8 Second-Order Product Density
8.1 Motivation
In the previous chapter, we introduced the intensity function, \(\lambda(u)\), which describes the “first-order structure” of a point process (how the expected number of points vary over space).
However intensity alone is not sufficient to describe spatial structure.
Two point processes may have the same intensity function, but exhibit very different spatial behavior:
- points may appear independently (as in a Poisson process),
- or may exhibit clustering or repulsion
To capture this, we need to study how pairs of points interact.
This leads us to the second-order product density.
8.2 Definition
Let \(X\) be a point process on a region \(W \subset \mathbb{R}^2\).
The second-order product density \(\lambda^{(2)}(u,v)\) is defined such that
\[ \mathbb{E}[N(du)N(dv)] = \lambda^{(2)}(u,v) \ du \ dv, \quad u \neq v \]
where:
- \(N(du)\) is the number of points in an infinitesimal region around \(u\),
- \(du, dv\) are the infinitesimal areas.
Strictly speaking, the definition applies for \(u \neq v\), since a (simple) point process does not place multiple points at exactly the same location.
8.3 Interpretation
The second order product density describes:
The expected joint occurrence of points near \(u\) and \(v\).
More precisely:
- \(\lambda^{(2)}(u,v) \ du \ dv\) is approximately the probability of observing both a point in a small region around \(u\) AND a point in a small region around \(v\).
Also note that \(\lambda^{(2)}(u,v)\) is symmetric:
\[ \lambda^{(2)}(u,v) = \lambda^{(2)}(v,u) \]
which reflects that the joint occurrence of points at \(u\) and \(v\) does not depend on order.
8.4 Relation to intensity
Recall that intensity satisfies:
\[ \mathbb{E}[N(du)] = \lambda(u) \ du \]
If points are completely independent, we would expect:
\[ \mathbb{E}[N(du)N(dv)] = \mathbb{E}[N(du)]\mathbb{E}[N(dv)] = \lambda(u)\lambda(v) \ du \ dv \]
That is,
\[ \lambda^{(2)}(u,v) = \lambda(u)\lambda(v) \]
This is exactly the case for a Poisson process.
The key idea is that:
\(\lambda(u)\) tells us how many points we expect.
\(\lambda^{(2)}(u,v)\) tells us how points co-occur.
Second-order density captures dependence between locations.
8.5 Detecting interaction
Comparing \(\lambda^{(2)}(u,v)\) to \(\lambda(u)\lambda(v)\) reveals spatial structure:
\(\lambda^{(2)}(u,v) > \lambda(u)\lambda(v)\) is indicative of clustering.
\(\lambda^{(2)}(u,v) < \lambda(u)\lambda(v)\) is indicative of repulsion.
\(\lambda^{(2)}(u,v) = \lambda(u)\lambda(v)\) is indicative of no interaction.
8.6 Limitations
While the second-order product density is fundamental, it is not always easy to directly interpret.
In particular:
- Its scale depends on the intensity \(\lambda(u)\)
- It is difficult to compare across different processes
- It does not provide an immediately intuitive mesure of interaction
Exemplify…
For example, even under independence, we have:
\[ \lambda^{(2)}(u,v) = \lambda(u)\lambda(v) \]
is constant unless the intensity is constant.
This makes it difficult to distinguish between:
- variation due to intensity and
- variation due to interaction.
8.7 Towards a normalised Measure
To better understand spatial interaction, we would like quantity that:
- removes the effect of first-oreder intensity, and
- isolates pure interaction between points.
This leads to the definition of the pair correlation function (PCF), which we introduce in the next section.
8.8 Summary
- The second-order product density \(\lambda^{(2)}(u,v)\) describes the joint occurrence of points at two locations.
- It generalizes the intensity to pairs of locations.
- Comparing \(\lambda^{(2)}(u,v)\) to \(\lambda(u)\lambda(v)\) reveals:
- clustering,
- repulsion,
- or independence.
- However, it is not easy to interpret directly, motivating a normalized measure of interaction.