We discuss the practice of directly or indirectly assuming a model for the number of points when modelling spatial point patterns even though it is rarely possible to validate such a model in practice because most point pattern data consist of only one pattern. We therefore explore the possibility to condition on the number of points instead when fitting and validating spatial point process models. In a simulation study with different popular spatial point process models, we consider model validation using global envelope tests based on functional summary statistics. We find that conditioning on the number of points will for some functional summary statistics lead to more narrow envelopes and thus stronger tests and that it can also be useful for correcting for some conservativeness in the tests when testing composite hypothesis. However, for other functional summary statistics, it makes little or no difference to condition on the number of points. When estimating parameters in popular spatial point process models, we conclude that for mathematical and computational reasons, it is convenient to assume a distribution for the number of points.

中文翻译：

---

在对空间点模式进行建模时，我们应该以点数为条件吗？

我们讨论了在对空间点模式进行建模时直接或间接假设点数模型的做法，尽管在实践中很少可能验证这样的模型，因为大多数点模式数据仅包含一个模式。因此，我们探索了在拟合和验证空间点过程模型时以点数为条件的可能性。在具有不同流行空间点过程模型的模拟研究中，我们考虑使用基于功能汇总统计的全局包络测试进行模型验证。我们发现，对某些功能汇总统计量的点数进行调节会导致更窄的包络，从而导致更强的测试，并且在测试复合假设时，它也可以用于纠正测试中的一些保守性。然而，对于其他功能汇总统计，以点数为条件几乎没有区别。在估计流行的空间点过程模型中的参数时，我们得出结论，出于数学和计算的原因，假设点数的分布很方便。