Rasmus Plenge Waagepetersen

Statistical inference and simulation for spatial point processes

EXAMPLES OF SPATIAL POINT PATTERNS INTRODUCTION TO POINT PROCESSES Point Processes on R^d Marked Point Processes and Multivariate Point Processes Unified Framework Space-Time Processes POISSON POINT PROCESSES Basic Properties Further Results Marked Poisson Processes SUMMARY STATISTICS First and Second Order Properties Summary Statistics Nonparametric Estimation Summary Statistics for Multivariate Point Processes Summary Statistics for Marked Point Processes COX PROCESSES Definition and Simple Examples Basic Properties Neyman-Scott Processes as Cox Processes Shot Noise Cox Processes Approximate Simulation of SNCPs Log Gaussian Cox Processes Simulation of Gaussian Fields and LGCPs Multivariate Cox Processes MARKOV POINT PROCESSES Finite Point Processes with a Density Pairwise Interaction Point Processes Markov Point Processes Extensions of Markov Point Processes to R^d Inhomogeneous Markov Point Processes Marked and Multivariate Markov Point Processes METROPOLIS-HASTINGS ALGORITHMS Description of Algorithms Background Material for Markov Chains Convergence Properties of Algorithms SIMULATION-BASED INFERENCE Monte Carlo Methods and Output Analysis Estimation of Ratios of Normalising Constants Approximate Likelihood Inference Using MCMC Monte Carlo Error Distribution of Estimates and Hypothesis Tests Approximate MissingData Likelihoods INFERENCE FOR MARKOV POINT PROCESSES Maximum Likelihood Inference Pseudo Likelihood Bayesian Inference INFERENCE FOR COX PROCESSES Minimum Contrast Estimation Conditional Simulation and Prediction Maximum Likelihood Inference Bayesian Inference BIRTH-DEATH PROCESSES AND PERFECT SIMULATION Spatial Birth-Death Processes Perfect Simulation APPENDICES History, Bibliography, and Software Measure Theoretical Details Moment Measures and Palm Distributions Perfect Simulation of SNCPs Simulation of Gaussian Fields Nearest-Neighbour Markov Point Processes Results for Spatial Birth-Death Processes References Subject Index Notation Index

Log Gaussian Cox processes

Planar Cox processes directed by a log Gaussian intensity process are investigated in the univariate and multivariate cases. The appealing properties of such models are demonstrated theoretically as well as through data examples and simulations. In particular, the first, second and third-order properties are studied and utilized in the statistical analysis of clustered point patterns. Also empirical Bayesian inference for the underlying intensity surface is considered.

Non-and semi-parametric estimation of interaction in inhomogeneous point patterns

We develop methods for analysing the ‘interaction’ or dependence between points in a spatial point pattern, when the pattern is spatially inhomogeneous. Completely non‐parametric study of interactions is possible using an analogue of the K‐function. Alternatively one may assume a semi‐parametric model in which a (parametrically specified) homogeneous Markov point process is subjected to (non‐parametric) inhomogeneous independent thinning. The effectiveness of these approaches is tested on datasets representing the positions of trees in forests.

Hierarchical spatial point process analysis for a plant community with high biodiversity

A complex multivariate spatial point pattern of a plant community with high biodiversity is modelled using a hierarchical multivariate point process model. In the model, interactions between plants with different post-fire regeneration strategies are of key interest. We consider initially a maximum likelihood approach to inference where problems arise due to unknown interaction radii for the plants. We next demonstrate that a Bayesian approach provides a flexible framework for incorporating prior information concerning the interaction radii. From an ecological perspective, we are able both to confirm existing knowledge on species’ interactions and to generate new biological questions and hypotheses on species’ interactions.

Selection for Environmental Variation : A Statistical Analysis and Power Calculations to Detect Response

Data from uterine capacity in rabbits (litter size) were analyzed to determine whether the environmental variance was partly genetically determined. The fit of a classical homogeneous variance mixed linear (HOM) model and that of a genetically structured heterogeneous variance mixed linear (HET) model were compared. Various methods to assess the quality of fit favor the HET model. The posterior mean (95% posterior interval) of the additive genetic variance affecting the environmental variance was 0.16 (0.10; 0.25) and the corresponding number for the coefficient of correlation between genes affecting mean and variance was −0.74 (−0.90;−0.52). It is argued that stronger support for the HET model than that derived from statistical analysis of data would be provided by a successful selection experiment designed to modify the environmental variance. A simple selection criterion is suggested (average squared deviation from the mean of repeated records within individuals) and its predicted response and variance under the HET model are derived. This is used to determine the appropriate size and length of a selection experiment designed to change the environmental variance. Results from the analytical expressions are compared with those obtained using simulation. There is good agreement provided selection intensity is not intense.

Packing densities and simulated tempering for hard core Gibbs point processes

Monotonicity and convergence properties of the intensity of hard core Gibbs point processes are investigated and compared to the closest packing density. For such processes simulated tempering is shown to be an efficient alternative to commonly used Markov chain Monte Carlo algorithms. Various spatial characteristics of the pure hard core process are studied based on samples obtained with the simulated tempering algorithm.

Geometric ergodicity of Metropolis-Hastings algorithms for conditional simulation in generalized linear mixed models

Conditional simulation is useful in connection with inference and prediction for a generalized linear mixed model. We consider random walk Metropolis and Langevin-Hastings algorithms for simulating the random effects given the observed data, when the joint distribution of the unobserved random effects is multivariate Gaussian. In particular we study the desirable property of geometric ergodicity, which ensures the validity of central limit theorems for Monte Carlo estimates.

Quantifying effects of habitat heterogeneity and other clustering processes on spatial distributions of tree species

Spatially explicit consideration of species distribution can significantly add to our understanding of species coexistence. In this paper, we evaluated the relative importance of habitat heterogeneity and other clustering processes (e.g., dispersal limitation, collectively called the non-habitat clustering process) in explaining the spatial distribution patterns of 341 tree species in three stem-mapped 25-50 ha plots of tropical, subtropical, and temperate forests. Their relative importance was estimated by a method that can take one mechanism into account when estimating the effects of the other mechanism and vice versa. Our results demonstrated that habitat heterogeneity was less important in explaining the observed species patterns than other clustering processes in plots with flat topography but was more important in one of the three plots that had a complex topography. Meanwhile, both types of clustering mechanisms (habitat or non-habitat) were pervasive among species at the 50-ha scale across the studied plots. Our analyses also revealed considerable variation among species in the relative importance of the two types of mechanism within each plot and showed that this species-level variation can be partially explained by differences in dispersal mode and growth form of species in a highly heterogeneous environment. Our findings provide new perspectives on the formation of species clustering. One important finding is that a significant species-habitat association does not necessarily mean that the habitat heterogeneity has a decisive influence on species distribution. The second insight is that the large species-level variation in the relative importance of the two types of clustering mechanisms should not be ignored. Non-habitat clustering processes can play an important role on species distribution.

A comparison of strategies for Markov chain Monte Carlo computation in quantitative genetics.

In quantitative genetics, Markov chain Monte Carlo (MCMC) methods are indispensable for statistical inference in non-standard models like generalized linear models with genetic random effects or models with genetically structured variance heterogeneity. A particular challenge for MCMC applications in quantitative genetics is to obtain efficient updates of the high-dimensional vectors of genetic random effects and the associated covariance parameters. We discuss various strategies to approach this problem including reparameterization, Langevin-Hastings updates, and updates based on normal approximations. The methods are compared in applications to Bayesian inference for three data sets using a model with genetically structured variance heterogeneity.

An Introduction to Simulation-Based Inference for Spatial Point Processes

Spatial point processes play a fundamental role in spatial statistics. In the simplest case they model “small” objects that may be identified by a map of points showing stores, towns, plants, nests, or cases of a disease observed in a two dimensional region or galaxies observed in a three dimensional region. The points may be decorated with marks (such as sizes or types) whereby marked point processes are obtained. The areas of applications are manifold: astronomy, geography, ecology, forestry, spatial epidemiology, image analysis, and many more. Currently spatial point processes is an active area of research, which probably will be of increasing importance for many new applications.