How to test if an intensity function is a conditional intensity function?

Question

I am trying to understand Hawkes process and I get that the conditional intensity is the expected number of events conditional on the past history.

I have an intensity function that comes from my application and I want to know if Hawkes process is the right one to apply.

Is there a test to verify if the intensity function is, in fact, conditional intensity? Like some sort of a recurrence relation that could be shown to illustrate the dependence behaviour? Thanks.

Answer 1

It's hard to verify whether the intensity function is in fact a conditional intensity function, when the intensity function is unknown, as in your case. Another complication arises because there are many forms of history dependence, see e.g. Cox or self-correcting processes.

Instead, it might be a question of how well your model fits the data (or at least its key features). There are a number of ways to check the goodness of fit of a point process model. For example, you could

1. simply compare the likelihood of different models given your data,
2. check how well you can make (probabilistic) forecasts using validation metrics and compare forecasts from different models to see which ones fit better (supervised forecasting setting),
3. use the random time change idea (see e.g. [1], p. 257ff) as a goodness of fit test by comparing the transformed process with a homogeneous Poisson process

There may also be theoretical, domain-specific reasons to assume a Hawkes process.

References:

• [1] Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. Vol. I: Elementary theory and methods., Probability and Its Applications. New York, NY: Springer. xxi, 469 p. (2003). ZBL1026.60061.