Is the Matérn covariance function associated with the Matérn cluster process?

Question

This question follows on from something I couldn't figure out in this question.

Is there a relationship between the Matérn cluster process and Matérn covariance function beyond both being attributed to Matérn? Specifically, does the construction given below associate this point process with Matérn covariance function? My very weak evidence for this is they both seem to be introduced in chapter 3 of the book "Spatial Variation", but I do not understand the book well enough to make much sense of this.

As I understand it, the cluster process works like this in $\mathbb{R}^2$. Select a square region, draw $n$ points from a homogeneous Poisson point process. Draw a circle of radius $r$ around each point. Within this circle select some more points according to a Poisson point process. The result seems to be something like this:

In chapter 3 beginning on page 28 of the book cited below, Matérn gives recipe for constructing stationary covariance function from a Poisson process. Assume you have a Poisson process with intensity $\lambda$. Let $dN(x)$ be the number of centers in the volume element $dx$ of $\mathbb{R}^n$. Suppose you specify a integrable function $q(x)$. Then he claims you get a stationary process $Z(x)$, given by $$ Z(x) = \int_{\mathbb{R}^n} q(x - y) dN(y). $$ Moreover the above process is a moving average model and $q$ is the weight function. The covariance function $c(u)$ of this stationary process is $$ c(u) = \lambda \int_{\mathbb{R}^n} q(u + y) q^*(y) dy $$ and mean $m$ is $$ m = \lambda \int_{\mathbb{R}^n} q(x) dx. $$ Here I'm having trouble with the notation and the construction is not particularly obvious to me. Matérn then claims that by various choices of $q$ you can construct processes with appropriate covariance functions (this is in Table 1, pg. 30). Some involve modified Bessel functions of the second kind, as are used in Matérn covariance.

Source: Matérn, Bertil. "Spatial variation, volume 36 of Lecture Notes in Statistics." (1986).

Answer 1

Is the Matérn covariance function associated with the Matérn cluster process?

No. Except from the fact that they are named after the same person.

The Matérn cluster process is treated in many books on point processes. I have coauthored one myself recently https://book.spatstat.org but another classic reference is Møller and Waagepetersen 2005.

The covariance function $c(u)$ you mention above is not a covariance function in the sense of real (or complex) valued stochastic processes on $R^d$ like the Matérn covariance function. It is a function which inters if you want to calculate the covariance of counts of points $N(A)$ and $N(B)$ occurring in the subsets $A$ and $B$.

The construction is quite simple:

1. Generate a homogeneous Poisson process of some given intensity $\kappa$ (parents).
2. For each generated point place a disc of radius $r$ around the points and place a Poisson number of iid. points (offspring) uniformly in the disc.
3. Delete the original (parent) points so you just observe a superposition of offspring points.

Answer 2

From my point of view these two approaches solve two different problems: 1. For Matern covariance function we consider the correlations between $y(x_1)$ and $y(x_2)$, where $y$ is a realization of a random (Gaussian) process. 2. For Matern cluster process we generate set of points.

So, these two different approaches solve two different problems, and seems to be related only by the same family in their names.