Point processes

Temporal point processes

Sometimes including spatiotemporal point processes, depending on mood.

In these, one has an arrow of time which simplifies things because you know that you “only need to consider the past of a process to understand its future”, which potentially simplifies many calculations about the conditional intensity processes; We consider only interactions from the past to the future, rather than some kind of mutual interaction.

In particular, for nice processes you can do fairly cheap likelihood calculations to estimate process parameters etc.

Using the regular point process representation of the probability density of the occurrences, we have the following joint log likelihood for all the occurrences

\[\begin{aligned} L_\theta(t_{1:N}) &:= -\int_0^T\lambda^*_\theta(t)dt + \int_0^T\log \lambda^*_\theta(t) dN_t\\ &= -\int_0^T\lambda^*_\theta(t)dt + \sum_{j} \log \lambda^*_\theta(t_j) \end{aligned}\]

I do a lot of this, for example, over at the branching processes notebook, and I have no use at the moment for other types of process, so I won’t say much about other cases for the moment.

See also change of time.

Spatial point processes

Processes without an arrow of time arise naturally, say as processes where we observe only snapshots of the dynamics, or where whatever dynamics that gave rise to the process being too slow to be considered as anything but static (e.g. location of trees in forests).

See spatial point processes.

References