Bayesian spatio-temporal inference of trace gas emissions using an integrated nested Laplacian approximation and Gaussian Markov random fields

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.

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.

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.

There is an increasing use of Markov chain Monte Carlo (MCMC) algorithms for fitting statistical models in psychometrics, especially in situations where the traditional estimation techniques are very difficult to apply. One of the disadvantages of using an MCMC algorithm is that it is not straightforward to determine the convergence of the algorithm. Using the output of an MCMC algorithm that has not converged may lead to incorrect inferences on the problem at hand. The convergence is not one to a point, but that of the distribution of a sequence of generated values to another distribution, and hence is not easy to assess; there is no guaranteed diagnostic tool to determine convergence of an MCMC algorithm in general. This article examines the convergence of MCMC algorithms using a number of convergence diagnostics for two real data examples from psychometrics. Findings from this research have the potential to be useful to researchers using the algorithms. For both the examples, the number of iterations required (suggested by the diagnostics) to be reasonably confident that the MCMC algorithm has converged may be larger than what many practitioners consider to be safe.

Numerous spatiotemporal species distribution modeling frameworks are now available to the ecological practitioner. This study compared three such frameworks accessible in the R programming language: generalized additive models with spatiotemporal smooths as implemented by mgcv, spatiotemporal generalized linear mixed models based on nearest neighbor Gaussian processes as implemented by starve, and spatiotemporal generalized linear mixed models based on the stochastic partial differential equations approach as implemented by sdmTMB. The primary focus was to compare the inferences obtained from applying these frameworks to the case study of the orange‐footed sea cucumber, Cucumaria frondosa, on the Scotian Shelf off Nova Scotia, Canada. Each model was fit to catch data (2000–2019) from Fisheries and Oceans Canada's annual Research Vessel and Snow Crab surveys. Environmental covariates were sourced from high‐resolution data layers, including physical oceanographic, bathymetric, and seafloor morphometric datasets. The three models captured variability in sea cucumber distribution that would have been overlooked without a spatiotemporal approach. Although their predictions were similar, including within C. frondosa spatial reserves, the models provided different inferences regarding covariate effects. This suggests that while practitioners primarily interested in mapping species distributions need only apply the most familiar framework, those most concerned with identifying predictive environmental covariates may benefit from comparing the output from multiple approaches. Employing multiple approaches can also serve as a validation technique.

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.

Spatial Poisson models for areal count data use nonstationary “intrinsic autoregressions,” also often referred to as “conditionally autoregressive” (CAR) models. Bayesian inference for these models has generally involved using single parameter updating Markov chain Monte Carlo algorithms, which often exhibit slow mixing (i.e., poor convergence) properties. These spatial models are richly parameterized and lend themselves to the structured Markov chain Monte Carlo (SMCMC) algorithms. SMCMC provides a simple, general, and flexible framework for accelerating convergence in an MCMC sampler by providing a systematic way to block groups of similar parameters while taking full advantage of the posterior correlation structure induced by the model and data. Among the SMCMC strategies considered here are blocking using different size blocks (grouping by geographical region), reparameterization, updating jointly with and without model hyperparameters, “oversampling” some of the model parameters, and “pilot adaptation” versus continuous tuning techniques for the proposal density. We apply the techniques presented here to datasets on cancer mortality and late detection in the state of Minnesota. We find that, compared to univariate sampling procedures, our techniques will typically lead to more accurate posterior estimates, and they are sometimes also far more efficient in terms of the number of effective samples generated per second.

The prevalence of Web 2.0 techniques has led to the boom of various online communities, where topics spread ubiquitously among user-generated documents. Working together with this diffusion process is the evolution of topic content, where novel contents are introduced by documents which adopt the topic. Unlike explicit user behavior (e.g., buying a DVD), both the diffusion paths and the evolutionary process of a topic are implicit, making their discovery challenging. In this paper, we track the evolution of an arbitrary topic and reveal the latent diffusion paths of that topic in a social community. A novel and principled probabilistic model is proposed which casts our task as an joint inference problem, which considers textual documents, social influences, and topic evolution in a unified way. Specifically, a mixture model is introduced to model the generation of text according to the diffusion and the evolution of the topic, while the whole diffusion process is regularized with user-level social influences through a Gaussian Markov Random Field. Experiments on both synthetic data and real world data show that the discovery of topic diffusion and evolution benefits from this joint inference, and the probabilistic model we propose performs significantly better than existing methods.

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.

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

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.

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.

Analysis of massive marked point patterns with stochastic partial differential equations

Image time series can be interpreted on a higher level. We use the information-bottleneck principle to demonstrate our approach.

Change-point detection for discrete time series is an important task with numerous applications. We develop a new hierarchical Bayesian framework for modelling inhomogeneous discrete time series with change-points. The distributions of different segments are modelled as variable-memory Markov chains, defining piece-wise homogeneous variable-memory chains. Building on the recently introduced Bayesian Context Trees framework, it is shown that the Context-Tree Weighting algorithm can be employed to compute the prior predictive likelihood of each segment, with all models and parameters integrated out. This is then used to develop a new class of effective Markov chain Monte Carlo algorithms for the posterior of the number and locations of change-points. These not only identify the most likely change-points, but also provide access to their entire posterior distribution. Estimates of the actual models in each segment can be obtained at negligible cost. Results on both synthetic and real-world data sets indicate that the proposed methodology performs better or as well as state-of-the-art techniques.