Space-time Multi Type Log Gaussian Cox Processes with a View to Modelling Weeds

Log Gaussian Cox processes as introduced in Moller et al. (1998) are extended to space-time models called log Gaussian Cox birth processes. These processes allow modelling of spatial and temporal heterogeneity in time series of increasing point processes consisting of different types of points. The models are shown to be easy to analyse yet flexible enough for a detailed statistical analysis of a particular agricultural experiment concerning the development of two weed species on an organic barley field. Particularly, the aspects of estimation, model validation and intensity surface prediction are discussed.     

Spatiotemporal prediction for log-Gaussian Cox processes

Space–time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems. We describe a flexible class of space–time point processes. Our models are Cox processes whose stochastic intensity is a space–time Ornstein–Uhlenbeck process. We develop moment-based methods of parameter estimation, show how to predict the underlying intensity by using a Markov chain Monte Carlo approach and illustrate the performance of our methods on a synthetic data set.     

Residual analysis for spatial point processes (with discussion)

Summary.  We define residuals for point process models fitted to spatial point pattern data, and we propose diagnostic plots based on them. The residuals apply to any point process model that has a conditional intensity; the model may exhibit spatial heterogeneity, interpoint interaction and dependence on spatial covariates. Some existing ad hoc methods for model checking (quadrat counts, scan statistic, kernel smoothed intensity and Berman's diagnostic) are recovered as special cases. Diagnostic tools are developed systematically, by using an analogy between our spatial residuals and the usual residuals for (non-spatial) generalized linear models. The conditional intensity λ plays the role of the mean response. This makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context, giving recommendations for diagnostic plots. A plot of smoothed residuals against spatial location, or against a spatial covariate, is effective in diagnosing spatial trend or co-variate effects. Q – Q -plots of the residuals are effective in diagnosing interpoint interaction.     

Two-step estimation for inhomogeneous spatial point processes

Summary.  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.     

Geometric Anisotropic Spatial Point Pattern Analysis and Cox Processes

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.     

Decomposition of Variance for Spatial Cox Processes

Abstract. Spatial Cox point processes is a natural framework for quantifying the various sources of variation governing the spatial distribution of rain forest trees. We introduce a general criterion for variance decomposition for spatial Cox processes and apply it to specific Cox process models with additive or log linear random intensity functions. We moreover consider a new and flexible class of pair correlation function models given in terms of normal variance mixture covariance functions. The proposed methodology is applied to point pattern data sets of locations of tropical rain forest trees.     

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

We consider the problem of parameter estimation for inhomogeneous space‐time shot‐noise Cox point processes. We explore the possibility of using a stepwise estimation method and dimensionality‐reducing techniques to estimate different parts of the model separately. We discuss the estimation method using projection processes and propose a refined method that avoids projection to the temporal domain. This remedies the main flaw of the method using projection processes – possible overlapping in the projection process of clusters, which are clearly separated in the original space‐time process. This issue is more prominent in the temporal projection process where the amount of information lost by projection is higher than in the spatial projection process. For the refined method, we derive consistency and asymptotic normality results under the increasing domain asymptotics and appropriate moment and mixing assumptions. We also present a simulation study that suggests that cluster overlapping is successfully overcome by the refined method.     

Practical Maximum Pseudolikelihood for Spatial Point Patterns(with Discussion)

This paper describes a technique for computing approximate maximum pseudolikelihood estimates of the parameters of a spatial point process. The method is an extension of Berman & Turner's (1992) device for maximizing the likelihoods of inhomogeneous spatial Poisson processes. For a very wide class of spatial point process models the likelihood is intractable, while the pseudolikelihood is known explicitly, except for the computation of an integral over the sampling region. Approximation of this integral by a finite sum in a special way yields an approximate pseudolikelihood which is formally equivalent to the (weighted) likelihood of a loglinear model with Poisson responses. This can be maximized using standard statistical software for generalized linear or additive models, provided the conditional intensity of the process takes an 'exponential family' form. Using this approach a wide variety of spatial point process models of Gibbs type can be fitted rapidly, incorporating spatial trends, interaction between points, dependence on spatial covariates, and mark information.     

Modern Statistics for Spatial Point Processes*

Abstract. We summarize and discuss the current state of spatial point process theory and directions for future research, making an analogy with generalized linear models and random effect models, and illustrating the theory with various examples of applications. In particular, we consider Poisson, Gibbs and Cox process models, diagnostic tools and model checking, Markov chain Monte Carlo algorithms, computational methods for likelihood-based inference, and quick non-likelihood approaches to inference.     

A Spatio-Temporal Model for Functional Magnetic Resonance Imaging Data – with a View to Resting State Networks

Abstract.  Functional magnetic resonance imaging (fMRI) is a technique for studying the active human brain. During the fMRI experiment, a sequence of MR images is obtained, where the brain is represented as a set of voxels. The data obtained are a realization of a complex spatio-temporal process with many sources of variation, both biological and technical. We present a spatio-temporal point process model approach for fMRI data where the temporal and spatial activation are modelled simultaneously. It is possible to analyse other characteristics of the data than just the locations of active brain regions, such as the interaction between the active regions. We discuss both classical statistical inference and Bayesian inference in the model. We analyse simulated data without repeated stimuli both for location of the activated regions and for interactions between the activated regions. An example of analysis of fMRI data, using this approach, is presented.     

A practical approach for assessing the effect of grouping in hierarchical spatio-temporal models

Hierarchical spatio-temporal models allow for the consideration and estimation of many sources of variability. A general spatio-temporal model can be written as the sum of a spatio-temporal trend and a spatio-temporal random effect. When spatial locations are considered to be homogeneous with respect to some exogenous features, the groups of locations may share a common spatial domain. Differences between groups can be highlighted both in the large-scale, spatio-temporal component and in the spatio-temporal dependence structure. When these differences are not included in the model specification, model performance and spatio-temporal predictions may be weak. This paper proposes a method for evaluating and comparing models that progressively include group differences. Hierarchical modeling under a Bayesian perspective is followed, allowing flexible models and the statistical assessment of results based on posterior predictive distributions. This procedure is applied to tropospheric ozone data in the Italian Emilia–Romagna region for 2001, where 30 monitoring sites are classified according to environmental laws into two groups by their relative position with respect to traffic emissions.     

Conditional intensity: A powerful tool for modelling and analysing point process data

The conditional intensity function of a spatial point process describes how the probability that a point of the process occurs ‘at’ a particular point in its carrier space depends on the realisation of the process in the remainder of the carrier space. Provided that the point process is simple, the conditional intensity determines all of the properties of the process, in particular its likelihood function. In this paper, we review the use of the conditional intensity function in the formulation of point process models and in making inferences from point process data, giving separate consideration to temporal, spatial and spatiotemporal settings. We argue that the conditional intensity function should take centre-stage in spatiotemporal point process modelling and analysis.     

A Non-Gaussian Spatial Process Model for Opacity of Flocculated Paper

ABSTRACT.  Product quality in the paper-making industry can be assessed by opacity of a linear trace through continuous production sheets, summarized in spectral form. We adopt a class of non-Gaussian stochastic models for continuous spatial variation to describe data of this type. The model has flexible covariance structure, physically interpretable parameters and allows several scales of variation for the underlying process. We derive the spectral properties of the model, and develop methods of parameter estimation based on maximum likelihood in the frequency domain. The methods are illustrated using sample data from a UK mill.     

A Stochastic Geometry Model for Functional Magnetic Resonance Images

In functional magnetic resonance imaging, spatial activation patterns are commonly estimated using a non-parametric smoothing approach. Significant peaks or clusters in the smoothed image are subsequently identified by testing the null hypothesis of lack of activation in every volume element of the scans. A weakness of this approach is the lack of a model for the activation pattern; this makes it difficult to determine the variance of estimates, to test specific neuroscientific hypotheses or to incorporate prior information about the brain area under study in the analysis. These issues may be addressed by formulating explicit spatial models for the activation and using simulation methods for inference. We present one such approach, based on a marked point process prior. Informally, one may think of the points as centres of activation, and the marks as parameters describing the shape and area of the surrounding cluster. We present an MCMC algorithm for making inference in the model and compare the approach with a traditional non-parametric method, using both simulated and visual stimulation data. Finally we discuss extensions of the model and the inferential framework to account for non-stationary responses and spatio-temporal correlation.     

A hierarchical point process with application to storm cell modelling

In environmetrics, interest often centres around the development of models and methods for making inference on observed point patterns assumed to be generated by latent spatial or spatio‐temporal processes, which may have a hierarchical structure. In this research, motivated by the analysis of spatio‐temporal storm cell data, we generalize the Neyman–Scott parent–child process to account for hierarchical clustering. This is accomplished by allowing the parents to follow a log‐Gaussian Cox process thereby incorporating correlation and facilitating inference at all levels of the hierarchy. This approach is applied to monthly storm cell data from the Bismarck, North Dakota radar station from April through August 2003 and we compare these results to simpler cluster processes to demonstrate the advantages of accounting for both levels of correlation present in these hierarchically clustered point patterns. The Canadian Journal of Statistics 47: 46–64; 2019 © 2019 Statistical Society of Canada     

Mean and Variance of Vacancy for Hard-Core Disc Processes and Applications

Abstract.  Hard-core Strauss disc processes with inhibition distance r and disc radius R are considered. The points in the Strauss point process are thought of as trees and the discs as crowns. Formulas for the mean and the variance of the vacancy (non-covered area) are derived. This is done both for the case of a fixed number of points and for the case of a random number of points. For tractability, the region is assumed to be a torus or, in one dimension, a circle in which case the discs are segments. In the one-dimensional case, the formulas are exact for all r . This case, although less important in practice than the two-dimensional case, has provided a lot of inspiration. In the two-dimensional case, the formulas are only approximate but rather accurate for r  <  R . Markov Chain Monte Carlo simulations confirm that they work well. For R  ≤  r  < 2 R , no formulas are presented. A forestry estimation problem, which has motivated the research, is briefly considered as well as another application in spatial statistics.     

Improving Ratio Estimators of Second Order Point Process Characteristics

Ripley's K function, the L function and the pair correlation function are important second order characteristics of spatial point processes. These functions are usually estimated by ratio estimators, where the numerators are Horvitz–Thompson edge corrected estimators and the denominators estimate the intensity or its square. It is possible to improve these estimators with respect to bias and estimation variance by means of adapted distance dependent intensity estimators. Further improvement is possible by means of refined estimators of the square of intensity. All this is shown by statistical analysis of simulated Poisson, cluster and hard core processes.     

Invariance principles and almost sure central limit theorem for the error variance estimator in linear models

A number of score statistics are derived for a heterogeneous spatial Poisson process which has a composite intensity. The intensity consists of a 'background' process which is estimated

from a control point process by kernel density estimation. The parametric form of the composite intensity yields score tests for particular spatial effects. A numerical example concerning respiratory cancer mortality is given.     

Modelling spatial intensity for replicated inhomogeneous point patterns in brain imaging

Summary.  Pharmacological experiments in brain microscopy study patterns of cellular activ- ation in response to psychotropic drugs for connected neuroanatomic regions. A typical ex- perimental design produces replicated point patterns having highly complex spatial variability. Modelling this variability hierarchically can enhance the inference for comparing treatments. We propose a semiparametric formulation that combines the robustness of a nonparametric kernel method with the efficiency of likelihood-based parameter estimation. In the convenient framework of a generalized linear mixed model, we decompose pattern variation by kriging the intensities of a hierarchically heterogeneous spatial point process. This approximation entails discretizing the inhomogeneous Poisson likelihood by Voronoi tiling of augmented point patterns. The resulting intensity-weighted log-linear model accommodates spatial smoothing through a reduced rank penalized linear spline. To correct for anatomic distortion between subjects, we interpolate point locations via an isomorphic mapping so that smoothing occurs relative to common neuroanatomical atlas co-ordinates. We propose a criterion for choosing the degree and spatial locale of smoothing based on truncating the ordered set of smoothing covariates to minimize residual extra-dispersion. Additional spatial covariates, experimental design factors, hierarchical random effects and intensity functions are readily accommodated in the linear predictor, enabling comprehensive analyses of the salient properties underlying replicated point patterns. We illustrate our method through application to data from a novel study of drug effects on neuronal activation patterns in the brain of rats.