  For point patterns observed in natura, spatial heterogeneity is more the rule
than the exception. In numerous applications, this can be mathematically
handled by the flexible class of log Gaussian Cox processes (LGCPs); in brief,
a LGCP is a Cox process driven by an underlying log Gaussian random field (log
GRF). This allows the representation of point aggregation, point vacuum and
intermediate situations, with more or less rapid transitions between these
different states depending on the properties of GRF. Very often, the covariance
function of the GRF is assumed to be stationary. In this article, we give two
examples where the sizes (that is, the number of points) and the spatial
extents of point clusters are allowed to vary in space. To tackle such
features, we propose parametric and semiparametric models of non-stationary
LGCPs where the non-stationarity is included in both the mean function and the
covariance function of the GRF. Thus, in contrast to most other work on
inhomogeneous LGCPs, second-order intensity-reweighted stationarity is not
satisfied and the usual two step procedure for parameter estimation based on
e.g. composite likelihood does not easily apply. Instead we propose a fast
three step procedure based on composite likelihood. We apply our modelling and
estimation framework to analyse datasets dealing with fish aggregation in a
reservoir and with dispersal of biological particles.
