  First-order separability of a spatio-temporal point process plays a
fundamental role in the analysis of spatio-temporal point pattern data. While
it is often a convenient assumption that simplifies the analysis greatly,
existing non-separable structures should be accounted for in the model
construction. We propose three different tests to investigate this hypothesis
as a step of preliminary data analysis. The first two tests are exact or
asymptotically exact for Poisson processes. The first test based on
permutations and global envelopes allows us to detect at which spatial and
temporal locations or lags the data deviate from the null hypothesis. The
second test is a simple and computationally cheap $\chi^2$-test. The third test
is based on statistical reconstruction method and can be generally applied for
non-Poisson processes. The performance of the first two tests is studied in a
simulation study for Poisson and non-Poisson models. The third test is applied
to the real data of the UK 2001 epidemic foot and mouth disease.
