# Prediction and Surveillance Sampling Assessment in Plant Nurseries and Fields
## 2 Method
### Research Object  
The utility of the proposed methodology is illustrated through a dataset obtained from a nursery, formed of 10,920 _Citrus macrophylla_ saplings. The 10,920 saplings are distributed in 40 rows of 273 saplings each. The saplings are placed on 20 mounds. Each mound is compound of two rows of saplings. The distance between any two saplings in the same row is between 15 and 18 cm; however, the final distance considered between every two saplings is the midpoint between 15 and 18 cm, i.e., 16.5 cm. Furthermore, the distance between two rows within a mound is 40 cm, and the distance between two adjacent rows of a different mound is 70 cm. The analysis was performed on 10,920 saplings in search of the _Citrus Tristeza virus_ (CTV; Family: _Closteroviridae_; Genus: _Closterovirus_). Figure 1 shows the distribution of the disease throughout the nursery. There is a total of 443 diseased saplings (red dots), representing an infection rate of 4.05%. The nursery is seen as a continuous region in which any sapling may become diseased with the virus at any point given the proximity of the saplings. This consideration can be due to the large number of saplings planted and to the low proportion of saplings infected with the citrus tristeza virus. Figure 1 only represents the triangulation underlying the SPDE approach that we have applied and which is based Bayesian kriging. Each vertex of the mesh is an observed point or a point prediction, with the red dots indicating infected saplings and black dots representing uninfected saplings. By applying the model defined in Section 2.2 over the observed dataset, we obtained the estimation of the posterior parameters of interest shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Mean** & **S.D.** & \(Q_{0.025}\) & \(Q_{0.50}\) & \(Q_{0.975}\) \\ \hline \(\beta_{0}\) (Intercept) & \(-3.13\) & \(0.31\) & \(-3.77\) & \(-3.14\) & \(-2.45\) \\ \(\kappa\) & \(0.00217\) & \(0.0063\) & \(0.00074\) & \(0.0019\) & \(0.0049\) \\ \(\tau\) (cm) & \(3.55\) & \(0.0153\) & \(1.198\) & \(3.12\) & \(8.38\) \\ \hline \hline \end{tabular} Q = Quantile; S.D. = standard desviation.  
\end{table} 
Table 1: Posterior distribution of the parameters using the complete dataset. Figure 1: Sampling locations for the presence and absence of CTV in the nursery—mesh. According to the definition of \(\phi\), we find that the range is equal to 1302.869 or approximately about 13 cm. Since this is the distance where the correlation is close to 0.10, it can be inferred that data are characterized by a strong correlation for distances less than or equal to 13 cm. Therefore, we can conclude that correlation decreases after this distance. Clearly the presence of the disease is determined by a spatial effect. In particular, infection occurs between plants located in the same row over distances of less than or equal to 13 cm in any of the mounds. Figure 1(a) shows the posterior mean of spatial effect \(W_{i}\). We observe how the spatial component reaches positive values to the north and south of the nursery, as well as negative values, and achieves values close to zero in the center. In this map, we may recognize the areas with higher risks. This can be explained by the action of the wind that introduces the aphids infected with the virus to the nursery and which carry the disease. The variance in random spatial effect \(\sigma_{W}^{2}\) is equal to 0.13. As the variability takes a small value, we can conclude that the spatial component reflects the pattern of contagion between close saplings located in the same row. In order to understand the behavior of disease in the nursery, maps were generated with an estimate of the probabilities \((\tau_{i}|Y)\) both at observed sites as well as unobserved sites. Figure 1(b) shows the posterior mean probability \(\tau_{i}|Y\), while Figure 2(a),b shows the quartiles of \(\pi_{i}|Y\). Thus, we obtain not only a point estimate of the probability of the disease of a subject but also an assessment of the uncertainty in this estimate.