K-function of a Three-Dimensional Point Pattern
Estimates the $K$-function from a three-dimensional point pattern.
K3est(X, ..., rmax = NULL, nrval = 128, correction = c("translation", "isotropic"), ratio=FALSE)
- Three-dimensional point pattern (object of class
- Optional. Maximum value of argument $r$ for which $K_3(r)$ will be estimated.
- Optional. Number of values of $r$ for which
$K_3(r)$ will be estimated. A large value of
nrvalis required to avoid discretisation effects.
- Optional. Character vector specifying the edge correction(s) to be applied. See Details.
TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.
For a stationary point process $\Phi$ in three-dimensional
space, the three-dimensional $K$ function
$$K_3(r) = \frac 1 \lambda E(N(\Phi, x, r) \mid x \in \Phi)$$
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]
- A function value table (object of class
"fv") that can be plotted, printed or coerced to a data frame containing the function values.
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.
X <- rpoispp3(42) Z <- K3est(X) if(interactive()) plot(Z)