  Conditional correlation networks, within Gaussian Graphical Models (GGM), are
widely used to describe the direct interactions between the components of a
random vector. In the case of an unlabelled Heterogeneous population,
Expectation Maximisation (EM) algorithms for Mixtures of GGM have been proposed
to estimate both each sub-population's graph and the class labels. However, we
argue that, with most real data, class affiliation cannot be described with a
Mixture of Gaussian, which mostly groups data points according to their
geometrical proximity. In particular, there often exists external co-features
whose values affect the features' average value, scattering across the feature
space data points belonging to the same sub-population. Additionally, if the
co-features' effect on the features is Heterogeneous, then the estimation of
this effect cannot be separated from the sub-population identification. In this
article, we propose a Mixture of Conditional GGM (CGGM) that subtracts the
heterogeneous effects of the co-features to regroup the data points into
sub-population corresponding clusters. We develop a penalised EM algorithm to
estimate graph-sparse model parameters. We demonstrate on synthetic and real
data how this method fulfils its goal and succeeds in identifying the
sub-populations where the Mixtures of GGM are disrupted by the effect of the
co-features.
