Analysis of massive marked point patterns with stochastic partial differential equations

Modeling spatial and spatio-temporal continuous processes is an important and challenging problem in spatial statistics. Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA describes in detail the stochastic partial differential equations (SPDE) approach for modeling continuous spatial processes with a Matern covariance, which has been implemented using the integrated nested Laplace approximation (INLA) in the R-INLA package. Key concepts about modeling spatial processes and the SPDE approach are explained with examples using simulated data and real applications. This book has been authored by leading experts in spatial statistics, including the main developers of the INLA and SPDE methodologies and the R-INLA package. It also includes a wide range of applications: * Spatial and spatio-temporal models for continuous outcomes * Analysis of spatial and spatio-temporal point patterns * Coregionalization spatial and spatio-temporal models * Measurement error spatial models * Modeling preferential sampling * Spatial and spatio-temporal models with physical barriers * Survival analysis with spatial effects * Dynamic space-time regression * Spatial and spatio-temporal models for extremes * Hurdle models with spatial effects * Penalized Complexity priors for spatial models All the examples in the book are fully reproducible. Further information about this book, as well as the R code and datasets used, is available from the book website at http://www.r-inla.org/spde-book. The tools described in this book will be useful to researchers in many fields such as biostatistics, spatial statistics, environmental sciences, epidemiology, ecology and others. Graduate and Ph.D. students will also find this book and associated files a valuable resource to learn INLA and the SPDE approach for spatial modeling.

Spatial pattern in prevalence of paratuberculosis infection diagnosed with misclassification in Danish dairy herds in 2009 and 2013

In this paper, we propose a structured additive regression (STAR) model for modeling the occurrence of a disease in fields or nurseries. The methodological approach involves a Gaussian field (GF) affected by a spatial process represented by an approximation to a Gaussian Markov random field (GMRF). This modeling allows the building of maps with prediction probabilities regarding the presence of a disease in plants using Bayesian kriging. The advantage of this modeling is its computational benefit when compared with known spatial hierarchical models and with the Bayesian inference based on Markov chain Monte Carlo (MCMC) methods. Inference through the use of the integrated nested Laplace approximation (INLA) with the stochastic partial differential equation (SPDE) approach facilitates the handling of large datasets in excellent computation times. Our approach allows the evaluation of different sampling strategies, from which we obtain inferences and prediction maps with similar behaviour to those obtained when we consider all subjects in the study population. The analysis of the different sampling strategies allows us to recognize the relevance of spatial components in the studied phenomenon. We demonstrate how Bayesian kriging can incorporate sources of uncertainty associated with the prediction parameters, which leads to more realistic and accurate estimation of the uncertainty. We illustrate the methodology with samplings of Citrus macrophylla affected by the tristeza virus (CTV) grown in a nursery.

A methodological approach for modelling the occurrence patterns of species for the purpose of fisheries management is proposed here. The presence/absence of the species is modelled with a hierarchical Bayesian spatial model using the geographical and environmental characteristics of each fishing location. Maps of predicted probabilities of presence are generated using Bayesian kriging. Bayesian inference on the parameters and prediction of presence/absence in new locations (Bayesian kriging) are made by considering the model as a latent Gaussian model, which allows the use of the integrated nested Laplace approximation ( INLA ) software (which has been seen to be quite a bit faster than the well-known MCMC methods). In particular, the spatial effect has been implemented with the stochastic partial differential equation (SPDE) approach. The methodology is evaluated on Mediterranean horse mackerel (Trachurus mediterraneus) in the Western Mediterranean. The analysis shows that environmental and geographical factors can play an important role in directing local distribution and variability in the occurrence of species. Although this approach is used to recognize the habitat of mackerel, it could also be for other different species and life stages in order to improve knowledge of fish populations and communities.

This paper presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in geostatistics. We show a general approach to construct stationary models related to a wide class of linear SPDEs, with applications to spatio-temporal models having non-trivial properties. Within the framework of Generalized Random Fields, a criterion for existence and uniqueness of stationary solutions for this class of SPDEs is proposed and proven. Their covariance are then obtained through their spectral measure. We present a result relating the covariance of the solution in the case of a White Noise source term with the covariance in a generic case through convolution. Then, we obtain a variety of SPDE-based stationary random fields. In particular, well-known results regarding the Matérn Model and Markovian models are recovered. A new relationship between the Stein model and a particular SPDE is obtained. New spatio-temporal models obtained from evolution SPDEs of arbitrary temporal derivative order are then obtained, for which properties of separability and symmetry can be controlled. We also obtain results concerning stationary solutions for physically inspired models, such as solutions to the heat equation, the advection-diffusion equation, some Langevin’s equations and the wave equation.

Any model for digital soil mapping suffers from different types of errors, including interpolation errors, so it is important to quantify the uncertainty associated with the maps produced. The most common approach is some form of regression kriging (RK) or variation involving geostatistical simulation. Another way of assessing the spatial uncertainty lies in the Bayesian approach where the uncertainty in the results is described by the posterior density. The aim of this paper is to present an example of a Bayesian approach for uncertainty estimation when mapping the topsoil organic matter content in the Grampian region of Scotland (UK, about 12,100 km2). The chosen approach uses (Bayesian) latent Gaussian models fitted using integrated nested Laplace approximation (INLA) and the stochastic partial differential equation (SPDE) models approach for coping with spatial correlation (INLA_SPDE). For practical comparison purposes, the results of INLA_SPDE were compared with the results of an extension of the scorpan kriging approach, i.e., (1) combining generalized additive models (GAM) with Gaussian simulations and (2) traditional RK. The results were assessed using in-sample and out-of-sample measures and compared for distribution similarity, spatial structure reproduction, computational load, and uncertainty ranges. We conclude that the Bayesian framework using INLA offers a viable alternative to existing methods and an improvement over traditional RK.

Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysis with three-dimensional spatial smoothing priors has been shown to produce state-of-the-art activity maps without pre-smoothing the data. The proposed inference algorithms are computationally demanding however, and the spatial priors used have several less appealing properties, such as being improper and having infinite spatial range. We propose a statistical inference framework for whole-brain fMRI analysis based on the class of Matérn covariance functions. The framework uses the Gaussian Markov random field (GMRF) representation of possibly anisotropic spatial Matérn fields via the stochastic partial differential equation (SPDE) approach of Lindgren et al. (2011). This allows for more flexible and interpretable spatial priors, while maintaining the sparsity required for fast inference in the high-dimensional whole-brain setting. We develop an accelerated stochastic gradient descent (SGD) optimization algorithm for empirical Bayes (EB) inference of the spatial hyperparameters. Conditionally on the inferred hyperparameters, we make a fully Bayesian treatment of the brain activity. The Matérn prior is applied to both simulated and experimental task-fMRI data and clearly demonstrates that it is a more reasonable choice than the previously used priors, using comparisons of activity maps, prior simulation and cross-validation.

This article will focus its discussion on accounting for the complex design when analyzing survey data, and on spatial modeling. With respect to the former, this article considers two approaches: direct use of the weights, and a model-based approach using a spatial model to acknowledge clustering. With respect to spatial modeling this article describes, the stochastic partial differential equations (SPDEs; Lindgren et al. 2011) approach to modeling.

Continuous technological growth and the corresponding environmental implications are triggering the enhancement of advanced environmental monitoring solutions, such as remote sensing. In this paper, we propose a new method for the spatial point patterns generation by classifying remote sensing images using convolutional neural network. To increase the accuracy, the training samples are extended by the suggested data augmentation scheme based on the similarities of images within the same part of the landscape for a limited observation time. The image patches are classified in accordance with the labels of previously classified images of the manually prepared training and test samples. This approach has improved the accuracy of image classification by 7% compared to current best practices of data augmentation. A set of image patch centers of a particular class is considered as a random point configuration, while the class labels are used as marks for every point. A marked point pattern is regarded as a combination of several subpoint patterns with the same qualitative marks. We analyze the bivariate point pattern to identify the relationships between points of different types using the features of a marked random point pattern.

Bayesian spatial modelling of soil properties and their uncertainty: The example of soil organic matter in Scotland using R-INLA

Two spatial data sets are considered to be similar if they originate from the same stochastic process in terms of their spatial structure. Many tests have been developed over recent years to test the similarity of certain types of spatial data, such as spatial point patterns, geostatistical data and images. This research proposes a generic spatial similarity test able to handle various types of spatial data, for example images (modelled spatially), point patterns, marked point patterns, geostatistical data and lattice patterns. A simulation study is done in order to test the method for each spatial data set. After the simulation study, it was concluded that the proposed spatial similarity test is not sensitive to the user-defined resolution of the pixel image representation. From the simulation study, the proposed spatial similarity test performs well on lattice data, some of the unmarked point patterns and the marked point patterns with discrete marks. We illustrate this test on property prices in the City of Cape Town and the City of Johannesburg, South Africa.

Basic methods of explorative statistical analysis for stationary and isotropic planar point processes are briefly and informally reviewed. At the explorative level, planar point patterns may be characterized in terms of the intensity, the K-function and the pair correlation function. These second-order functions enable one to classify a given point process as completely random, clustering or repulsive. The repulsive behaviour may be quantified by an estimate of the hard-core distance. In the exploratory approach, the statistics are essentially free from model assumptions. Second-order spatial functions have been estimated to characterize genuine planar point processes in the macroscopic domain, for example in forestry, geography and epidemiology. For light microscopy and transmission electron microscopy, two situations are distinguished, which may be summarized as the genuine planar case and the stereological case. In the genuine planar case, a direct interpretation of the results of spatial statistics is feasible. Here, monolayers in cell culture, intramembranous particles on freeze fracture specimens and amacrine cells of the retina are mentioned as examples. In the stereological case, point patterns are generated by sections through 3D structures. Here the observed point patterns may arise as the centres of sectional profiles of particles, or as centres of sectional profiles of spatial fibre processes. In both situations, exploratory spatial point process statistics allow a quantitative characterization of sectional images for the purposes of group comparisons and classification. Moreover, for spatial fibre processes it has recently been shown that the observed pair correlation function of the centres of the fibre profiles is an estimate of the reduced pair correlation function of the fibre process in 3D. Hence for fibre processes a stereological interpretation of point process statistics obtained from sections is an additional option.

Seabirds are facing increasing threats in both marine and terrestrial habitats, and many populations have experienced dramatic declines over past decades. Fisheries bycatch is the most pervasive at-sea threat and is of increasing concern in fisheries management and marine conservation. We predicted spatial and temporal heterogeneities of seabird bycatch probability in the US Atlantic pelagic longline fishery (PLL) through an interactive Barrier model based on observer data from the National Marine Fisheries Service Pelagic Observer Program. The Barrier model prevents bias caused by physical barriers such as coastlines by defining the spatial correlation function as a collection of paths between points and eliminating any paths across physical barriers. The integrated nested Laplace approximations methodology and stochastic partial differential equations approach were applied to fit the model, greatly reducing execution time. Seabird bycatch had a hotspot of high bycatch probability in the mid-Atlantic bight in most years, and the hotspot varied in presence and location yearly. The inter-annual variations in bycatch hotspot are correlated with Gulf Stream meanders. Special area and time fishing restrictions predicted by relationships with Gulf Stream positions might enable the US Atlantic PLL to avoid peak areas and periods of seabird bycatch and thereby support seabird conservation.

<title>Abstract</title> Species distribution models (SDMs) are vital tools in ecology and conservation. The integration of increasingly available citizen science data with planned survey data offers a significant opportunity to enhance species distribution estimates. While integrated SDMs often combine presence-only and abundance data, the interdependence between the conditional distributions of these outcomes remains to be elucidated. This study proposes a Bayesian spatial fusion modelling framework to jointly analyse presence-only and abundance data for the African baobab in Benin. The aim was to understand and map the spatial variation in the species’ distribution. We briefly reviewed process-based models for count and point process data and explored various data fusion strategies using Integrated Nested Laplace Approximations (INLA) and Stochastic Partial Differential Equations (SPDE) for inference. The results revealed a heterogeneous baobab distribution across Benin, characterised by a spatial autocorrelation range of 34.4 km (95% Bayesian credible interval, BCI = 27.59-42.52). Key drivers of this distribution include environmental factors such as annual temperature, rainfall of the driest month, soil texture (silt/clay fractions), and slope. A spatial fusion model incorporating shared latent components and common covariates' effects demonstrated the highest performance level, surpassing alternative fusion approaches. The model achieved the highest mean composite scores for the Area Under the ROC Curve (AUC) (0.85±0.02), accuracy (0.77±0.02), and True Skill Statistics (TSS) (0.66±0.05). The shared component model has the capacity to explain datasets' interdependence, estimate covariate effects missed by separate models, and enhance prediction precision. Despite relying on the assumption of an identically shared spatial signal across target responses, this research underscores the potential of spatial fusion modelling for integrating disparate data sources. The findings contribute to advancing SDM inference, particularly in data-limited contexts, and have wider applicability to spatial regression problems involving multisource outcomes.

Spatiotemporal modelling of soil organic matter changes in Jiangsu, China between 1980 and 2006 using INLA-SPDE