# Spatial point process and their statistics

The intersection of point processes and spatial processes. Popular in, e.g, earthquake modelling.

The references here are , especially the latter.

Here is a non-exhaustive taxonomy of ones of interest to me.

## Gibbs point processes

Processes where the points themselves notionally interact.

Processes with undirected interactions and no time index. Such processes without an arrow of time arise naturally, say, where you observe only snapshots of the dynamics, (where you captured those koalas that one time you got fieldwork funding) aggregates of the dynamics (as with classical statmech) or where whatever dynamics that gave rise to the process were too slow to be considered as anything but static on the time-scale you can model (locations of trees).

This is fiddly because with only undirected interactions, inference for these processes is painful, because you have to, “regress everything against everything else” Gibbs ensemble-style, which is slow and boring, which is the general problem with Markov random fields.

It is, however an interesting introduction to statistical mechanics]. In the Gibbs interaction process example (Daley and Vere-Jones 2003 Example 5.3c) they explains to probabilists what physicists mean by “grand canonical ensembles” in under two pages, which is something I have not seen a physicist do yet. (Daley and Vere-Jones 2008 Ch 15) gives an overarching view of the tools of this trade such Papengelou intensities and pseudolikelihoods as does with some other connections to physics.

## Determinantal point processes

Determinantal point processes are a tractable special case of Gibbs point processes. They have their own notebook.

## Cox processes

A.k.a. doubly stochastic point processes. Processes where the latent Poisson intensity is not dependent upon interactions between the points but around some latent field which is itself stochastic.

Typically this a latent gaussian field transformed by some link function into a positive Poisson intensity,generalized linear models-style. I am sure other models are possible but I have not seen them.

### Log Gaussian point process

Cox process using the classic exponential link function.

## Permanental point processes

Permanental point processes assume a squared gaussian latent intensity, supposedly?

## Spatio-temporal point processes

Some of the problems with Gibbs point processes seem to me artificial. Everything we witness is the result of dynamics occurring in time, or we’d still be at the big bang. So any non-causal Markov random field we sample is presumably sampled from a causal process, which means that the interactions are directed over time. Indeed, e.g. Markov Chain Monte Carlo sampling works makes it easier to sample from weird random fields by constructing a miniature physics with an artificial time arrow to sample from possible configurations of random processes. Intuitively this makes sense. Particles, at least in classical physics, don’t teleport themselves into a particular configuration with a probability inversely proportional to the exponential of some potential energy field; rather they have particular trajectory that got them there. However, maybe the time-wise dynamics are even more complex and we wish to average over all possible such configurations or something. Welcome to statistical mechanics!

Anyway, practicalities.

## Statistical theory

### Unconditional intensity estimation via pseudolikelihood

discusses two major intensity estimators — KDEs and Delaunay tessellation-based local estimates. discuss intensity estimation questions generally. gives inhomogeneous Poisson asymptotics.

Originally Besag defined the pseudolikelihood of a finite set of random variables $$X_1, \dots, X_n$$ as the product of the conditional likelihoods of each $$X_i$$ given the other variables $$\{X_j, j \neq i\}$$. This was extended ([; Besag et al., 1982) to point processes, for which it can be viewed as an infinite product of infinitesimal conditional probabilities.

### Model estimation via GLM

Berman and Turner show how to use quadrature and generalised linear models to fit models without a form for the integrated influence kernel (which I don’t care about) using a linear model (which I do).

From the spatstat documentation:

Models are currently fitted by the method of maximum pseudolikelihood, using a computational device developed by Berman and Turner which we adapted to pseudolikelihoods in . […] it has the virtue that we can implement it in software with great generality. […] Disadvantages of maximum pseudolikelihood (MPL) include its small-sample bias and inefficiency relative to maximum likelihood estimators (MLE).

## Implementations of methods

Hmm.

Presumably other spatial statistical software?

## References

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Baddeley, Adrian J, Jesper Møller, and Rasmus Plenge Waagepetersen. 2000. Statistica Neerlandica 54 (3): 329–50.
Baddeley, Adrian, Pablo Gregori, Jorge Mateu, Radu Stoica, and Dietrich Stoyan. 2006. Case Studies in Spatial Point Process Modeling. Vol. 185. Springer.
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