As a crucial step in the SLCA, the set of models needs to be filtered for essential genes which represent neither underfitted nor overfitted models. From the PCA, we know that the hyperparameter 𝑛𝑛 𝑑𝑑 should not exceed 3, otherwise, the model is overfitted. However, this information alone is not sufficient to identify the essential models. The R 2 values in group III still vary from 54.0% to 99.8%, meaning that some models are underfitted while others are overfitted. Removing the models with low R 2 to avoid underfitting within group III is straightforward. Excluiding overfitted models is less obvious. Using R 2 alone can not distinguish between over-fitted and just well-fitted models. Therefore, an additional criterion is applied on the models of group III, whereby a model is considered overfitted when R 2 is high, but not rooted in group hierarchy: adding one more element from group to group must improve model linearity. Figure shows two families originating from S and P, selected by way of example for their highest R 2 among the group I models. On the second level of the family trees, only those models (group II) are considered decendant from S (or P) which (i) include S (or P) and (ii) have higher R 2 than their parent. On the third and any higher level, in addition to (i-ii), the ratio and signs of parent elements must reflect in model stoichiometry (iii). For example, all of the five possible group II decendants of S have increased R 2 values, therefore, they are all children of S. Each of these five children has four further decendants in group III, however, the majority of those second-level decendants is excluded because of the stoichiometry criterion (iii), or because of an insignificant increase in R 2 over the group II ancestor (indicating overfitting). As a result, we identify four essential genes, and a limited number of uncertain models (for which exclusion based on deviating stoichiometry is less clear; the main reason for such uncertainty is with the limited size of the dataset relative to the strong variation in the elemental concentration of S). conductivity. In Figure , the essential models [Na/S/O] and [P/F/Al] are used for this demonstration. They do not contain any common element and, consequently, seem to conjugate in their contribution to ionic conductivity.