# Prediction and Surveillance Sampling Assessment in Plant Nurseries and Fields
###### Abstract  
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. Bayesian kriging; Bayesian hierarchical models; Gaussian Markov random field (GMRF); integrated nested Laplace approximation (INLA); stochastic partial differential equation (SPDE)  
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Generalized mixed linear models have enjoyed a growing popularity due to their ability to model correlated observations. Their application range can go beyond the popular generalized linear models, but this involves more complex and difficult calculations. Various inference procedures have been proposed, among them the Bayesian approach through Markov chain Monte Carlo methods. Bayesian hierarchical geostatistical models can be seen as a particular case of STAR models, where we assign Gaussian priors to all components of the additive predictor. In this case, we have latent Gaussian models [4], where variables are part of a latent Gaussian field. Gaussian fields play a dominant role in spatial statistics and in the geostatistical field [5; 6; 7] and are an important component in current spatial hierarchical models [1], since they provide one appropriate multivariate model with an explicit normalization constant and with good analytical properties. The Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial dependence and spatio-temporal dependence, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph-based models [8]. The increasing popularity of Bayesian hierarchical models has made this situation very important because of the need to perform repeated simulations for model fitting, which may be impractical. Various methodologies have been proposed to address these limitations. Specifically, in this paper, we use the approach proposed by [9], in which a Gaussian field is approximated to a Gaussian Markov random field [4; 10; 11]. The main advantage of this representation is the remarkable improvement in calculation times and reduction in numerical difficulties associated with the analysis of generalized mixed linear models since the Gaussian Markov random fields are defined on sparse matrices and not on dense arrays. This work describes how to use the integrated nested Laplace approximation (INLA) with the stochastic partial differential equation (SPDE) approach via Bayesian hierarchical models. In addition, we illustrate the proposed methodology in an agricultural context and discuss the advantages of this methodology when choosing samples from a large population.