Estimates the \(K\)-function from a three-dimensional point pattern.

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

A function value table (object of class `"fv"`

) that can be
plotted, printed or coerced to a data frame containing the function values.

- 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.

- ratio
Logical. If

`TRUE`

, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rana Moyeed.

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:

`"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).

Alternatively `correction="all"`

selects all options.

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"`

).

`pcf3est`

,
`F3est`

,
`G3est`

for other summary functions of
a three-dimensional point pattern.

`Kest`

to estimate the \(K\)-function of
point patterns in two dimensions or other spaces.

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

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