  We observe $n$ inhomogeneous Poisson processes with covariates and aim at
estimating their intensities. We assume that the intensity of each Poisson
process is of the form $s (\cdot, x)$ where $x$ is the covariate and where $s$
is an unknown function. We propose a model selection approach where the models
are used to approximate the multivariate function $s$. We show that our
estimator satisfies an oracle-type inequality under very weak assumptions both
on the intensities and the models. By using an Hellinger-type loss, we
establish non-asymptotic risk bounds and specify them under several kind of
assumptions on the target function $s$ such as being smooth or a product
function. Besides, we show that our estimation procedure is robust with respect
to these assumptions.
