Robust PCA and MIC statistics of baryons in early mini-haloes

We present a novel approach, based on robust principal components analysis (RPCA) and maximal information coefficient (MIC), to study the redshift dependence of halo baryonic properties. Our data are composed of a set of different physical quantities for primordial minihaloes: dark-matter mass ($M_{\mathrm{dm}}$), gas mass ($M_{\mathrm{gas}}$), stellar mass ($M_{\mathrm{star}}$), molecular fraction (${\mathrm{x_{mol}}}$), metallicity ($Z$), star formation rate (SFR) and temperature. We find that $M_{\mathrm{dm}}$ and $M_{\mathrm{gas}}$ are dominant factors for variance, particularly at high redshift. Nonetheless, with the emergence of the first stars and subsequent feedback mechanisms, ${\mathrm{x_{mol}}}$, SFR and $Z$ start to have a more dominant role. Standard PCA gives three principal components (PCs) capable to explain more than 97 per cent of the data variance at any redshift (two PCs usually accounting for no less than 92 per cent), whilst the first PC from the RPCA analysis explains no less than 84 per cent of the total variance in the entire redshift range (with two PCs explaining $\gtrsim 95$ per cent anytime). Our analysis also suggests that all the gaseous properties have a stronger correlation with $M_{\mathrm{gas}}$ than with $M_{\mathrm{dm}}$, while $M_{\mathrm{gas}}$ has a deeper correlation with ${\mathrm{x_{mol}}}$ than with $Z$ or SFR. This indicates the crucial role of gas molecular content to initiate star formation and consequent metal pollution from Population III and Population II/I regimes in primordial galaxies. Finally, a comparison between MIC and Spearman correlation coefficient shows that the former is a more reliable indicator when halo properties are weakly correlated.