The classic Hawkes process assumes the baseline intensity to be constant and the triggering kernel to be a parametric function. Differently, we present a generalization of the parametric Hawkes process by using a Bayesian nonparametric model called quadratic Gaussian Hawkes process. We model the baseline intensity and trigger kernel as the quadratic transformation of random trajectories drawn from a Gaussian process (GP) prior. We derive an analytical expression for the EM-variational inference algorithm by augmenting the latent branching structure of the Hawkes process to embed the variational Gaussian approximation into the EM framework naturally. We also use a series of schemes based on the sparse GP approximation to accelerate the inference algorithm. The results of synthetic and real data experiments show that the underlying baseline intensity and triggering kernel can be recovered efficiently and our model achieved superior performance in fitting capability and prediction accuracy compared to the state-of-the-art approaches.

经典的 Hawkes 过程假设基线强度是恒定的，触发内核是一个参数函数。不同的是，我们通过使用称为二次高斯霍克斯过程的贝叶斯非参数模型对参数霍克斯过程进行了概括. 我们将基线强度和触发内核建模为从高斯过程 (GP) 先验绘制的随机轨迹的二次变换。我们通过增加霍克斯过程的潜在分支结构以将变分高斯近似自然地嵌入到 EM 框架中，从而推导出 EM 变分推理算法的解析表达式。我们还使用了一系列基于稀疏 GP 近似的方案来加速推理算法。合成和真实数据实验的结果表明，与最先进的方法相比，可以有效地恢复基础基线强度和触发内核，并且我们的模型在拟合能力和预测精度方面取得了优异的性能。