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.

更新的Hawkes流程是一个新兴的流程模型，可以概括Hawkes流程。尽管已显示出强大的应用潜力，但将更新的Hawkes流程适应于数据仍然是一项艰巨的任务，尤其是在较大的数据集上。本文通过提供两种方法来显着减少更新霍克斯流程所需的时间，从而解决了这一挑战。由于基于导数的优化方法通常比无导数的方法收敛更快，因此我们的第一种方法是派生用于评估对数似然函数的梯度和Hessian的算法，然后使用基于导数的方法，例如牛顿-拉夫森法（Newton-Raphson method）代替了目前使用的无导数法，以最大化似然性。我们的第二种方法是寻找线性时间算法，该算法可以对似然函数产生精确的近似值，然后直接优化对数似然函数的近似值。我们的仿真实验表明，Newton-Raphson方法将计算时间减少了约30％。此外，与基于精确似然度的方法相比，近似似然法可以产生同样准确的估计值，并且在具有约10,000个事件的数据集上，其近似速度要快20-40倍。最后，我们分析了几种货币相对于美元的价格变化。与基于精确似然性的方法相比，近似似然法可产生同样准确的估计值，并且在具有约10,000个事件的数据集上，其近似速度要快20-40倍。最后，我们对几种货币相对于美元的价格变化进行了分析。与基于精确似然性的方法相比，近似似然法可产生同样准确的估计值，并且在具有约10,000个事件的数据集上，其近似速度要快20-40倍。最后，我们分析了几种货币相对于美元的价格变化。