positively associated with S. portulacastrum and CWD, whereas seedlings were located around tree crowns. The density of fruit‐bearing trees was positively associated with sapling density, indicating that regeneration relied on locally dispersed propagules.ConclusionsHerbaceous vegetation and CWD have an important ecological function in degraded mangroves by retaining tidally dispersed propagules. Here, we show that herbaceous vegetation does not suppress the growth of seedlings but facilitates mangrove recolonization. Due to limited tidal dispersal, regeneration relies on local propagule supply. In addition to hydrological restoration, the observed vegetation patterns suggest that, in the absence of propagule‐retaining vegetation, restoration of high‐intertidal mangroves can be facilitated by establishing nuclei of planted trees and installing secured logs.

Estimating intensity functions of spatial inhomogeneous Poisson point processes via a Stein estimator

We develop a global Stein estimator through Malliavin derivatives to estimate the multi-variable intensity functions of general d-dimensional spatial inhomogeneous Poisson point processes (IPPPs). The gain performance between our Stein estimator and a tested maximum likelihood (ML) estimator is analyzed via Malliavin calculus. In comparison with existing homogeneous studies, our inhomogeneous cases are more complex in a global way and more real in practice. In the numerical simulation experiments, our estimated intensity functions via the Stein estimator well match the true one, which outperform the ones obtained by the ML estimator with an average gain larger than 30% concerning mean squared errors (MSEs). Hence, our Stein method is more robust with smaller MSEs. We also establish the mean-squared convergence with error estimation for our Stein estimator. Comparing with the existing study, our observation window for an IPPP is a d-dimensional rectangle and its boundary is not -smooth. This non-smoothness introduces additional complexity to our convergence analysis. Thus, a weak convergence approach is employed to overcome this difficulty.

A systematic approach and software for the analysis of point patterns on river networks

AbstractMany geomorphic phenomena such as bank failures, landslide dams, riffle‐pool sequences and knickpoints can be modelled as spatial point processes. However, as the locations of these phenomena are constrained to lie on or alongside rivers, their analysis must account for the geometry and topology of river networks. Here, we introduce a new numeric class in TopoToolbox called Point Pattern on Stream networks (PPS), which supports exploratory analysis, statistical modelling, simulation and visualization of point processes. We present three case studies that aim at inferring processes and factors that control the spatial density of geomorphic phenomena along river networks: analysis of a synthetic dataset of points on a stream network, the analysis of knickpoints in river profiles, and modelling spatial locations of beaver dams based on topographic metrics. The case studies rely on exploratory analysis and statistical inference using inhomogeneous Poisson point processes. Thereby, statistical and probabilistic procedures implemented in PPS provide a systematic approach for treating and quantifying uncertainties. PPS offers a consistent numeric framework for modelling point processes on river networks with a wide range of applications in fluvial geomorphology, but also other disciplines such as ecology.

A numerical method for computing interval distributions for an inhomogeneous Poisson point process modified by random dead times

The inhomogeneous Poisson point process is a common model for time series of discrete, stochastic events. When an event from a point process is detected, it may trigger a random dead time in the detector, during which subsequent events will fail to be detected. It can be difficult or impossible to obtain a closed-form expression for the distribution of intervals between detections, even when the rate function (often referred to as the intensity function) and the dead-time distribution are given. Here, a method is presented to numerically compute the interval distribution expected for any arbitrary inhomogeneous Poisson point process modified by dead times drawn from any arbitrary distribution. In neuroscience, such a point process is used to model trains of neuronal spikes triggered by the detection of excitatory events while the neuron is not refractory. The assumptions of the method are that the process is observed over a finite observation window and that the detector is not in a dead state at the start of the observation window. Simulations are used to verify the method for several example point processes. The method should be useful for modeling and understanding the relationships between the rate functions and interval distributions of the event and detection processes, and how these relationships depend on the dead-time distribution.