## 3 Methodology
### Prediction Models
Thus, decomposing \(Z(\mathbf{x})\) into a mean \(m(\mathbf{x})\) plus residual \(R(\mathbf{x})\) component; \(R(\mathbf{x})\) is modelled as a stationary random function with \(\mathrm{E}\{R(\mathbf{x})\}=0\) and covariogram \(\mathrm{COV}\{R(\mathbf{x}),R(\mathbf{x}+\mathbf{h})\}=\mathrm{E}\{R(\mathbf{x }),R(\mathbf{x}+\mathbf{h})\}=C_{R}(\mathbf{h})\), where \(\mathbf{h}\) is the separation distance vector, \(\mathbf{h}=\mathbf{x}_{\alpha}-\mathbf{x}_{\beta}\). For details, see, for example, [PERSON] [55]. For KED, \(m(\mathbf{x})\) is modelled as an unknown using the linear function \(m(\mathbf{x})=\sum_{k=0}^{K}\chi_{k}\,y_{k}(\mathbf{x})\) (_i.e._, a MLR fit with \(y_{0}(\mathbf{x})=1\)), where it is filtered from the linear predictor by using constraints to give: \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{\begin{tabular}{c} **Are Spatial Effects** \\ **Modelled?** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **How are Spatial Relationships** \\ **Modelled?** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Is Spatial Autocorrelation** \\ **Accounted for?** \\ \end{tabular} } \\ \hline MLR & No & Stationary & No \\ KED-GN & Yes & Stationary & Yes \\ GWR & Yes & Non-stationary & No \\ GWRK & Yes & Non-stationary & Yes \\ KED-LN & Yes & Non-stationary & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Core properties of this study’s prediction models. \[\hat{z}_{{}_{KED}}\big{(}\mathbf{x}\big{)}\!=\!\sum_{\alpha=1}^{n}\lambda_{\alpha }^{KED}\big{(}\mathbf{x}\big{)}z\big{(}\mathbf{x}_{\alpha}\big{)} \tag{4}\]  
as a KED prediction at \(\mathbf{x}\). If \(m(\mathbf{x})\) is modelled as an unknown constant, then an ordinary kriging (OK) prediction at \(\mathbf{x}\) results: \(\hat{z}_{{}_{OK}}(\mathbf{x})=\sum_{\alpha=1}^{n}\lambda_{\alpha}^{OK}\ (\mathbf{x})z\big{(}\mathbf{x}_{\alpha}\big{)}\). The GWRK predictor at a target location \(\mathbf{x}\) is the sum of the GWR prediction from Equation (3) and the OK prediction of the residual:  
\[\hat{z}_{{}_{GWRK}}\big{(}\mathbf{x}\big{)}\!=\!\hat{z}_{{}_{GWR}}\big{(} \mathbf{x}\big{)}\!+\hat{r}_{\alpha K}\big{(}\mathbf{x}\big{)} \tag{5}\]  
GWRK is an explicit model where the mean and residual processes are dealt with separately in a two-stage procedure. KED deals with the mean and residual processes in an implicit fashion where all model equations are solved at once. Kriging is always defined with the residual covariogram \(\mathcal{C}_{R}(\mathbf{h})\), but when the mean is taken as some constant with OK, only the raw data covariogram \(\mathcal{C}(\mathbf{h})\) is needed. Furthermore, and as is common practice, we find variograms \(\gamma(\mathbf{h})\) instead of covariograms (where \(\gamma(\mathbf{h})=\mathcal{C}(0)-\mathcal{C}(\mathbf{h})\) is used to relate the two). Restricted maximum likelihood (REML) is used to find relatively unbiased variogram parameters for KED. Parameterization via REML is not suited to GWRK and as such, the residual variogram \(\gamma_{R}(\mathbf{h})\) from the GWR fit is first estimated with the usual classical estimator (e.g., [56], pp.153-154) and then modelled using a WLS fitting approach. After some initial experimentation, only exponential variogram model-types are considered, _i.e._, \(\gamma(h)=c_{0}+c_{1}(1-\exp(-h/a))\); which caters for a (small-scale) nugget variance \(c_{0}\); a (large-scale) structural variance \(c_{1}\) (where \(c_{0}+c_{1}=\sigma^{2}\)); and a correlation range \(\alpha\). All forms of kriging can be applied using local prediction neighbourhoods as opposed to a unique, global prediction neighbourhood; an approximation that can account for local mean fluctuations and ease computational burden. In this study, KED is not only applied in its correct and unbiased form using a global neighbourhood (KED-GN), but also applied using local neighbourhoods (KED-LN), as this usefully caters for relationships that may vary across space (as now the MLR component fit is also local).