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Thinning distance between point process realisations

The paper "Wasserstein Learning of Deep Generative Point Process Models" published at the NIPS 2017 conference has some interesting ideas in it, connecting generative deep learning - which is mostly used for dense data such as pixels - together with point processes, which are useful for "spiky" timestamp events.

They use the Wasserstein distance (aka the "earth-mover's distance") to compare sequences of spikes, and they do acknowledge that this has advantages and disadvantages. It's all about pushing things around until they match up - e.g. move a spike a few seconds earlier in one sequence, so that it lines up with a spike in the other sequence. It doesn't nicely account for insertions or deletions, which is tricky because it's quite common to have "missing" spikes for added "clutter" in data coming from detectors, for example. It'd be better if this method could incorporate more general "edit distances", though that's non-trivial.

So I was thinking about distances between point processes. More reading to be done. But a classic idea, and a good way to think about insertions/deletions, is called "thinning". It's where you take some data from a point process and randomly delete some of the events, to create a new event sequence. If you're using Poisson processes then thinning can be used for example to sample from a non-stationary Poisson process, essentially by "rejection sampling" from a stationary one.

Thinning is a probabilistic procedure: in the simplest case, take each event, flip a coin, and keep the event only if the coin says heads. So if we are given one event sequence, and a specification of the thinning procedure, we can define the likelihood that this would have produced any given "thinned" subset of events. Thus, if we take two arbitrary event sequences, we can imagine their union was the "parent" from which they were both derived, and calculate a likelihood that the two were generated from it. (Does it matter if the parent process actually generated this union list, or if there were unseen "extra" parent events that were actually deleted from both? In simple models where the thinning is independent for each event, no: the deletion process can happen in any order, and so we can assume those common deletions happened first to take us to some "common ancestor". However, this does make it tricky to compare distances across different datasets, because the unseen deletions are constant multiplicative factors on the true likelihood.)

We can thus define a "thinning distance" between two point process realisations as the negative log-likelihood under this thinning model. Clearly, the distance depends entirely on the number of events the two sequences have in common, and the numbers of events that are unique to them - the actual time positions of the events has no effect, in this simple model, it's just whether they line up or not. It's one of the simplest comparisons we can make. It's complementary to the Wasserstein distance which is all about time-position and not about insertions/deletions.

This distance boils down to:

```NLL = -( n1 * log(n1/nu)  +  n2 * log(n2/nu)  +  (nu-n1) * log(1 - n1/nu)  +  (nu-n2) * log(1 - n2/nu) )
```

where "n1" is the number of events in seq 1, "n2" in seq 2, and "nu" in their union.

Does this distance measure work? Yes, at least in limited toy cases. I generated two "parent" sequences (using the same rate for each) and separately thinned each one ten times. I then measured the thinning distance between all pairs of the child sequences, and there's a clear separation between related and unrelated sequences:

```Distances between distinct children of same process:
Min 75.2, Mean 93.3, Median 93.2, Max 106.4
Distances between children of different processes:
Min 117.3, Mean 137.7, Median 138.0, Max 167.3
```

This is nice because easy to calculate, etc. To be able to do work like in the paper I cited above, we'd need to be able to optimise against something like this, and even better, to be able to combine it into a full edit distance, one which we can parameterise according to situation (e.g. to balance the relative cost of moves vs. deletions).

This idea of distance based on how often the spikes coincide relates to "co-occurrence metrics" previously described in the literature. So far, I haven't found a co-occurrence metric that takes this form. To relax the strict requirement of events hitting at the exact same time, there's often some sort of quantisation or binning involved in practice, and I'm sure that'd help for direct application to data. Ideally we'd generalise over the possible quantisations, or use a jitter model to allow for the fact that spikes might move.