  Complex systems are characterised by a tight, nontrivial interplay of their
constituents, which gives rise to a multi-scale spectrum of emergent
properties. In this scenario, it is practically and conceptually difficult to
identify those degrees of freedom that mostly determine the behaviour of the
system and separate them from less prominent players. Here, we tackle this
problem making use of three measures of statistical information: resolution,
relevance, and mapping entropy. We address the links existing among them,
taking the moves from the established relation between resolution and relevance
and further developing novel connections between resolution and mapping
entropy; by these means we can identify, in a quantitative manner, the number
and selection of degrees of freedom of the system that preserve the largest
information content about the generative process that underlies an empirical
dataset. The method, which is implemented in a freely available software, is
fully general, as it is shown through the application to three very diverse
systems, namely a toy model of independent binary spins, a coarse-grained
representation of the financial stock market, and a fully atomistic simulation
of a protein.
