# How does a covariance intensity function measure clustering?

I was taught in a class on spatial statistics that the covariance intensity function (defined below) measured clustering and inhibition in a point process, but isn't used because good test statistics for it don't exist.

$$c(\mathbf{x},\mathbf{y})=\lim_{|d\mathbf{x}|\to0,|d\mathbf{y}|\to 0} \frac{\mathrm{Cov}\{N(d\mathbf{x}), N(d\mathbf{y})\}}{|d\mathbf{x}||d\mathbf{y}|}$$

Where $$\mathbf{x}, \mathbf{y}$$ are positions on the domain and $$d\mathbf{x}, d\mathbf{y}$$ are regions around them with areas given by $$|d\mathbf{x}|, |d\mathbf{y}|$$, and $$N(R)$$ represents the random variable corresponding to the number of events in a region.

But I can't see how this measures non-homogeneity at all -- if one starts with a process that is described by an intensity function -- any intensity function -- this function should necessarily be zero, as the existence of an intensity function means a point turning up at point $$\mathbf{x}$$ is independent of a point turning up at intensity $$\mathbf{y}$$. And you can certainly have intensity functions that exhibit clustering.

The only way that this function can be non-zero as I see it is if there are correlations within a realisation, e.g. if "everything is clustered to one side" and "everything is clustered to the other side" are the possibilities, or something i.e. if you don't have an intensity function at all, but rather some sort of "entangled state".

What am I missing?

• If c(x,y)>0 then the number of points near x and the number of points near y have positive correlation E.g. consider lots of locations x near to y - if they have positive correlation then the intensity is similar in an area around y, this can account for repulsion or clustering. Independence between points is not an assumption of all point processes. Terminology can be difficult here - heterogeneity in intensity is often not what is meant by clustering and repulsion - these are inter-point effects – ASeaton Jul 19 '19 at 13:48