\mathbf{y}_*}$ is strictly diagonal.

Reference methods
-----------------

The GPR point estimates are compared with a state-of-the-art kNN algorithm. We select ten predictors from the (transformed) data using a simulated annealing -based optimization approach of [@Packalen2012] and use the most similar neighbor (MSN) method for selecting the neighbors. The number of neighbors is chosen to be $k=5$, as in [@packalen2009; @Packalen2012]. The predictor selection is done using the whole data set and leave-one-out cross-validation.

The prediction credible intervals provided by GPR are compared with the Bayesian inference approach [@varvia]. In the Bayesian approach the posterior predictive density: $$\label{alsextposterior}
\pi(\mathbf{y}_*|\mathbf{x}) \propto \begin{cases} \mathcal{N}(\mathbf{x}|\hat{\mathbf{A}}\boldsymbol{\phi}(\mathbf{y}_*)+\hat{\boldsymbol{\mu}}_{\mathbf{e}|\mathbf{y}},\hat{\boldsymbol{\Gamma}}_{\mathbf{e}|\mathbf{y}}) &\\
\qquad\qquad\quad\;\cdot\:\mathcal{N}(\mathbf{y}_*|\hat{\boldsymbol{\mu}}_{\boldsymbol{\theta}},\hat{\boldsymbol{\Gamma}}_{\mathbf{y}}), & \mathbf{y}_*\geq 0 \\
0, & \mathbf{y}_*< 0, \end{cases}$$ is constructed based on the training data and the new measurement. The model matrix $\hat{\mathbf{A}}$, conditional (residual) error statistics $\hat{\boldsymbol{\mu}}_{\mathbf{e}|\mathbf{y}}$ and $\hat{\boldsymbol{\Gamma}}_{\mathbf{e}|\mathbf{y}}$, and the prior statistics $\hat{\boldsymbol{\mu}}_{\boldsymbol{\theta}}$ and $\hat{\boldsymbol{\Gamma}}_{\mathbf{y}}$ are learned from the training data. The density is then sampled using a Markov chain Monte Carlo method. The point estimate and 95% credible intervals are then calculated from the samples.

![image](rmsesrev.png){width="\textwidth"}

Performance assessment
----------------------

The proposed GPR method is first evaluated using leave-one-out cross-validation (i.e. $n_t=492$). From the results, relative root mean square error (RMSE%), relative bias (bias%), and credible interval coverage (CI%) are calculated. Credible interv