# K3est

##### K-function of a Three-Dimensional Point Pattern

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

- Keywords
- spatial, nonparametric

##### Usage

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

##### 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.
- ratio
- Logical.
If
`TRUE`

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

##### 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]

##### Value

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

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

##### 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

##### Examples

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

*Documentation reproduced from package spatstat, version 1.42-2, License: GPL (>= 2)*