Example of Bayesian Uncertainty for Digital Soil Mapping

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

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.

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

Analysis of massive marked point patterns with stochastic partial differential equations

<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.

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.

Estuaries are under increasing pressure from urbanization, including through the release of domestic wastes such as anthropogenic debris. Growing concern over environmental contamination has led to management interventions, aimed at reducing the further release of debris. Effective strategies, however, require an understanding of the factors influencing the abundance and distribution of debris. Estuaries pose a challenge to assess debris trends due to the myriad of factors that influence debris patterns. Here we present a Bayesian approach to assess spatial patterns in debris across four estuaries of New South Wales, Australia. We provide an overview of debris composition and employ Integrated Nested Laplace Approximation (INLA) alongside Stochastic Partial Differential Equation (SPDE) to produce three models testing predictors of debris abundance: (i) non spatial, (ii) a model accounting for spatial autocorrelation and (iii) accounting for spatial autocorrelation and introducing the estuarine foreshore as a barrier. Data was sourced from cleanups reported within the Australian Marine Debris Initiative (AMDI), a national citizen science program coordinated by the Tangaroa Blue Foundation. The data, from 169 sampling events between 2012 and 2017 was used. All models tested predictors against the abundance of hard plastic, expanded plastic, plastic bags, and plastic drink bottles. Plastic was the dominant component, ranging from 69% to 82% of total debris within the study estuaries. Important predictors differed across debris types and included ‘distance to estuary mouth’, access to foreshore, population, and site substrate. Specific factors differed in importance when accounting for spatial autocorrelation, demonstrating the need for spatial approaches when assessing debris trends within estuaries. The present Bayesian approach could be used to better understand drivers of estuarine debris and inform interventions for effective management.

Spatiotemporal variation of soil organic matter (SOM) contents was significant to research on global warming, greenhouse effect, and ecosystem health and quality. However, the spatiotemporal modeling for soil properties in most studies focused on a discrete-time calibration and validation, and faced the problems of missing observations. Integrated Nested Laplace Approximation with the Stochastic Partial Differential Equation (INLA-SPDE) model that is robust to missing data and unbalanced sampling design was proposed as a potential model for spatiotemporal soil modeling. This study presented an application of INLA-SPDE for spatiotemporal modeling of SOM (2006, 2010, 2018) using 924 samples and 8 environmental covariates in Lishu County, Northeast China. The results demonstrated that the INLA-SPDE model incorporating spatiotemporal information generally outperformed the two-phase methods based on Cubist and Random Forest, particularly in years requiring greater temporal extrapolation (2006 and 2018), while achieving comparable performance in 2010. This superiority can be attributed to its comprehensive consideration of various sources of uncertainty. Furthermore, the posterior distributions derived from the model provided valuable insights into the effects of environmental covariates on SOM spatiotemporal variation, with clay content showing the strongest positive influence and annual precipitation exhibiting a notable negative effect. The spatial pattern of SOM consistently exhibited higher values in the east and lower values in the west. After an overall decline from 2006 to 2010, mean SOM content increased from 17.97 g kg-1 to 20.85 g kg-1 between 2010 and 2018 (a total increase of 2.88 g kg-1 at an annual rate of 0.36 g kg-1 yr-1), with notable recovery in central and eastern areas, likely associated with the implementation of straw returning practices. In addition to prediction accuracy, computational complexity, and uncertainty analysis, the study evaluated the model from new perspectives, including covariate interpretability and flexibility. This research provides a promising spatiotemporal modeling framework for digital soil mapping.

A number of opportunistic content sharing services were developed that exploit device-to-device contacts for infrastructure-less operation. All of them depend, like geo-based ad-hoc routing protocols, on mobile devices knowing their respective positions to accurately perform data replication. In this paper, we explore the impact of different types of location errors on the performance of such a service. We use a GPS error distribution for mobiles derived from real-world measurements, consider different frequencies for GPS readings, and account for only subsets of mobile devices actively using GPS. We carry out extensive simulation studies using synthetic mobility models as well as real-world traces to assess the impact of different types of errors. We find that, overall, opportunistic content sharing is quite robust provided that a sufficient number of nodes support GPS and allow the others to have a rough estimate of where they are. Whether or not the GPS position is prone to errors affects some scenarios and is almost negligible in others.

In the last two decades, machine learning (ML) methods have been widely used in digital soil mapping (DSM), but the regression kriging (RK) model which combines the advantages of the ML and kriging methods has rarely been used in DSM. In addition, due to the limitation of a single-model structure, many ML methods have poor prediction accuracy in undulating terrain areas. In this study, we collected the SOC content of 115 soil samples in a hilly farming area with continuous undulating terrain. According to the theory of soil-forming factors in pedogenesis, we selected 10 topographic indices, 7 vegetation indices, and 2 soil indices as environmental covariates, and according to the law of geographical similarity, we used ML and RK methods to mine the relationship between SOC and environmental covariates to predict the SOC content. Four ensemble models-random forest (RF), Cubist, stochastic gradient boosting (SGB), and Bayesian regularized neural networks (BRNNs)-were used to fit the trend of SOC content, and the simple kriging (SK) method was used to interpolate the residuals of the ensemble models, and then the SOC and residual were superimposed to obtain the RK prediction result. Moreover, the 115 samples were divided into calibration and validation sets at a ratio of 80%, and the tenfold cross-validation method was used to fit the optimal parameters of the model. From the results of four ensemble models: RF performed best in the calibration set (R2c = 0.834) but poorly in the validation set (R2v = 0.362); Cubist had good accuracy and stability in both the calibration and validation sets (R2c = 0.693 and R2v = 0.445); SGB performed poorly (R2c = 0.430 and R2v = 0.336); and BRNN had the lowest accuracy (R2c = 0.323 and R2v = 0.282). The results showed that the R2 of the four RK models in the validation set were 0.718, 0.674, 0.724, and 0.625, respectively. Compared with the ensemble models without superimposed residuals, the prediction accuracy was improved by 0.356, 0.229, 0.388, and 0.343, respectively. In conclusion, Cubist has high prediction accuracy and generalization ability in areas with complex topography, and the RK model can make full use of trends and spatial structural factors that are not easy to mine by ML models, which can effectively improve the prediction accuracy. This provides a reference for soil survey and digital mapping in complex terrain areas.

Here we present several refinements to a model of feedback control for the suppression of epileptic seizures. We utilize a stochastic partial differential equation (SPDE) model of the human cortex. First, we verify the strong convergence of numerical solutions to this model, paying special attention to the sharp spatial changes that occur at electrode edges. This allows us to choose appropriate step sizes for our simulations; because the spatial step size must be small relative to the size of an electrode in order to resolve its electrical behavior, we are able to include a more detailed electrode profile in the simulation. Then, based on evidence that the mean soma potential is not the variable most closely related to the measurement of a cortical surface electrode, we develop a new model for this. The model is based on the currents flowing in the cortex and is used for a simulation of feedback control. The simulation utilizes a new control algorithm incorporating the total integral of the applied electrical potential. Not only does this succeed in suppressing the seizure-like oscillations, but it guarantees that the applied signal will be charge-balanced and therefore unlikely to cause cortical damage.

Digital soil mapping of soil particle-size fractions (PSFs) using log-ratio methods is a widely used technique. As a hybrid interpolator, regression kriging (RK) provides a way to improve prediction accuracy. However, there have been few comparisons with other techniques when RK is applied for compositional data, and it is not known if its performance based on different balances of isometric log-ratio (ILR) transformation is robust. Here, we compared the generalized linear model (GLM), random forest (RF), and their hybrid patterns (RK) using different transformed data based on three ILR balances, with 29 environmental covariables (ECs) for the prediction of soil PSFs in the upper reaches of the Heihe River Basin (HRB), China. The results showed that the RF performed best, with more accurate predictions, but the GLM produced a more unbiased prediction. As a hybrid interpolator, RK was recommended because it widened the data ranges of the prediction values, and modified the bias and accuracy of most models, especially the RF. The prediction maps generated from RK revealed more details of the soil sampling points than the other models. Different data distributions were produced for the three ILR balances. Using the most abundant component of the compositional data as the first component of the permutations was not considered to be the right choice because it produced the worst performance. Based on the relative abundance of the components, we recommend that the focus should be on data distribution. This study provides a reference for the mapping of soil PSFs combined with transformed data at the regional scale.

To model biochemical systems in which both noise in the chemical reaction process and spatial movement of molecules is important, both the reaction-diffusion master equation (RDME) and Smoluchowski diffusion-limited reaction (SDLR) partial differential equation (PDE) models have been used. In previous work we showed that the solution to the RDME may be interpreted as an asymptotic approximation in the reaction radius to the solution of the SDLR PDE [S. A. Isaacson, SIAM J. Appl. Math. 70, 77 (2009)]. The approximation was shown to be divergent in the limit that the lattice spacing in the RDME approached zero. In this work we expand upon these results for the special case of the two-molecule annihilation reaction, A+B-->Ø. We first introduce a third stochastic reaction-diffusion PDE model that incorporates a pseudopotential based bimolecular reaction mechanism. The solution to the pseudopotential model is then shown to be an asymptotic approximation to the solution of the SDLR PDE for small reaction radii. We next illustrate how the RDME may be obtained by a formal discretization of the pseudopotential model, motivating why the RDME is itself an asymptotic approximation of the SDLR PDE. Finally, we give a more detailed numerical analysis of the difference between solutions to the RDME and SDLR PDE models as a function of both the reaction-radius and the lattice spacing (in the RDME).