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
Based on the preceding expression for the product density and a planar version of Moller and Waagepetersen (2004), can be written as
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 , ; 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.