Introduction

The K-function is a method used in spatial Point Pattern Analysis
(PPA) to inspect the spatial distribution of a set of points. It allows
the user to assess if the set of points is more or less clustered that
what we could expect from a given distribution. Most of the time, the
set of point is compared with a random distribution.

The empirical K-function for a specified radius \(r\) is calculated with the following
formula:

\[\hat{K}(r)=\frac{1}{n(n-1)}
\sum_{i=1}^{n} \sum_{j=1 \atop j \neq i}^{n} \mathbf{1}\left\{d_{i j}
\leq r\right\}\] Basically, the K-function calculates for a
radius \(r\) the proportion of cells
with a value bellow \(r\) in the
distance matrix between all the points \(D_{ij}\). In other words, the K-function
estimates “the average number of neighbours of a typical random point”
(Baddeley, Rubak, and Turner 2015).

A modified version of the K-function is the G-function (Pair
Correlation Function) (Stoyan and Stoyan
1996). The regular K-function is calculated for subsequent
disks with increasing radii and thus is cumulative in nature. The
G-function uses rings instead of disks and permits the analysis of the
points concentrations at different geographical scales.

When the points are located on a network, the use of the Euclidean
distance systematically underestimates the real distance between points.
These two functions can be extended for network spaces by using the
network distance instead of the Euclidean distance. The value of the
empirical k function on a network is calculated with the following
formula:

\[\hat{K}(r)=\frac{1}{(n-1)/Lt}
\sum_{i=1}^{n} \sum_{j=1 \atop j \neq i}^{n} \mathbf{1}\left\{d_{i j}
\leq r\right\}\]

With \(Lt\) the total length of the
network and \(n\) the number of
events.