Hawkes processes are point process models that have been used to capture self-excitatory behavior in social interactions, neural activity, earthquakes and viral epidemics. They can model the occurrence of the times and locations of events. Here we develop a new class of spatiotemporal Hawkes processes that can capture both triggering and clustering behavior and we provide an efficient method for performing inference. We use a log-Gaussian Cox process (LGCP) as prior for the background rate of the Hawkes process which gives arbitrary flexibility to capture a wide range of underlying background effects (for infectious diseases these are called endemic effects). The Hawkes process and LGCP are computationally expensive due to the former having a likelihood with quadratic complexity in the number of observations and the latter involving inversion of the precision matrix which is cubic in observations. Here we propose a novel approach to perform MCMC sampling for our Hawkes process with LGCP background, using pre-trained Gaussian Process generators which provide direct and cheap access to samples during inference. We show the efficacy and flexibility of our approach in experiments on simulated data and use our methods to uncover the trends in a dataset of reported crimes in the US.
翻译:霍克斯过程是用来捕捉社会互动、神经活动、地震和病毒流行病中自我探索行为的点点样过程模型,用于捕捉社会互动、神经活动、地震和病毒流行病中的自我探索行为,它们可以模拟事件发生的时间和地点。我们在这里开发了一种新的波形瞬时霍克斯过程,可以捕捉触发和组合行为,我们为进行推断提供了一种有效的方法。我们使用对地对地对地计算过程的背景速率(LGCP),作为以前霍克斯过程的背景率的对地对地,它提供了任意的灵活性,可以捕捉广泛的基本背景影响(对传染病来说,这些称为地方性影响)。霍克斯过程和LGCP过程计算费用昂贵,因为前者在观测数量上有可能具有二次复杂度,而后者则涉及对精确矩阵的转换,而这是三分立的。我们在这里提出了一个新的方法,用LGCP背景对我们的霍克斯过程进行MC取样,使用经过预先训练的戈斯过程生成器,在推断期间提供直接和廉价的样品。我们展示了模拟数据实验方法的效力和灵活性,并使用我们所报告的美国犯罪趋势。