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.

本文介绍了一种检测空间点图案的一阶结构差异的方法。所提出的方法通过引入针对空间点过程量身定制的近似版本，以一种新颖的方式利用了内核均值嵌入。虽然原始的嵌入是无限维的和隐式的，但我们的近似嵌入是有限维的，并带有显式的封闭式公式。借助于它，我们将模式比较问题简化为欧几里得空间中的均值比较。假设检验基于$Ť$-测试嵌入的各个维度，并将结果合并 $p$-值使用最近引入的一种 $p$值组合技术。如果需要，可以在所有测试中计算相应的贝叶斯因子并将其平均，以量化针对无效值的证据。所提方法的主要优点是，它既可以应用于单模式比较又可以应用于复制模式比较，并且不需要引导程序或置换程序即可获得或校准该模式。$p$值。我们的实验表明，生成的测试功能强大且$p$-值已正确校准；介绍了对现实世界数据的两种应用。