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. Three different tests are proposed 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 one 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 stochastic 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.¹

时空点过程的一阶可分离性在时空点模式数据的分析中起着基本作用。虽然通常是一个方便的假设，可以大大简化分析，但在模型构建中应考虑到现有的不可分离的结构。提出了三种不同的检验方法来研究这一假设，作为初步数据分析的一个步骤。前两个检验对于泊松过程是精确的或渐近精确的。基于排列和全局包络的第一个检验使人们能够检测出数据偏离零假设的空间和时间位置或滞后。第二项测试简单且计算便宜${\chi }^{\mathrm{2个}}$-测试。第三次测试基于随机重构方法，通常可用于非泊松过程。前两个测试的性能在针对Poisson和非Poisson模型的仿真研究中进行了研究。第三次检验应用于英国2001年流行的口蹄疫的真实数据。^1个