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.

- X
Three-dimensional point pattern (object of class

`"pp3"`

).- ...
Ignored.

- rmax
Optional. Maximum value of argument \(r\) for which \(g_3(r)\) will be estimated.

- nrval
Optional. Number of values of \(r\) for which \(g_3(r)\) will be estimated.

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

- delta
Optional. Half-width of the Epanechnikov smoothing kernel.

- adjust
Optional. Adjustment factor for the default value of

`delta`

.- biascorrect
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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