## 4 Discussion  
In order to extract a biochemically relevant explanation from the results of the pattern recognition procedure, for both experiments, we computed over the extracted granules (simplices), namely small peptides located into the protein structure, the main chemico-physical parameters at the amino-acid residue level according to the results presented in Ref. [88]. Each information granule (simplex) has been mapped with 6 real values indicating the average and standard deviation of polarity, volume and hydrophilicity evaluated amongst the amino-acids forming the simplex. The chemico-physical properties of each information granule have been correlated with a score ranging from 1 to 5, namely the number of times said granule has been selected across the five runs: the higher the score, the higher the confidence about its discrimination importance for the classification problem. Let us discuss the solubility classification problem first. The score assigned to each simplex has been discretised according to the following rules: all scores greater than 2 have been considered  
Figure 5: Average Performance on Test Set for Experiment #2 (\(\ell_{1}\)-SVM, \(\alpha=1\)). Figure 6: Average Performance on Test Set for Experiment #2 (\(\ell_{1}\)-SVM, \(\alpha=0.5\)). 'positives', all scores equal to 0 have been considered 'negatives' and all other simplices have been discarded. Statistical tests show that, despite the huge number of samples (approx. 11000 simplices), the average volume is not statistically significant (\(p\)-value approx. 0.11). This is perfectly coherent if we consider that the volume of a simplex (usually less than 5 residues) is very unlikely to endow biological meaning in terms of the overall protein solubility. On the other hand, the standard deviation volume has been shown to be statistically significant (\(p\)-value \(<0.0001\)). This interesting result shows that simplices composed of'similar amino-acids' (small standard deviation) show better solubility. Nonetheless, it is important to note that, for a given chemico-physical property (e.g., volume in this case) the standard deviation and the average value shall be treated independently and do not show any correlation. This latter aspect of average and standard deviation carrying different information has also been confirmed by analysing the two other properties (polarity and hydrophilicity). Polarity and hydrophilicity not only show statistical significance (all \(p\)-values are less than 0.0001) but also show a strong correlation (\(>\)0.99) in terms of both mean values and standard deviations, as shown in Table 4, yet mean values and standard deviations are not correlated with each other (as per the volume case). This perfectly fits with current biochemical knowledge and, specifically, this is consistent with the well-known importance of 'hydrophobic interaction' in protein folding (residues with hydrophobicity/hydrophilicity values tend to aggregate [89]). Similar analyses have been carried for the EC classification problem. All of the seven statistical models show statistical significance, mainly thanks to the large number of samples (more than 12,000 simplices). Table 5 summarises their main characteristics. Alongside the statistical significance, it is interesting to note that _all_ of the seven models have \(R^{2}\approx 0.02\), meaning that they explain 2% of the overall variance. Furthermore, also in this experiment, hydrophilicity has been shown to be the most important predictor according to linear discriminant analysis [90] and completely superimposable results are obtained for average polarity, which is strictly related to hydrophilicity. Table 6 shows the main characteristics of the seven models where hydrophilicity is concerned and Table 7 is its counterpart as regards polarity.