9  Pair Correlation Function

9.1 Motivation

In the previous section, we introduced the second-order product density \(\lambda^{(2)}(u, v)\), which describes the joint occurrences of points at two locations.

However, this quantity is difficult to interpret directly:

  • its magnitude depends on the intensity \(\ambda(u)\),
  • it is not scale invariant
  • and it does not isolate pure interaction between points.

To address this, we introduct a normalized version.

9.2 Definition

The pair correlation function (PCF) is defined as

\[ g(u, v) = \frac{\lambda^{(2)}(u, v)}{\lambda(u) \lambda(v)}, \quad u \neq v \] ## Interpretation

The PCF compares the observed joint occurrence of points at \(u\) and \(v\), to what would be expected under independence.

In particular:

  • \(g(u, v) = 1 \Longrightarrow\) no interaction (Poisson-like)

  • \(g(u, v) > 1 \Longrightarrow\) clustering (points occur together more often than expected)

  • \(g(u, v) > 1 \Longrightarrow\) repulsion (points inhibit each other)

The PCF removes the effect of intensity. It isolates the pure spatial ineraction between points.

9.3 Relation to the Poisson process

For a Poisson process, we have:

\[ \lambda^{(2)}(u, v) = \lambda(u) \lambda(v) \]

and hence:

\[ g(u, v) = 1 \]

Thus, the PCF can be interpreted as a measure of deviation from complete spatial randomness.

9.4 Interpretation as a ratio

Another useful way to interpret the PCF is:

\[ g(u,v) \approx \frac{\text{probability of observing a pair at } (u,v)} {\text{probability under independence}} \]

So:

  • \(g(u,v) = 2\) means that pairs occur twice as often as expected

  • \(g(u, v) = 0.5\) means that pairs occur half as often as expected

9.5 Distance based form

In many situations, interaction depends only on the distance between points.

In such cases, we write:

\[ r = \|u - v\| \]

and express the PCF as a function of distance, \(g(r)\).

This reduces the problem from studying interactions between all pairs of locations to understanding how interaction varies with separation distance.

This \(g(r)\) form is most commonly used in practice.

9.6 Examples (conceptual)

An example of a clustered process:

par(mfrow = c(1, 2))
plot(longleaf)
plot(pcf(longleaf))

And an example of a regular process:

par(mfrow = c(1, 2))
plot(hamster)
plot(pcf(hamster))

9.7 Summary

  • The PCF is defined as: \[ g(u,v) = \frac{\lambda^{(2)}(u,v)}{\lambda(u)\lambda(v)}. \]
  • It measures deviation from independence.
  • Values of:
    • \(g > 1\) indicate clustering,
    • \(g < 1\) indicate repulsion.
  • When expressed as \(g(r)\), it describes how interaction varies with distance.

9.8 Next Step

We now explore another second-order summary: the \(K\)-function, which aggregates interaction over distance and provides a complementary perspective.