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.
