Self-exciting models are statistical models of count data where the probability of an event occurring is influenced by the history of the process. In particular, self-exciting spatio-temporal models allow for spatial dependence as well as temporal self-excitation. For large spatial or temporal regions, however, the model leads to an intractable likelihood. An increasingly common method for dealing with large spatio-temporal models is by using Laplace approximations (LA). This method is convenient as it can easily be applied and is quickly implemented. However, as we will demonstrate in this manuscript, when applied to self-exciting Poisson spatial–temporal models, Laplace Approximations result in a significant bias in estimating some parameters. Due to this bias, we propose using up to sixth-order corrections to the LA for fitting these models. We will demonstrate how to do this in a Bayesian setting for self-exciting spatio-temporal models. We will further show there is a limited parameter space where the extended LA method still has bias. In these uncommon instances we will demonstrate how a more computationally intensive fully Bayesian approach using the Stan software program is possible in those rare instances. The performance of the extended LA method is illustrated with both simulation and real-world data.

自激模型是计数数据的统计模型，其中事件发生的概率受到过程历史的影响。特别是，自激励时空模型允许空间依赖性以及时间自激励。然而，对于大的空间或时间区域，该模型会导致难以处理的可能性。处理大型时空模型的一种越来越常见的方法是使用拉普拉斯近似 (LA)。该方法应用方便，实施速度快。然而，正如我们将在本手稿中演示的那样，当应用于自激泊松时空模型时，拉普拉斯近似会导致估计某些参数时出现显着偏差。由于这种偏差，我们建议使用 LA 的六阶修正来拟合这些模型。我们将演示如何在贝叶斯设置中执行此操作用于自激时空模型。我们将进一步证明，在有限的参数空间中，扩展 LA 方法仍然存在偏差。在这些不常见的情况下，我们将演示如何在这些罕见的情况下使用 Stan 软件程序实现计算强度更大的完全贝叶斯方法。扩展 LA 方法的性能通过仿真和实际数据进行了说明。