$\hat{r}(\bm{a})$, using the root mean-square error (RMSE), namely the interval $$\begin{aligned}
  C = \left(\hat{r}(\bm{a}) - \Phi^{-1}(1 - \alpha / 2) \hat{\sigma}(\bm{a}), \hat{r}(\bm{a}) + \Phi^{-1}(1 - \alpha / 2) \hat{\sigma}(\bm{a})\right)
  \label{eq:confidence_interval}\end{aligned}$$ where $\Phi$ is the CDF for the univariate normal distribution and $\alpha$ is the critical value. If there are no errors associated with the training data then the RMSE of the estimator is zero at the training points and increases as the distance of the test point from a training point increases. If the errors on the sample are constant, then we expect the RMSE to be approximately constant (except at the boundaries where it will be greater on account of boundary bias).

Validating the emulator {#sec:validating_the_emulator}
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The machinery of Gaussian-process regression assumes that the covariance function, $k$, is known. In practice, it never is. We must chose an approximation to the covariance, invariably from a list of standard covariance functions, as discussed above. We would like to know that this covariance function has been well chosen, i.e. we would like to assess the performance of the emulator given our choice of covariance function. We may do this using leave-one-out cross-validation (LOOCV) [@wasserman_all_2007]. We note that the estimator (equation 