Statistics that are based on second-order characteristics include Ripley’s K function, Besag’s L function, and the pair correlation function (also called the g function). To understand why these functions are based on second-order characteristics, see Illian et al. (2008, p. 223-243). These functions usually involve computation of pairwise distances between points.
The K function of a stationary point process is defined such that is the expected number of points within a distance of r from an arbitrary point of the process. The empirical K function
of a set of points is the weighted and renormalized empirical distribution function of the set of pairwise distances between
points. The empirical K function can be written as
where is the border edge correction that is described in the section Border Edge Correction for Distance Functions.
For a homogeneous Poisson process, can be written as
Exploratory analysis usually involves computing both the empirical K function, , and the K function for a Poisson process,
. A comparison of
and
might indicate clustering or regularity depending on whether
or
.
Besag’s L function is a transformation of the K function and is defined as
For a homogeneous Poisson process, .
The pair correlation function, g(r), can also be expressed as a transformation of the K function:
Illian et al. (2008), Stoyan (1987), and Fiksel (1988) suggest an alternative expression for :
where is the second-order product density function. Cressie and Collins (2001) provides an expression for
as
where can be written as a kernel estimate,
where a is the area, , and
is a kernel such as the uniform kernel or the Epanechnikov kernel (Silverman 1986). PROC SPP uses the version that is based on the uniform kernel; for more information about the uniform kernel, see the section
Nonparametric Intensity Estimation. Based on the formula for the second-order product density
in terms of the kernel estimate, Stoyan (1987) gives an edge-corrected kernel estimate for
as
Dividing by
gives the pair correlation function
as
where indicates the translation of the study area window W by the distance
from its origin. The above expression for
was given by Stoyan and Stoyan (1994) using the translation edge correction.
A border-edge-corrected version of can be written as
where and
are points within the boundary at a distance greater than or equal to r; where
is the distance of
to the boundary of W,
; and where
for a kernel
, such as the uniform kernel or the Epanechnikov kernel. For more information about the uniform kernel, see the section Nonparametric Intensity Estimation. For a homogeneous Poisson process,
. For any point pattern, values of
greater than 1 indicate clustering or attraction at distance r, whereas values of
less than 1 indicate regularity.