## 4 Description of the Application Problem and Employed Multi-Fidelity Models
### Surrogate Modeling for Space Design Exploration
#### 5.3.4 Calculation of Error Metrics and Assessment of Surrogate Models  
The error metrics described in Section 3.4 were finally calculated for the 15 combinations of surrogated models and variables above, as shown in Table 7. Coefficients of determination \(R^{2}\) (see Equation (7)) were obtained to assess the fitting of each surrogate model to the corresponding variable in the training dataset. This adjustment is also evaluated for intermediate variables when the corresponding surrogate represents the target variable \(z_{l}\). Generally speaking, good values in terms of \(R^{2}\) were obtained for most combinations, as Table 7 displays. The analysis showed perfect goodness of fit for the estimations employing radial basis function models and kriging models. However, when such surrogates employ the ratio of condensations \(\varphi\) between HF and LF models, their fitting to the training dataset deteriorates when the results are translated into values of HF condensation mass flow rate \(\hat{m}_{cond,3D}\), which is the target variable. Polynomial Response Surface models also provide excellent (but not perfect) calibrations, except for the ratio of condensations \(\varphi\). The ultimate test for the surrogate models is their ability to predict results at points not used for their training, i.e., the verification stage. To do so, several metrics based on the Mean Square Error (MAE) of Equation (8) were identified in Section 3.4, such as the MAE retaining 95% of data with the smallest residuals and the normalized \(\widetilde{MAE}\) (Equation (9)) and \(\widetilde{MAE}_{\sigma}\) (Equation (10)). Finally, Table 7 includes a qualitative assessment of the predictive capability of each metamodel based on the last two indicators, in accordance with the work of [PERSON] et al. (2016). Table 7 shows that 12 combinations of surrogate models and variables provide good predictive capability; 2 obtain a moderate only one presents a lousy performance. Thekriging model is the only surrogate that provides evaluation deemed as \"good\" for all metamodeled variables. Interestingly, at least one MF variable performs better than the target variable for each surrogate model. Still, not all MF variables necessarily provide better agreement than the HF target variable (such as \(\varphi\) for PRS and \(\Delta\) and \(\eta\) for Kriging model). This idea was already noted by [PERSON] et al. [76], who identify that MF models \"may be less accurate than a surrogate built using only HF training data points\". The results included in Table 7 support the current work of assessing not only MF but also HF intermediate variables. Indeed, the HF intermediate variable \(w_{cond\_3D}\) performs better than the HF target variable \(\dot{m}_{cond\_3D}\) for all surrogate models when applied to the problem of volume condensation due to mixing. Furthermore, \(w_{cond\_3D}\) is the best intermediate variable to be surrogated using a Radial Basis Function since it presents the lowest \(\widetilde{MAE}\) and \(\widetilde{MAE}_{\sigma}\) amongst all assessed variables, including the MF ones. For the Kriging model, there is not an MF variable that presents lower values of both \(\widetilde{MAE}\) and \(\widetilde{MAE}_{\sigma}\) than those of \(w_{cond\_3D}\), so the best intermediate variable to be metamodeled using a Kriging model will depend on the weighting of such error metrics. Finally, the best overall combination of surrogate and variable is the Polynomial Response Surface of the MF variable \(\Delta\), by the values of \(\widetilde{MAE}\) and \(\widetilde{MAE}_{\sigma}\) shown in Table 7.  
To visualize the results underlying Table 7 and exemplify the implications of the multi-fidelity proposal, an analysis of the surrogate model based on a polynomial response surface acting over condensation difference (Equation (16)) is conducted since this combination is the one providing the lowest error, as aforementioned.