  This paper proposes a log-linear model for the latent intensity functions of
a replicated spatio-temporal point process. By simultaneously fitting
correlated spatial and temporal Karhunen-Lo\`eve expansions, the model produces
spatial and temporal components that are usually easy to interpret and capture
the most important modes of variation and spatio-temporal correlation of the
process. The asymptotic distribution of the estimators is derived. The finite
sample properties are studied by simulations. As an example of application, we
analyze bike usage patterns on the Divvy bike sharing system of the city of
Chicago.
