Linear multivariate Hawkes processes (MHP) are a fundamental class of point processes with self-excitation. When estimating parameters for these processes, a difficulty is that the two main error functionals, the log-likelihood and the least squares error (LSE), as well as the evaluation of their gradients, have a quadratic complexity in the number of observed events. In practice, this prohibits the use of exact gradient-based algorithms for parameter estimation. We construct an adaptive stratified sampling estimator of the gradient of the LSE. This results in a fast parametric estimation method for MHP with general kernels, applicable to large datasets, which compares favourably with existing methods.

翻译：线性多变量霍克斯进程( MHP) 是自引点进程的基本分类。 在估算这些进程的参数时,一个困难是,两个主要错误功能,即日志相似性和最小方差(LSE),以及对其梯度的评估,在观测到的事件数量上具有二次复杂性。在实践中,这禁止使用精确的梯度算法进行参数估计。我们为 LSE 梯度建立了一个适应性分层抽样估计仪。这导致对大数据集适用的通用内核对MHP的快速参数估计方法,与现有方法相比更为有利。