7.4 Kriging

Having obtained estimates of $\beta$, $\sigma$, $\varphi$ and $\tau$, we can proceed to the prediction step. Suppose we wish to predict $S$ at some new locations, denoted $\tilde{S}$, we do this by computing $𝔼(\tilde{S}|Y)$ and $𝕍(\tilde{S}|Y)$. Suppose the length of $\tilde{S}$ is $m$ i.e. we are interested in predicting the spatial process at $m$ new locations in space. The $(m+n)$-column vector, ${[\tilde{S}\mathit{\hspace{1em}}Y]}^T$, is jointly multivariate Gaussian. For given $\beta$, $\sigma$, $\tau$ and $\varphi$

$$\mathrm{cov}(\tilde{S},Y)=\mathrm{cov}(\tilde{S},Z\beta +S+\epsilon )=\mathrm{cov}(\tilde{S},S)={\mathrm{\Psi }}_{\tilde{S},S},$$

so the joint density function of ${[\tilde{S}\mathit{\hspace{1em}}Y]}^T$ is,

In the above ${\mathrm{\Psi }}_{\tilde{S},S}$ is available directly by evaluating the covariance function ${\sigma }^2\rho (\parallel {\tilde{x}}_i-x_j\parallel )$ between all prediction locations ${\tilde{x}}_i$ and all data locations $x_j$ and similarly $\tilde{\mathrm{\Sigma }}$ is computed as ${\sigma }^2\rho (\parallel {\tilde{x}}_i-{\tilde{x}}_j\parallel )$. The conditional density of interest, $\pi (\tilde{S}|Y)$, is also Gaussian with mean and variance available using standard results:

	$𝔼(\tilde{S}|Y)$	$=$	${\mathrm{\Psi }}_{\tilde{S},S}{[{\mathrm{\Sigma }}_{\sigma ,\varphi }+{\tau }^{2}I]}^{-1}(Y-Z\beta )$
	$𝕍(\tilde{S}|Y)$	$=$	$\tilde{\mathrm{\Sigma }}-{\mathrm{\Psi }}_{\tilde{S},S}{[{\mathrm{\Sigma }}_{\sigma ,\varphi }+{\tau }^{2}I]}^{-1}{\mathrm{\Psi }}_{\tilde{S},S}^T$

An example of the prediction surface for the magnesium example is shown in Figure 7.5. Sometimes only the diagonal entries of $𝕍(\tilde{S}|Y)$ are returned by computer software, but minimally, this enables us to compute pointwise standard errors.