  This paper introduces a new approach to inferring the second order properties
of a multivariate log Gaussian Cox process (LGCP) with a complex intensity
function. We assume a semi-parametric model for the multivariate intensity
function containing an unspecified complex factor common to all types of
points. Given this model we exploit the availability of several types of points
to construct a second-order conditional composite likelihood to infer the pair
correlation and cross pair correlation functions of the LGCP. Crucially this
likelihood does not depend on the unspecified part of the intensity function.
We also introduce a cross validation method for model selection and an
algorithm for regularized inference that can be used to obtain sparse models
for cross pair correlation functions. The methodology is applied to simulated
data as well as data examples from microscopy and criminology. This shows how
the new approach outperforms existing alternatives where the intensity
functions are estimated non-parametrically.
