# Spatial point process statistics

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

Processes without an arrow of time arise naturally, say, where you observe only snaphosts of the dynamics, (where you captured those koalas that 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 timescale you can model (locations of trees).

This is fiddly because without directed interaction over time, and where the points in your process interact in an undirected fashion, these processes a pain in the arse, 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. All the convenient features of point processes are dead to you.

In a sense, this is an artificial problem. 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 you sample is presumably sampled from a causal process. Indeed, e.g. Bayesian Monte Carlo sampling works by constructing a miniature physics with an artificial time arrow to sample from possible configurations of such random processes. Particles, at least in classical physics, don’t just teleport themselves into a particular configuration with a probability inversely proportional to the exponential of some potential energy field. However, maybe the time-wise dynamics are even more complex and you wish to average over all possible such configurations or something. Welcome to statistical mechanics.

There is a broad and fairly deep introduction in the two-volume volume Daley and Jones epic. (DaVe03, DaVe08) By the way, DaVe03 Example 5.3(c) is a Gibbs interaction process and explains what the hell physicists mean with their “grand canonical ensembles” for proabilists in under two pages, which is much easier than sitting through a semester of thermodynamics. If you want to add phase transitions into the same formalism, see MøWa07. DaVe08 Ch 15 give an overarching view of Papengelou intensities and other tools for spatial point processes.

Anyway, practicalities.

## Statistical theory

### unconditional intensity estimation via pseudolikelihood

Lies11 discusses two major intensity estimators – KDEs and Delaunay tessellation-based local estimates. MøWa07 discuss intensity estimation questions generally. RaCr94 gives inhomogeneous Poisson asymptotics.

From BaTu00:

Originally Besag (Besa75, Besa77) 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 (Besa75, Besa77; 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 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 (BeTu92) which we adapted to pseudolikelihoods in Baddeley and Turner (BaTu00). […] 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 (Besa77; JeMø91; JeKü94_) relative to maximum likelihood estimators (MLE).

Hmm.