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This tutorial provides an introduction to Palm distributions for spatial point processes. Initially, in the context of finite point processes, we give an explicit definition of Palm distributions in terms of their density functions. Then we review Palm distributions in the general case. Finally, we discuss some examples of Palm distributions for specific models and some applications.

Geostatistics involves the fitting of spatially continuous models to spatially discrete data. Preferential sampling arises when the process that determines the data locations and the process being modelled are stochastically dependent. Conventional geostatistical methods assume, if only implicitly, that sampling is non-preferential. However, these methods are often used in situations where sampling is likely to be preferential. For example, in mineral exploration, samples may be concentrated in areas that are thought likely to yield high grade ore. We give a general expression for the likelihood function of preferentially sampled geostatistical data and describe how this can be evaluated approximately by using Monte Carlo methods. We present a model for preferential sampling and demonstrate through simulated examples that ignoring preferential sampling can lead to misleading inferences. We describe an application of the model to a set of biomonitoring data from Galicia, northern Spain, in which making allowance for preferential sampling materially changes the results of the analysis.

ABSTRACT The study of the spatial point patterns in ecology, such as the records of the observed locations of trees, shrubs, nests, burrows, or documented animal presence, relies on multivariate point process models. This study aims to compare the efficacy and applicability of two prominent multivariate point process models, the multivariate log Gaussian Cox process (MLGCP), and the saturated pairwise interaction Gibbs point process model (SPIGPP), highlighting their respective strengths and weaknesses when prior knowledge of the underlying mechanisms driving the patterns is lacking. Using synthetic and real datasets, we assessed both models based on their predictive accuracy of the empirical K function. Our analysis revealed that both MLGCP and SPIGPP effectively identify and capture mild to moderate clustering and regulations. MLGCP struggles to capture repulsive associations as they intensify. In contrast, SPIGPP can well estimate both the direction and magnitude of interactions even when the model is misspecified. Both models present unique advantages: MLGCP is particularly effective when there is a need to account for complex, unobserved heterogeneities that vary across space, while SPIGPP is suitable when interactions between points are the primary focus. The choice between these models should be guided by the specific needs of the research question and data characteristics. This study is the first to systematically compare the multivariate log‐Gaussian Cox process and the saturated pairwise interaction Gibbs point process, providing key insights into their application in ecological modelling. Our findings offer practical guidance for researchers, helping them select and apply these models effectively.

Integrated distribution models (IDMs) predict where species might occur using data from multiple sources, a technique thought to be especially useful when data from any individual source are scarce. Recent advances allow us to fit such models with latent terms to account for dependence within and between data sources, but they are computationally challenging to fit. We propose a fast new methodology for fitting integrated distribution models using presence/absence and presence‐only data, via a spatial random effects approach combined with automatic differentiation. We have written an R package (called scampr ) for straightforward implementation of our approach. We use simulation to demonstrate that our approach has comparable performance to INLA —a common framework for fitting IDMs—but with computation times up to an order of magnitude faster. We also use simulation to look at when IDMs can be expected to outperform models fitted to a single data source, and find that the amount of benefit gained from using an IDM is a function of the relative amount of additional information available from incorporating a second data source into the model. We apply our method to predict 29 plant species in NSW, Australia, and find particular benefit in predictive performance when data from a single source are scarce and when compared to models for presence‐only data. Our faster methods of fitting IDMs make it feasible to more deeply explore the model space (e.g. comparing different ways to model latent terms), and in future work, to consider extensions to more complex models, for example the multi‐species setting.

Multivariate log-Gaussian Cox processes are flexible models for multivariate point patterns. However, they have so far been applied in bivariate cases only. We move beyond the bivariate case to model multispecies point patterns of tree locations. In particular we address the problems of identifying parsimonious models and of extracting biologically relevant information from the models fitted. The latent multivariate Gaussian field is decomposed into components given in terms of random fields common to all species and components which are species specific. This allows a decomposition of variance that can be used to quantify to what extent the spatial variation of a species is governed by common or species-specific factors. Cross-validation is used to select the number of common latent fields to obtain a suitable trade-off between parsimony and fit of the data. The selected number of common latent fields provides an index of complexity of the multivariate covariance structure. Hierarchical clustering is used to identify groups of species with similar patterns of dependence on the common latent fields.

We consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of 'structured' anisotropy : the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.

Modelling spatial and temporal patterns in ecology is imperative to understand the complex processes inherent in ecological phenomena. Log‐Gaussian Cox processes are a popular choice among ecologists to describe the spatiotemporal distribution of point‐referenced data. In addition, point pattern models where events instigate others nearby (i.e., self‐exciting behaviour) are becoming increasingly popular to infer the contagious nature of events (e.g., animal sightings). While there are existing R packages that facilitate fitting spatiotemporal point processes and, separately, self‐exciting models, none incorporate both. We present an R package, stelfi, that fits spatiotemporal self‐exciting and log‐Gaussian Cox process models using Template Model Builder through a range of custom‐written C++ templates. We illustrate the use of stelfi's functions fitting models to Sasquatch (bigfoot) sightings data within the USA. The structure of these data is typical of many seen in ecology studies. We show, from a temporal Hawkes process to a spatiotemporal self‐exciting model, how the models offered by the package enable additional insights into the temporal and spatial progression of point pattern data. We present extensions to these well‐known models that include spatiotemporal self‐excitation and joint likelihood models, which are better suited to capture the complex mechanisms inherent in many ecological data. The package stelfi offers user‐friendly functionality, is open source, and is available from CRAN. It offers the implementation of complex spatiotemporal point process models in R for applications even beyond the field of ecology. We introduce the R package stelfi, available from the Comprehensive R Archive Network. This package allows users to fit temporal self‐exciting Hawkes models, spatial and spatiotemporal log‐Gaussian Cox process models and self‐exciting spatiotemporal models. The functionality of stelfi is illustrated using Sasquatch (bigfoot) sightings data shipped with the package.

We propose a new summary statistic for inhomogeneous intensity-reweighted moment stationarity spatio-temporal point processes. The statistic is defined in terms of the n-point correlation functions of the point process, and it generalizes the J-function when stationarity is assumed. We show that our statistic can be represented in terms of the generating functional and that it is related to the spatio-temporal K-function. We further discuss its explicit form under some specific model assumptions and derive ratio-unbiased estimators. We finally illustrate the use of our statistic in practice.

The paper is concerned with parameter estimation for inhomogeneous spatial point processes with a regression model for the intensity function and tractable second-order properties (K-function). Regression parameters are estimated by using a Poisson likelihood score estimating function and in the second step minimum contrast estimation is applied for the residual clustering parameters. Asymptotic normality of parameter estimates is established under certain mixing conditions and we exemplify how the results may be applied in ecological studies of rainforests.

This paper studies two new models for a telegraph process: Cox-based and elliptical telegraph processes. The paper deals with the stochastic motion of a particle on a straight line and on an ellipse with random positive velocity and two opposite directions of motion, which is governed by a telegraph–Cox switching process. A relevant result of our analysis on the straight line is obtaining a linear Volterra integral equation of the first kind for the characteristic function of the probability density function (PDF) of the particle position at a given time. We also generalize Kac’s condition for the telegraph process to the case of a telegraph–Cox switching process. We show some examples of random velocity where the distribution of the coordinate of a particle is expressed explicitly. In addition, we present some novel results related to the switched movement evolution of a particle according to a telegraph–Cox process on an ellipse. Numerical examples and applications are presented for a telegraph–Cox-based process (option pricing formulas) and elliptical telegraph process.