Finding the MLE for a univariate exponential Hawkes process

The univariate exponential Hawkes process is a self-exciting point process with an event arrival rate of:

$ \lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$

where $ t_1,..t_n $ are the event arrival times.

The log likelihood function is

$ – t_n \mu + \frac{\alpha}{\beta} \sum{( e^{-\beta(t_n-t_i)}-1 )} + \sum\limits_{i<j}{\ln(\mu+\alpha e^{-\beta(t_j-t_i)})} $

which can be calculated recursively:

$ – t_n \mu + \frac{\alpha}{\beta} \sum{( e^{-\beta(t_n-t_i)}-1 )} + \sum{\ln(\mu+\alpha R(i))} $

$ R(i) = e^{-\beta(t_i-t_{i-1})} (1+R(i-1)) $

$ R(1) = 0 $

What numerical methods can I use to find the MLE? What is the simplest practical method to implement?

Answer

The Nelder-Mead simplex algorithm seems to work well.. It is implemented in Java by the Apache Commons Math library at https://commons.apache.org/math/ . I’ve also written a paper about the Hawkes processes at Point Process Models for Multivariate High-Frequency Irregularly Spaced Data .

felix, using exp/log transforms seems to ensure positivity of the parameters. As for the small alpha thing, search the arxiv.org for a paper called “limit theorems for nearly unstable hawkes processes”

Attribution
Source : Link , Question Author : Dave Anderson , Answer Author : crow

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