Let \(Z\) be a real Gaussian space-time process with zero mean and full-rank separable covariance function:

\[
\operatorname{cov}\left(Z(x, t), Z\left(x^{\prime}, t^{\prime}ight)ight)=K\left(x, x^{\prime}ight) V\left(t, t^{\prime}ight)
\]

Let \(\mathbf{x}=\left\{x_{1}, \ldots, x_{n}ight\}\) be a spatial configuration, \(\mathbf{t}=\left\{t_{1}, \ldots, t_{k}ight\}\) a temporal configuration, and let \(Z[\mathbf{x} \times \mathbf{t}]\) be the values on the Cartesian product set. Show that the conditional expected value of \(Z\left(x_{0}, t_{0}ight)\) given \(Z[\mathbf{x} \times \mathbf{t}]\) satisfies

\[
E\left(Z\left(x_{0}, t_{0}ight) \mid Z[\mathbf{x} \times \mathbf{t}]ight)=\sum_{i j} \sum r s K\left(x_{0}, x_{i}ight) V\left(t_{0}, t_{r}ight) K[\mathbf{x}]_{i j}^{-1} V[\mathbf{t}]_{r s}^{-1} Z\left(x_{j}, t_{s}ight)
\]

Show also that if the prediction site belongs to \(\mathbf{x}\), say \(x_{0}=x_{1}\), the conditional expectation reduces to the linear combination

\[
E\left(Z\left(x_{0}, t_{0}ight) \mid Z[\mathbf{x} \times \mathbf{t}]ight)=\sum_{r s} V\left(t_{0}, t_{r}ight) V[\mathbf{t}]_{r s}^{-1} Z\left(x_{0}, t_{s}ight)
\]

depending only on the values at \(\left(x_{0}, \mathbf{t}ight)\). In other words, if the model is separable, the spatial covariance is irrelevant for temporal prediction.