Mapping Escolar ( Lepidocybium flavobrunneum ) in Motion: Oceanographic Forces Shaping Its Habitat in the Southwestern South Atlantic, With Insights From Fishers' Perceptions

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.

Visceral Leishmaniasis is a very dangerous form of leishmaniasis and, shorn of appropriate diagnosis and handling, it leads to death and physical disability. Depicting the spatiotemporal pattern of disease is important for disease regulator and deterrence strategies. Spatiotemporal modeling has distended broad veneration in recent years. Spatial and spatiotemporal disease modeling is extensively used for the analysis of registry data and usually articulated in a hierarchical Bayesian framework. In this study, we have developed the hierarchical spatiotemporal Bayesian modeling of the infected rate of Visceral leishmaniasis in Human (VLH). We applied the Stochastics Partial Differential Equation (SPDE) approach for a spatiotemporal hierarchical model for Visceral leishmaniasis in human (VLH) that involves a GF and a state process is associated with an autoregressive order one temporal dynamics and the spatially correlated error term, along with the effect of land shield, metrological, demographic, socio-demographic and geographical covariates in an endemic area of Amhara regional state, Ethiopia. The model encompasses a Gaussian Field (GF), affected by an error term, and a state process described by a first-order autoregressive dynamic model and spatially correlated innovations. A hierarchical model including spatially and temporally correlated errors was fit to the infected rate of Visceral leishmaniasis in human (VLH) weekly data from January 2015 to December 2017 using the R package R-INLA, which allows for Bayesian modeling using the stochastic partial differential equation (SPDE) approach. We found that the mean weekly temperature had a significant positive association with infected rate of VLH. Moreover, net migration rate, clean water coverage, average number of households, population density per square kilometer, average number of persons per household unit, education coverage, health facility coverage, mortality rate, and sex ratio had a significant association with the infected rate of visceral leishmaniasis (VLH) in the region. In this study, we investigated the dynamic spatiotemporal modeling of Visceral leishmaniasis in Human (VLH) through a stochastic partial differential equation approach (SPDE) using integrated nested Laplace approximation (INLA). Our study had confirmed both metrological, demographic, sociodemographic and geographic covariates had a significant association with the infected rate of visceral leishmaniasis (VLH) in the region.

Gridded observational datasets of the main climate variables are essential in climate science. However, common interpolation approaches (e.g., classical kriging-based methods), often lack of a proper propagation and representation of uncertainty. In this study, a Bayesian spatio-temporal regression model based on the Integrated Nested Laplace Approximation (INLA) and the Stochastic Partial Differential Equation (SPDE) is introduced. Although the effective use of INLA and SPDE is documented in several envitonmental studies, their use among climate practitioners is still quite limited. Here, based on high-resolution monthly 2-meter maximum (Tmax) and minimum (Tmin) air temperature, we employ INLA and SPDE to derive gridded monthly temperature climatologies for Italy both for the most recent standard 30-year period (1991–2020) and three previous standard periods (1961-1990, 1971-2000, 1981-2010). Our regression model includes three spatial predictors (elevation, latitude and distance to sea) and a linear time effect accounting for the  temporal trend in the observed monthly temperatures. A Matern field is used to capture the residual spatio-temporal correlation. Because of the large space-time domain of our study, the regression analysis is run separately for each month (January-December) and for each variable (Tmax/Tmin). Despite its simplicity, this approach provides a flexible model to produce accurate continuous gridded surfaces equipped with model-based uncertainties. Through simulation, we generate  a distribution of plausible gridded surfaces of Tmax and Tmin monthly means, which we summarize through measures of central tendency (posterior mean)  and variability (standard deviation). We use the standard deviation maps to investigate how uncertainty affects our estimates of the 1991-2020 monthly climatologies and where, and the 95% credible intervals maps to assess the regions where the  1991-2020 period is significantly warmer than the previous 30-year standard periods.    

In this work, we consider a hierarchical spatio-temporal model for particulate matter (PM) concentration in the North-Italian region Piemonte. The model involves a Gaussian Field (GF), affected by a measurement error, and a state process characterized by a first order autoregressive dynamic model and spatially correlated innovations. This kind of model is well discussed and widely used in the air quality literature thanks to its flexibility in modelling the effect of relevant covariates (i.e. meteorological and geographical variables) as well as time and space dependence. However, Bayesian inference—through Markov chain Monte Carlo (MCMC) techniques—can be a challenge due to convergence problems and heavy computational loads. In particular, the computational issue refers to the infeasibility of linear algebra operations involving the big dense covariance matrices which occur when large spatio-temporal datasets are present. The main goal of this work is to present an effective estimating and spatial prediction strategy for the considered spatio-temporal model. This proposal consists in representing a GF with Matern covariance function as a Gaussian Markov Random Field (GMRF) through the Stochastic Partial Differential Equations (SPDE) approach. The main advantage of moving from a GF to a GMRF stems from the good computational properties that the latter enjoys. In fact, GMRFs are defined by sparse matrices that allow for computationally effective numerical methods. Moreover, when dealing with Bayesian inference for GMRFs, it is possible to adopt the Integrated Nested Laplace Approximation (INLA) algorithm as an alternative to MCMC methods giving rise to additional computational advantages. The implementation of the SPDE approach through the R-library INLA ( www.r-inla.org ) is illustrated with reference to the Piemonte PM data. In particular, providing the step-by-step R-code, we show how it is easy to get prediction and probability of exceedance maps in a reasonable computing time.

The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field u on R d as the solution of an elliptic SPDE L β u = W where L is a second-order differential operator, 2 β ∈ N is a positive parameter that controls the smoothness of u and W is Gaussian white noise. A few approaches have been suggested in the literature to extend the approach to allow for any smoothness parameter satisfying β > d / 4 . Even though those approaches work well for simulating SPDEs with general smoothness, they are less suitable for Bayesian inference since they do not provide approximations which are Gaussian Markov random fields (GMRFs) as in the original SPDE approach. We address this issue by proposing a new method based on approximating the covariance operator L − 2 β of the Gaussian field u by a finite element method combined with a rational approximation of the fractional power. This results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference. A rigorous convergence analysis of the method is performed and the accuracy of the method is investigated with simulated data. Finally, we illustrate the approach and corresponding implementation in the R package rSPDE via an application to precipitation data which is analyzed by combining the rSPDE package with the R-INLA software for full Bayesian inference. Supplementary materials for this article are available online.

Bayesian spatial modelling of geostatistical data using INLA and SPDE methods: A case study predicting malaria risk in Mozambique

Spatial modelling of hydrothermal mineralization-related geochemical patterns using INLA+SPDE and local singularity analysis

Spatial species distribution models: Using Bayes inference with INLA and SPDE to improve the tree species choice for important European tree species

This study proposes a Bayesian hierarchical model to analyze spatiotemporal patterns of extreme rainfall frequencies across Denmark, with a focus on understanding the underlying factors driving these patterns. Covariates and spatial information are incorporated through a generalized additive modeling framework. To handle the spatial correlation in the data, the model employs a Markovian representation of the Matérn covariance function via the stochastic partial differential equations (SPDE) approach. The model fits within the framework of latent Gaussian models, enabling efficient inference using the Integrated Nested Laplace Approximation (INLA). Results indicate that extreme rainfall frequencies are clustered in space, and there are both meteorological factors (i.e., atmospheric water content, vertical instability) and topographical features (i.e., distance to open sea) that explain a significant part of the spatial variability. Our method explicitly models the spatiotemporal correlation structure inherent in extreme rainfall data while remaining computationally efficient, improving our understanding of extreme rainfall patterns.

The ever‐growing popularity of citizen science, as well as recent technological and digital developments, have allowed the collection of data on species' distributions at an extraordinary rate. In order to take advantage of these data, information of varying quantity and quality needs to be integrated. Point process models have been proposed as an elegant way to achieve this for estimates of species distributions. These models can be fitted efficiently using Bayesian methods based on integrated nested Laplace approximations (INLA) with stochastic partial differential equations (SPDEs). This approach uses an efficient way to model spatial autocorrelation using a Gaussian random field and a triangular mesh over the spatial domain. The mesh is constructed by user‐defined variables, so effectively represents a free parameter in the model. However, there is a lack of understanding about how to set these mesh parameters, and their effect on model performance. Here, we assess how mesh parameters affect predictions and model fit to estimate the distribution of the serotine bat, Eptesicus serotinus , in Great Britain. A Bayesian INLA model was fitted using five meshes of varying densities to a dataset comprising both structured observations from a national monitoring programme and opportunistic records. We demonstrate that mesh density impacted spatial predictions with a general loss of accuracy with increasing mesh coarseness. However, we also show that the finest mesh was unable to overcome spatial biases in the data. In addition, the magnitude of the covariate effects differed markedly between meshes. This confirms that mesh parameterisation is an important and delicate process with implications for model inference. We discuss how species distribution modellers might adapt their use of INLA in the light of these findings.

Most previous studies have reported a decrease in global tropical cyclone (TC) genesis frequency (TCGF) under anthropogenic warming. However, little attention has been drawn to the influence of sea surface temperature (SST) warming patterns on TCGF changes. Here, we investigate the impacts of three distinct SST warming patterns on global TCGF: uniform SST warming, nonuniform (El Niño-like) SST warming, and a combination of both. Results show that spatio-uniform SST warming has a limited impact on global TCGF, instead redistributing the TC genesis locations. Conversely, nonuniform SST warming significantly suppresses global TCGF. The combined warming produces a similar decrease in TCGF to nonuniform warming albeit with differences in spatial distribution. This indicates the dominant role of nonuniform SST warming in affecting TCGF and highlights the nonlinearity of the process. Further analysis shows that these differences in TCGF primarily stem from the distinct responses of tropical circulations to the three warming patterns.

<p>The assumption of spatial and temporal stationarity does not hold for many ecological and environmental processes. This is particularly the case for many soil processes like carbon sequestration, often driven by factors such as biological dynamics, climate change and anthropogenic influences. For better understanding and predicting such phenomena, we develop a Bayesian inference framework that combines the integrated nested Laplace approximation (INLA) with the stochastic partial differential equation approach (SPDE). We put focus on modeling complex temporal trends varying through space with an accurate assessment of uncertainties, and on spatio-temporal mapping of processes that are only partially observed.</p><p>We model observed data through a latent (i.e., unobserved) smooth process whose additive components are endowed with Gaussian process priors. We use the SPDE approach to implement flexible sparse-matrix approximations of the Matérn covariance for spatial fields. The separate specification of the spatially varying linear trend allows us to conduct component-specific statistical inferences (range and variance estimates, standard errors, confidence bounds), and to provide maps to stakeholders for time-invariant spatial patterns, spatial patterns in slopes of time trends, and the associated uncertainties. For observed data following a Gaussian distribution, we add independent measurement errors, but more general response distributions of the data can be implemented. We also include in our model covariate information on parent material, climate and seasonality.</p><p>The INLA method and its implementation in the R-INLA library provide a rich toolbox for statistical space-time modelling while sidestepping typical convergence problems arising with simulation-based techniques using Markov Chain Monte–Carlo codes for large and complex hierarchical models such as ours. Uncertainties arising in model parameters and in pointwise spatio-temporal predictions are naturally captured in the posterior distributions computed through INLA using appropriate approximation techniques, and we can communicate on them through maps of various properties. Moreover, INLA also allows for direct simulation from the estimated posterior model, such that we can conduct statistical inferences on more complex functionals of the multivariate predictive distributions by analogy with MCMC frameworks.</p><p>Soil organic carbon is a major compartment of the global carbon cycle and small variations of its level can largely impact atmospheric CO<sub>2</sub> concentrations. In the context of global climate change, it is important to be able to quantify and explain spatial and temporal variability of SOC in order to forecast future changes. In this work, we used this approach to study possible trends in space and time of soil carbon stock of three agricultural fields in France. Fitted models reveal significant temporal trends with strong spatial heterogeneity. The Matérn model and SPDE approach provide a flexible framework with respect to field design.</p>

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.

Modeling wildfire activity is crucial for informing science-based risk management and understanding the spatiotemporal dynamics of fire-prone ecosystems worldwide. Models help disentangle the relative influences of different factors, understand wildfire predictability, and provide insights into specific events. Here, we develop Firelihood, a two-component, Bayesian, hierarchically structured, probabilistic model of daily fire activity, which is modeled as the outcome of a marked point process: individual fires are the points (occurrence component), and fire sizes are the marks (size component). The space-time Poisson model for occurrence is adjusted to gridded fire counts using the integrated nested Laplace approximation (INLA) combined with the stochastic partial differential equation (SPDE) approach. The size model is based on piecewise-estimated Pareto and generalized Pareto distributions, adjusted with INLA. The Fire Weather Index (FWI) and forest area are the main explanatory variables. Temporal and spatial residuals are included to improve the consistency of the relationship between weather and fire occurrence. The posterior distribution of the Bayesian model provided 1,000 replications of fire activity that were compared with observations at various temporal and spatial scales in Mediterranean France. The number of fires larger than 1ha across the region was coarsely reproduced at the daily scale, and was more accurately predicted on a weekly basis or longer. The regional weekly total number of larger fires (10-100ha) was predicted as well, but the accuracy degraded with size, as the model uncertainty increased with event rareness. Local predictions of fire numbers or burned areas also required a longer aggregation period to maintain model accuracy. The estimation of fires larger than 1ha was also consistent with observations during the extreme fire season of the 2003 unprecedented heat wave, but the model systematically underrepresented large fires and burned areas, which suggests that the FWI does not consistently rate the actual danger of large fire occurrence during heat waves. Firelihood enabled a novel analysis of the stochasticity underlying fire hazard, and offers a variety of applications, including fire hazard predictions for management and projections in the context of climate change.

Spatial models with explanatory variables in the dependence structure