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Spatial statistics is a powerful tool for analyzing data that are illustrated as points or positions in a regular or non-regular state space. Techniques that are proposed to investigate the spatial association between neighboring positions are based on the point process analysis. One of the main goals is to simulate real data positions (such as antenna base stations) using the type of point process that most closely matches the data. Spatial patterns could be detailed describing the observed positions and appropriate models were proposed to simulate these patterns. A common model to simulate spatial patterns is the Poisson point process. In this work analyses of the Poisson point process—as well as modified types such as inhibition point process and determinantal Poisson point process—are presented with simulated data close to the true data (i.e., antenna base station positions). Investigation of the spatial variation of the data led us to the spatial association between positions by applying Ripley’s K-functions and L-Function.

In this paper, we propose a new comparison tool for spatial homogeneity of point processes, based on the joint examination of void probabilities and factorial moment measures. We prove that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure. We provide some motivating results on percolation and coverage processes, and preview further ones on other stochastic geometric models, such as minimal spanning forests, Lilypond growth models, and random simplicial complexes, showing that the new tool is relevant for a systemic approach to the study of macroscopic properties of non-Poisson point processes. This new comparison is also implied by the directionally convex ordering of point processes, which has already been shown to be relevant to the comparison of the spatial homogeneity of point processes. For this latter ordering, using a notion of lattice perturbation, we provide a large monotone spectrum of comparable point processes, ranging from periodic grids to Cox processes, and encompassing Poisson point processes as well. They are intended to serve as a platform for further theoretical and numerical studies of clustering, as well as simple models of random point patterns to be used in applications where neither complete regularity nor the total independence property are realistic assumptions.

We study the general problem of estimating a \"hidden\" point process X, given the realization of an \"observed\" point process Y (possibly defined in different spaces) with known joint distribution. We characterize the posterior distribution of X under marginal Poisson and Gauss-Poisson priors and when the transformation from X to Y includes thinning, displacement, and augmentation with extra points. These results are then applied in a filtering context when the hidden process evolves in discrete time in a Markovian fashion. The dynamics of X considered are general enough for many target tracking applications. [PUBLICATION ABSTRACT]

Many species distribution models (SDMs) are built with precise but geographically restricted presence–absence data sets (e.g., a country) where only a subset of the environmental conditions experienced by a species across its range is considered (i.e., spatial niche truncation). This type of truncation is worrisome because it can lead to incorrect predictions e.g., when projecting to future climatic conditions belonging to the species niche but unavailable in the calibration area. Data from citizen-science programs, species range maps or atlases covering the full species range can be used to capture those parts of the species’ niche that are missing regionally. However, these data usually are too coarse or too biased to support regional management. Here, we aim to (1) demonstrate how varying degrees of spatial niche truncation affect SDMs projections when calibrated with climatically truncated regional data sets and (2) test the performance of different methods to harness information from larger-scale data sets presenting different spatial resolutions to solve the spatial niche truncation problem. We used simulations to compare the performance of the different methods, and applied them to a real data set to predict the future distribution of a plant species (Potentilla aurea) in Switzerland. SDMs calibrated with geographically restricted data sets expectedly provided biased predictions when projected outside the calibration area or time period. Approaches integrating information from larger-scale data sets using hierarchical data integration methods usually reduced this bias. However, their performance varied depending on the level of spatial niche truncation and how data were combined. Interestingly, while some methods (e.g., data pooling, downscaling) performed well on both simulated and real data, others (e.g., those based on a Poisson point process) performed better on real data, indicating a dependency of model performance on the simulation process (e.g., shape of simulated response curves). Based on our results, we recommend to use different data integration methods and, whenever possible, to make a choice depending on model performance. In any case, an ensemble modeling approach can be used to account for uncertainty in how niche truncation is accounted for and identify areas where similarities/dissimilarities exist across methods.

Population-level estimates of species' distributions can reveal fundamental ecological processes and facilitate conservation. However, these may be difficult to obtain for mobile species, especially colonial central-place foragers (CCPFs; e.g., bats, corvids, social insects), because it is often impractical to determine the provenance of individuals observed beyond breeding sites. Moreover, some CCPFs, especially in the marine realm (e.g., pinnipeds, turtles, and seabirds) are difficult to observe because they range tens to ten thousands of kilometers from their colonies. It is hypothesized that the distribution of CCPFs depends largely on habitat availability and intraspecific competition. Modeling these effects may therefore allow distributions to be estimated from samples of individual spatial usage. Such data can be obtained for an increasing number of species using tracking technology. However, techniques for estimating population-level distributions using the telemetry data are poorly developed. This is of concern because many marine CCPFs, such as seabirds, are threatened by anthropogenic activities. Here, we aim to estimate the distribution at sea of four seabird species, foraging from approximately 5,500 breeding sites in Britain and Ireland. To do so, we GPS-tracked a sample of 230 European Shags Phalacrocorax aristotelis, 464 Black-legged Kittiwakes Rissa tridactyla, 178 Common Murres Uria aalge, and 281 Razorbills Alca torda from 13, 20, 12, and 14 colonies, respectively. Using Poisson point process habitat use models, we show that distribution at sea is dependent on (1) density-dependent competition among sympatric conspecifics (all species) and parapatric conspecifics (Kittiwakes and Murres); (2) habitat accessibility and coastal geometry, such that birds travel further from colonies with limited access to the sea; and (3) regional habitat availability. Using these models, we predict space use by birds from unobserved colonies and thereby map the distribution at sea of each species at both the colony and regional level. Space use by all four species' British breeding populations is concentrated in the coastal waters of Scotland, highlighting the need for robust conservation measures in this area. The techniques we present are applicable to any CCPF.

In this paper we introduce two new classes of stationary random simplicial tessellations, the so-called$\\beta$- and$\\beta^{\\prime}$-Delaunay tessellations. Their construction is based on a space–time paraboloid hull process and generalizes that of the classical Poisson–Delaunay tessellation. We explicitly identify the distribution of volume-power-weighted typical cells, establishing thereby a remarkable connection to the classes of$\\beta$- and$\\beta^{\\prime}$-polytopes. These representations are used to determine the principal characteristics of such cells, including volume moments, expected angle sums, and cell intensities.

We consider a network model where small base stations (SBSs) have caching capabilities as a means to alleviate the backhaul load and satisfy users’ demand. The SBSs are stochastically distributed over the plane according to a Poisson point process (PPP) and serve their users either (i) by bringing the content from the Internet through a finite rate backhaul or (ii) by serving them from the local caches. We derive closed-form expressions for the outage probability and the average delivery rate as a function of the signal-to-interference-plus-noise ratio (SINR), SBS density, target file bitrate, storage size, file length, and file popularity. We then analyze the impact of key operating parameters on the system performance. It is shown that a certain outage probability can be achieved either by increasing the number of base stations or the total storage size. Our results and analysis provide key insights into the deployment of cache-enabled small cell networks (SCNs), which are seen as a promising solution for future heterogeneous cellular networks.

Consider the distances R ~ o and R o from the nucleus to a uniformly random point in the 0-cell and the typical cell, respectively, of the d -dimensional Poisson–Voronoi (PV) tessellation. The main objective of this paper is to characterize the exact distributions of R ~ o and R o . First, using the well-known relationship between the 0-cell and the typical cell, we show that the random variable R ~ o is equivalent in distribution to the contact distance of the Poisson point process. Next, we derive a multi-integral expression for the exact distribution of R o . Further, we derive a closed-form approximate expression for the distribution of R o , which is the contact distribution with a mean corrected by a factor equal to the ratio of the mean volumes of the 0-cell and the typical cell. An additional outcome of our analysis is a direct proof of the well-known spherical property of the PV cells having a large inball.

We consider a random geometric graph G(χn, rn), given by connecting two vertices of a Poisson point process χn of intensity n on the d-dimensional unit torus whenever their distance is smaller than the parameter rn . The model is conditioned on the rare event that the number of edges observed, |E|, is greater than (1 + δ)𝔼(|E|), for some fixed δ > 0. This article proves that upon conditioning, with high probability there exists a ball of diameter rn which contains a clique of at least √2δ𝔼(|E|)(1 − ε) vertices, for any given ε > 0. Intuitively, this region contains all the “excess” edges the graph is forced to contain by the conditioning event, up to lower order corrections. As a consequence of this result, we prove a large deviations principle for the upper tail of the edge count of the random geometric graph. The rate function of this large deviation principle turns out to be nonconvex.

Species distribution models (SDMs) are usually based on a single data type, such as presence‐only (PO), presence‐absence (PA) or abundance (AA). Results from SDMs using single sources of data will suffer from inherent biases and limitations to that data type. For example, PO data contain sampling‐bias and PA/AA data are often less expansive and more sparse. Integrated SDMs (ISDMs) combine multiple data types and have recently emerged as a way to leverage strengths and minimise weaknesses of the different data types. They pose a common (distribution) model and separate observation models for each of the data types. The ‘RISDM' package for the R environment (www.r‐project.org) provides access to this modelling framework using functions for preparation, fitting, interpreting and diagnosing models. The functionality of the package is demonstrated here using synthetic data sets.