Estimates the pair correlation function from a three-dimensional point pattern.
pcf3est(X, ..., rmax = NULL, nrval = 128,
correction = c("translation", "isotropic"),
delta=NULL, adjust=1, biascorrect=TRUE)A function value table (object of class "fv") that can be
plotted, printed or coerced to a data frame containing the function
values.
Additionally the value of delta is returned as an attribute
of this object.
Three-dimensional point pattern (object of class "pp3").
Ignored.
Optional. Maximum value of argument \(r\) for which \(g_3(r)\) will be estimated.
Optional. Number of values of \(r\) for which \(g_3(r)\) will be estimated.
Optional. Character vector specifying the edge correction(s) to be applied. See Details.
Optional. Half-width of the Epanechnikov smoothing kernel.
Optional. Adjustment factor for the default value of delta.
Logical value. Whether to correct for underestimation due to truncation of the kernel near \(r=0\).
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rana Moyeed.
For a stationary point process \(\Phi\) in three-dimensional
space, the pair correlation function is
$$
g_3(r) = \frac{K_3'(r)}{4\pi r^2}
$$
where \(K_3'\) is the derivative of the
three-dimensional \(K\)-function (see K3est).
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.
Kernel smoothing is applied to these distance values (weighted by
an edge correction factor) and the result is
renormalised to give the estimate of \(g_3(r)\).
The available edge corrections are:
"translation":the Ohser translation correction estimator (Ohser, 1983; Baddeley et al, 1993)
"isotropic":the three-dimensional counterpart of Ripley's isotropic edge correction (Ripley, 1977; Baddeley et al, 1993).
Kernel smoothing is performed using the Epanechnikov kernel
with half-width delta. If delta is missing, the
default is to use the rule-of-thumb
\(\delta = 0.26/\lambda^{1/3}\) where
\(\lambda = n/v\) is the estimated intensity, computed
from the number \(n\) of data points and the volume \(v\) of the
enclosing box. This default value of delta is multiplied by
the factor adjust.
The smoothing estimate of the pair correlation \(g_3(r)\)
is typically an underestimate when \(r\) is small, due to
truncation of the kernel at \(r=0\).
If biascorrect=TRUE, the smoothed estimate is
approximately adjusted for this bias. This is advisable whenever
the dataset contains a sufficiently large number of points.
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.
pp3 to create a three-dimensional point
pattern (object of class "pp3").
F3est,
G3est,
K3est for other summary functions of
a three-dimensional point pattern.
pcf to estimate the pair correlation function of
point patterns in two dimensions or other spaces.
X <- rpoispp3(250)
Z <- pcf3est(X)
Zbias <- pcf3est(X, biascorrect=FALSE)
if(interactive()) {
opa <- par(mfrow=c(1,2))
plot(Z, ylim.covers=c(0, 1.2))
plot(Zbias, ylim.covers=c(0, 1.2))
par(opa)
}
attr(Z, "delta")
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