# Prediction and Surveillance Sampling Assessment in Plant Nurseries and Fields
## 2 Method
### Implementation of Bayesian Kriging with INLA  
The basic idea of the modeling proposal is to realize that the hierarchical models (1) can be seen as structured additive regression (STAR) models. In other words, they are models in which the mean of the response variable \(Y_{i}\) is linked to a structured predictor which represents the effects of different covariates in an additive way. Specifically, these spatial hierarchical Bayesian models are the so-called latent Gaussian models [4], where Gaussian priors are assigned to all the components of the additive predictor. From this perspective, all latent Gaussian variables can be part of a vector and form the latent Gaussian field. The approach based on integrated nested Laplace approximation (INLA) is a methodology introduced by [4; 15] for statistical inference in latent Gaussian models. INLA provides a quick and efficient method of performing approximations to the marginal posterior density of the hyperparameters \(\hat{\pi}(\theta|y)\) and to the full conditionals of the posterior marginals of the latent variables \(\hat{\pi}(x_{i}|\theta,y)\), \(i=1,\ldots,n\). The key to this new approach of inference is in approximations to the marginal posterior of xi by nested approximations of  
\[\hat{\pi}(x_{i}|y)=\int\hat{\pi}(x_{i}|\theta,y)\hat{\pi}(\theta|y)d\theta \approx\sum_{k=1}^{K}\hat{\pi}(x_{i}|\theta_{k},y)\hat{\pi}(\theta_{k}|y)\Delta _{k} \tag{5}\]  
where \(\hat{\pi}(\cdot|\cdot)\) is an approximate conditional density. The approximations of (5) are calculated by approximations \(\pi(\theta|y)\) y \(\pi(x_{i}|\theta,y)\) using numerical integration (finite sum) on \(\theta\). The marginal posterior for the hyperparameters \(\hat{\pi}(\theta_{i}|y)\), \(j=1,\ldots,m\) are determined similarly. The inference is based on the approximation \(\pi(\theta|y)\) of the marginal posterior of \(\theta\):  
\[\tilde{\pi}(\theta|y)\propto\frac{\pi(x,\theta,y)}{\hat{\pi}_{G}(x|\theta,y) }|_{x=x^{*}(\theta)} \tag{6}\]  
where \(\hat{\pi}_{G}(x|\theta,y)\) is the Gaussian approximation of the full conditionals \(x\), and \(x^{*}(\theta)\) is the mode of the full conditional of \(x\) for a given \(\theta\). The sign of proportionality is due to the fact that the normalization constant for \(\pi(x,\theta|Y)\) is unknown. This expression is equivalent to a Laplace approximation [16] and this suggests that the approximation error is relative and of order \(O(n^{-3/2})\) after renormalization. Note that \(\tilde{\pi}(\theta|y)\) tends to stray too far from Gaussianity; therefore, this approach determines approximations of \(\tilde{\pi}(\theta|y)\) and \(\tilde{\pi}(x_{i}|y)\) in a non-parametric way. The main tool for inference is in the application of the Laplace approximation to \(\pi(x_{i}|\theta,y)\). The Matern covariance function appears naturally in various scientific fields [17]. However, Ref. [9] establish an approximation between the Gaussian field and the Matern covariance function using a weak stochastic approximation to a stochastic partial differential equation (SPDE) as follows:  
\[(\kappa^{2}-\Delta)^{\alpha/2}x(u)=W(u),\ u\in\Re^{d} \tag{7}\] \[\alpha=\
u+d/2,\ \kappa>0,\ \
u>0\]  
where \((\kappa^{2}-\Delta)^{\alpha/2}\) is a pseudo-differential operator. \(W\) has a Gaussian distribution of white noise with unitary variance; \(\Delta\) is the Laplacian  
\[\Delta=\sum_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}} \tag{8}\]and marginal variance  
\[\sigma^{2}=\frac{\Gamma(\
u)}{\Gamma(\
u+d/2)(4\pi)^{d/2}\kappa^{2\
u}} \tag{9}\]  
Hereinafter, any solution of the form (7) is called a Matern field. The solutions limit the SPDE approach when \(\kappa\to 0\) or \(\
u\rightarrow\)0 do not have Matern covariance functions. However, there is a solution when \(\kappa=0\) or \(\
u=0\) whether the random measures are well defined. When \(\alpha\geq 2\), the null space of the differential operator is not trivial and contains, for example, the functions \(\exp(\kappa e^{T}u)\) for all \(\|e\|=1\).