We propose a constructive approach to building temporal point processes that incorporate dependence on their history. The dependence is modeled through the conditional density of the duration, i.e., the interval between successive event times, using a mixture of first-order conditional densities for each one of a specific number of lagged durations. Such a formulation for the conditional duration density accommodates high-order dynamics, and it thus enables flexible modeling for point processes with memory. The implied conditional intensity function admits a representation as a local mixture of first-order hazard functions. By specifying appropriate families of distributions for the first-order conditional densities, with different shapes for the associated hazard functions, we can obtain either self-exciting or self-regulating point processes. From the perspective of duration processes, we develop a method to specify a stationary marginal density. The resulting model, interpreted as a dependent renewal process, introduces high-order Markov dependence among identically distributed durations. Furthermore, we provide extensions to cluster point processes. These can describe duration clustering behaviors attributed to different factors, thus expanding the scope of the modeling framework to a wider range of applications. Regarding implementation, we develop a Bayesian approach to inference, model checking, and prediction. We investigate point process model properties analytically, and illustrate the methodology with both synthetic and real data examples.

翻译：我们提出了一种构建时序点过程的构造性方法，该方法能够纳入对历史过程的依赖性。这种依赖性通过持续时间的条件密度（即连续事件时间之间的间隔）进行建模，采用针对特定数量滞后持续时间的一阶条件密度混合形式。这种条件持续时间密度的表述方式能够适应高阶动态特性，从而为具有记忆性的点过程提供灵活的建模能力。隐含的条件强度函数可表示为局部混合的一阶风险函数。通过为具有不同形状关联风险函数的一阶条件密度指定适当的分布族，我们可以获得自激励或自调节的点过程。从持续时间过程的角度出发，我们开发了一种方法来设定平稳边际密度。所得模型可解释为依赖更新过程，在相同分布的持续时间之间引入了高阶马尔可夫依赖性。此外，我们还提供了对聚类点过程的扩展。这些扩展能够描述由不同因素引起的持续时间聚类行为，从而将建模框架的应用范围扩展到更广泛的领域。在实施方面，我们开发了用于推断、模型检验和预测的贝叶斯方法。我们通过解析方法研究了点过程的模型特性，并利用合成数据和实际数据示例对该方法进行了说明。