10 K-function
10.1 Motivation
The PCF \(g(r)\) describes how interaction between points varies with distance.
However, \(g(r)\) is a local measure - it focuses on behavior at specific distance \(r\).
In many situations, it is useful to have a summary that captures the total interaction up to a given distance.
This leads to the \(K\)-function.
10.2 Definition
The \(K\)-function, denoted \(K(r)\), is defined such that
\[ \lambda K(r) \]
is the expected number of further points within distance \(r\) of a typical point.
The \(K\)-function answers the question:
Given a point of the process, how many other points do we expect to find within distance \(r\)?
More precisely:
- \(K(r)\) describes the cumulative interaction up to distance \(r\).
- It aggregates behavior over all distances from \(0\) to \(r\).
Just remember:
- \(g(r)\) tells us what happens AT distance \(r\)
- \(K(r)\) tells us what happens WITHIN distance \(r\)
10.3 Relation to the Poisson process
For a Poisson process with intensity \(\lambda\), we have:
\[ K(r) = \pi r^2 \]
No explanation why??????? Are you fucking serious dog
This corresponds to the area of a disc of radius \(r\), reflecting complete spatial randomness.
Shitty ahh explanation.
10.4 Detecting interaction
Comparing \(K(r)\) to \(\pi r^2\) reveals the spatial structure:
\(K(r) > \pi r^2 \Longrightarrow\) clustering
\(K(r) < \pi r^2 \Longrightarrow\) repulsion
\(K(r) = \pi r^2 \Longrightarrow\) no interaction
10.5 Relationship to the PCF
The \(K\)-function and the PCF are closely related.
In particular, the \(K\)-function can be expressed as
\[ K(r) = \int_0^r 2\pi s\ g(s)\ ds \]
Thus:
- the PCF \(g(r)\) describes local interaction
- the \(K\)-function aggregates this into a cumulative measure.
The \(K\)-function is essentially a smoothed, accumulated version of the PCF.
10.6 Advantages and Limitations
Advantages
Easier to estimate in practice (less noisy than the PCF)
Provides a stable summary of interaction (is this not just the same as the last point).
Widely used in exploratory analysis
Limitations
Being cumulative, it can mask local features
Different processes can produce very similar \(K(r)\)
Interpretation is less direct than for \(g(r)\)
10.7 Summary
- The \(K\)-function describes the expected number of neighbours within distance \(r\)
- It provides a cumulative measure of interaction
- It is related to the PCF via: \[ K(r) = \int_0^r 2\pi s \, g(s)\, ds \]
- Comparing \(K(r)\) to \(\pi r^2\) reveals clustering or repulsion
10.8 Next Step
We now consider how second-order quantities such as the PCF and \(K\)-function can be estimated from observed data.