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


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.

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