标题： 用时间反转检验霍克斯过程的因果性 显示英文标题

标题： Testing the causality of Hawkes processes with time reversal

摘要： 我们证明了一元和对称多元霍克斯过程只是弱因果的：真实事件时间向量和反转事件时间向量的真实对数似然几乎相等，因此通过最大似然进行参数估计仅在时间箭头方向上弱相关。 在理想（合成）条件下，参数拟合优度检验明确拒绝反向事件时间，这意味着从时间对称量（如事件率的自协方差）推断核函数仅在少数情况下产生统计显著的拟合。 最后，我们发现使用多参数核函数拟合金融数据，对于同一事件时间向量，可能在两个时间箭头方向都产生显著的拟合，有时更倾向于反向时间方向。 这表明，对真实数据使用灵活核函数的霍克斯过程的显著拟合并不意味着确定的时间箭头，除非进行测试。 显示英文摘要

摘要： We show that univariate and symmetric multivariate Hawkes processes are only weakly causal: the true log-likelihoods of real and reversed event time vectors are almost equal, thus parameter estimation via maximum likelihood only weakly depends on the direction of the arrow of time. In ideal (synthetic) conditions, tests of goodness of parametric fit unambiguously reject backward event times, which implies that inferring kernels from time-symmetric quantities, such as the autocovariance of the event rate, only rarely produce statistically significant fits. Finally, we find that fitting financial data with many-parameter kernels may yield significant fits for both arrows of time for the same event time vector, sometimes favouring the backward time direction. This goes to show that a significant fit of Hawkes processes to real data with flexible kernels does not imply a definite arrow of time unless one tests it.