# localK

##### Neighbourhood density function

Computes the neighbourhood density function, a local version of the $K$-function or $L$-function, defined by Getis and Franklin (1987).

- Keywords
- spatial, nonparametric

##### Usage

```
localK(X, ..., correction = "Ripley", verbose = TRUE, rvalue=NULL)
localL(X, ..., correction = "Ripley", verbose = TRUE, rvalue=NULL)
```

##### Arguments

- X
- A point pattern (object of class
`"ppp"`

). - ...
- Ignored.
- correction
- String specifying the edge correction to be applied.
Options are
`"none"`

,`"translate"`

,`"translation"`

,`"Ripley"`

,`"isotropic"`

or`"best"`

. Only one correction may be - verbose
- Logical flag indicating whether to print progress reports during the calculation.
- rvalue
- Optional. A
*single*value of the distance argument $r$ at which the function L or K should be computed.

##### Details

The command `localL`

computes the *neighbourhood density function*,
a local version of the $L$-function (Besag's transformation of Ripley's
$K$-function) that was proposed by Getis and Franklin (1987).
The command `localK`

computes the corresponding
local analogue of the K-function.

Given a spatial point pattern `X`

, the neighbourhood density function
$L_i(r)$ associated with the $i$th point
in `X`

is computed by
$$L_i(r) = \sqrt{\frac a {(n-1) \pi} \sum_j e_{ij}}$$
where the sum is over all points $j \neq i$ that lie
within a distance $r$ of the $i$th point,
$a$ is the area of the observation window, $n$ is the number
of points in `X`

, and $e_{ij}$ is an edge correction
term (as described in `Kest`

).
The value of $L_i(r)$ can also be interpreted as one
of the summands that contributes to the global estimate of the L
function.

By default, the function $L_i(r)$ or
$K_i(r)$ is computed for a range of $r$ values
for each point $i$. The results are stored as a function value
table (object of class `"fv"`

) with a column of the table
containing the function estimates for each point of the pattern
`X`

.

Alternatively, if the argument `rvalue`

is given, and it is a
single number, then the function will only be computed for this value
of $r$, and the results will be returned as a numeric vector,
with one entry of the vector for each point of the pattern `X`

.

Inhomogeneous counterparts of `localK`

and `localL`

are computed by `localKinhom`

and `localLinhom`

.

##### Value

- If
`rvalue`

is given, the result is a numeric vector of length equal to the number of points in the point pattern.If

`rvalue`

is absent, the result is an object of class`"fv"`

, see`fv.object`

, which can be plotted directly using`plot.fv`

. Essentially a data frame containing columns r the vector of values of the argument $r$ at which the function $K$ has been estimated theo the theoretical value $K(r) = \pi r^2$ or $L(r)=r$ for a stationary Poisson process - together with columns containing the values of the
neighbourhood density function for each point in the pattern.
Column
`i`

corresponds to the`i`

th point. The last two columns contain the`r`

and`theo`

values.

##### References

Getis, A. and Franklin, J. (1987)
Second-order neighbourhood analysis of mapped point patterns.
*Ecology* **68**, 473--477.

##### See Also

##### Examples

```
data(ponderosa)
X <- ponderosa
# compute all the local L functions
L <- localL(X)
# plot all the local L functions against r
plot(L, main="local L functions for ponderosa", legend=FALSE)
# plot only the local L function for point number 7
plot(L, iso007 ~ r)
# compute the values of L(r) for r = 12 metres
L12 <- localL(X, rvalue=12)
# Spatially interpolate the values of L12
# Compare Figure 5(b) of Getis and Franklin (1987)
X12 <- X %mark% L12
Z <- smooth.ppp(X12, sigma=5, dimyx=128)
plot(Z, col=topo.colors(128), main="smoothed neighbourhood density")
contour(Z, add=TRUE)
points(X, pch=16, cex=0.5)
```

*Documentation reproduced from package spatstat, version 1.31-2, License: GPL (>= 2)*