  This paper introduces an approach for detecting differences in the
first-order structures of spatial point patterns. The proposed approach
leverages the kernel mean embedding in a novel way by introducing its
approximate version tailored to spatial point processes. While the original
embedding is infinite-dimensional and implicit, our approximate embedding is
finite-dimensional and comes with explicit closed-form formulas. With its help
we reduce the pattern comparison problem to the comparison of means in the
Euclidean space. Hypothesis testing is based on conducting t-tests on each
dimension of the embedding and combining the resulting p-values using one of
the recently introduced p-value combination techniques. If desired,
corresponding Bayes factors can be computed and averaged over all tests to
quantify the evidence against the null. The main advantages of the proposed
approach are that it can be applied to both single and replicated pattern
comparisons and that neither bootstrap nor permutation procedures are needed to
obtain or calibrate the p-values. Our experiments show that the resulting tests
are powerful and the p-values are well-calibrated; two applications to real
world data are presented.
