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Accelerating the estimation of renewal Hawkes self-exciting point processes

机译：加速续签鹰自我激动点流程的估计

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摘要

The renewal Hawkes process is a nascent point process model that generalizes the Hawkes process. Although it has shown strong application potential, fitting the renewal Hawkes process to data remains a challenging task, especially on larger datasets. This article tackles this challenge by providing two approaches that significantly reduce the time required to fit renewal Hawkes processes. Since derivative-based methods for optimization, in general, converge faster than derivative-free methods, our first approach is to derive algorithms for evaluating the gradient and Hessian of the log-likelihood function and then use a derivative-based method, such as the Newton-Raphson method, in maximizing the likelihood, instead of the derivative-free method currently being used. Our second approach is to seek linear time algorithms that produce accurate approximations to the likelihood function, and then directly optimize the approximation to the log-likelihood function. Our simulation experiments show that the Newton-Raphson method reduces the computational time by about 30%. Furthermore, the approximate likelihood methods produce equally accurate estimates compared to the methods based on the exact likelihood and are about 20-40 times faster on datasets with about 10,000 events. We conclude with an analysis of price changes of several currencies relative to the US Dollar.
机译：续订鹰过程是一个新的点进程模型，概括了鹰过程。虽然它已经表现出强大的应用潜力，但拟合续订鹰过程到数据仍然是一个具有挑战性的任务，尤其是在较大的数据集上。本文通过提供两种方法来解决这一挑战，这些方法显着减少了适合续订鹰过程所需的时间。由于基于衍生的优化方法，通常会收敛于无衍生方法，我们的第一种方法是导出用于评估日志似然函数的渐变和Hessian的算法，然后使用基于衍生的方法，如Newton-Raphson方法，最大化可能性，而不是目前正在使用的衍生方法。我们的第二种方法是寻求线性时间算法，其为似然函数产生准确的近似，然后直接优化对日志似然函数的近似。我们的仿真实验表明，牛顿-Raphson方法将计算时间降低约30％。此外，与基于精确可能性的方法相比，近似似然方法产生同样准确的估计，并且在具有大约10,000个事件的数据集上的速度速度快到约20-40倍。我们得出结论，分析了几种货币相对于美元的价格变化。

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