# What is the general idea of Spatial Cox Process?

I can't seem to understand the whole idea of Cox Process in spatial point pattern. Can someone explain it in layman's term? thank you

## 1 Answer

I'm going to assume you understand what an intensity function of a point process is.

This is from p380 of Statistical Analysis and Modelling of Spatial Point Patterns by Illian et al (2008).

recall the inhomogeneous Poisson process, based on a non-constant intensity function $\lambda(x)$ [$x \in \mathbb{R_2}$ for example] ... In a stationary Cox process the intensity function is replaced by a stationary random field with non-negative values. The realisations of this random field are functions which are treated as intensity functions of inhomogeneous Poisson processes

You can think of a random field as like a random variable, except each realisation of the random field is itself a function of space. So for random field $\{\Lambda(x)\}$, a specific realisation is some $\lambda(x)$ intensity function. Just like the number $x = 0$ might be a realisation of some random variable $X$. Except here it is a function of space.

So a Cox process replaces an inhomogenous $\lambda(x)$ with a whole family of possible intensity functions that are described by the random field $\{\Lambda(x)\}$. There are various options for what $\{\Lambda(x)\}$ actually is. I work with a log-Gaussian random field with some spatial structure (so you get some smoothness). The log is to avoid negative values which don't make sense for an intensity function.