spatstat (version 1.23-1)

K3est: K-function of a Three-Dimensional Point Pattern

Description

Estimates the $K$-function from a three-dimensional point pattern.

Usage

K3est(X, ..., rmax = NULL, nrval = 128, correction = c("translation", "isotropic"))

Arguments

X

Three-dimensional point pattern (object of class "pp3").

...

Ignored.

rmax

Optional. Maximum value of argument $r$ for which $K_3(r)$ will be estimated.

nrval

Optional. Number of values of $r$ for which $K_3(r)$ will be estimated. A large value of nrval is required to avoid discretisation effects.

correction

Optional. Character vector specifying the edge correction(s) to be applied. See Details.

Value

• A function value table (object of class "fv") that can be plotted, printed or coerced to a data frame containing the function values.

Details

For a stationary point process $\Phi$ in three-dimensional space, the three-dimensional $K$ function is $$K_3(r) = \frac 1 \lambda E(N(\Phi, x, r) \mid x \in \Phi)$$ where $\lambda$ is the intensity of the process (the expected number of points per unit volume) and $N(\Phi,x,r)$ is the number of points of $\Phi$, other than $x$ itself, which fall within a distance $r$ of $x$. This is the three-dimensional generalisation of Ripley's $K$ function for two-dimensional point processes (Ripley, 1977). The three-dimensional point pattern X is assumed to be a partial realisation of a stationary point process $\Phi$. The distance between each pair of distinct points is computed. The empirical cumulative distribution function of these values, with appropriate edge corrections, is renormalised to give the estimate of $K_3(r)$.

The available edge corrections are: [object Object],[object Object]

References

Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. (1993) Analysis of a three-dimensional point pattern with replication. Applied Statistics 42, 641--668.

Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 -- 71.

Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society, Series B, 39, 172 -- 212.

See Also

F3est, G3est, pcf3est

Examples

X <- rpoispp3(42)
  Z <- K3est(X)
  if(interactive()) plot(Z)