Hawkes process models are used in settings where past events increase the likelihood of future events occurring. Many applications record events as counts on a regular grid, yet discrete-time Hawkes models remain comparatively underused and are often constrained by fixed-form baselines and excitation kernels. In particular, there is a lack of flexible, nonparametric treatments of both the baseline and the excitation in discrete time. To this end, we propose the Gaussian Process Discrete Hawkes Process (GP-DHP), a nonparametric framework that places Gaussian process priors on both the baseline and the excitation and performs inference through a collapsed latent representation. This yields smooth, data-adaptive structure without prespecifying trends, periodicities, or decay shapes, and enables maximum a posteriori (MAP) estimation with near-linear-time \(O(T\log T)\) complexity. A closed-form projection recovers interpretable baseline and excitation functions from the optimized latent trajectory. In simulations, GP-DHP recovers diverse excitation shapes and evolving baselines. In case studies on U.S. terrorism incidents and weekly Cryptosporidiosis counts, it improves test predictive log-likelihood over standard parametric discrete Hawkes baselines while capturing bursts, delays, and seasonal background variation. The results indicate that flexible discrete-time self-excitation can be achieved without sacrificing scalability or interpretability.

翻译：霍克斯过程模型适用于历史事件增加未来事件发生概率的场景。许多应用将事件记录为规则网格上的计数，但离散时间霍克斯模型仍相对使用不足，且常受限于固定形式的基线函数和激励核。特别是，离散时间下对基线和激励均缺乏灵活的非参数化处理方法。为此，我们提出高斯过程离散霍克斯过程（GP-DHP），该非参数框架对基线和激励均赋予高斯过程先验，并通过坍缩隐表示进行推断。该方法无需预先指定趋势、周期性或衰减形态即可获得平滑的数据自适应结构，并能以近线性时间 \(O(T\log T)\) 复杂度实现最大后验（MAP）估计。通过闭式投影可从优化后的隐轨迹中恢复可解释的基线函数与激励函数。在模拟实验中，GP-DHP 成功还原了多样化的激励形态和时变基线。在美国恐怖袭击事件和隐孢子虫病周发病数的案例研究中，相较于标准参数化离散霍克斯基线模型，本方法在捕获突发模式、延迟效应和季节性背景变化的同时，提升了测试预测对数似然。结果表明，灵活的离散时间自激励建模可在不牺牲可扩展性或可解释性的前提下实现。