for a multitype point pattern, computes the cross-type version of the local K function.

```
localKcross(X, from, to, ..., rmax = NULL,
correction = "Ripley", verbose = TRUE, rvalue=NULL)
localLcross(X, from, to, ..., rmax = NULL, correction = "Ripley")
```

If `rvalue`

is given, the result is a numeric vector
of length equal to the number of points in the point pattern
that belong to type `from`

.

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
of type `from`

.
The last two columns contain the `r`

and `theo`

values.

- X
A multitype point pattern (object of class

`"ppp"`

with marks which are a factor).- ...
Further arguments passed from

`localLcross`

to`localKcross`

.- rmax
Optional. Maximum desired value of the argument \(r\).

- from
Type of points from which distances should be measured. A single value; one of the possible levels of

`marks(X)`

, or an integer indicating which level.- to
Type of points to which distances should be measured. A single value; one of the possible levels of

`marks(X)`

, or an integer indicating which level.- correction
String specifying the edge correction to be applied. Options are

`"none"`

,`"translate"`

,`"translation"`

,`"Ripley"`

,`"isotropic"`

or`"best"`

. Only one correction may be specified.- 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.

Ege Rubak rubak@math.aau.dk and Adrian Baddeley Adrian.Baddeley@curtin.edu.au.

Given a multitype spatial point pattern `X`

,
the local cross-type \(K\) function `localKcross`

is the local version of the multitype \(K\) function
`Kcross`

.
Recall that `Kcross(X, from, to)`

is a sum of contributions
from all pairs of points in `X`

where
the first point belongs to `from`

and the second point belongs to type `to`

.
The *local* cross-type \(K\)
function is defined for each point `X[i]`

that belongs to
type `from`

, and it consists of all the contributions to
the cross-type \(K\) function that originate from point `X[i]`

:
$$
K_{i,from,to}(r) = \sqrt{\frac a {(n-1) \pi} \sum_j e_{ij}}
$$
where the sum is over all points \(j \neq i\)
belonging to type `to`

, 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 \(K_{i,from,to}(r)\)
can also be interpreted as one
of the summands that contributes to the global estimate of the
`Kcross`

function.

By default, the function \(K_{i,from,to}(r)\)
is computed for a range of \(r\) values
for each point \(i\) belonging to type `from`

.
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`

belonging to type `from`

.

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`

belonging to type `from`

.

The local cross-type \(L\) function `localLcross`

is computed by applying the transformation
\(L(r) = \sqrt{K(r)/(2\pi)}\).

`Kcross`

,
`Lcross`

,
`localK`

,
`localL`

.

Inhomogeneous counterparts of `localK`

and `localL`

are computed by `localKcross.inhom`

and
`localLinhom`

.

```
X <- amacrine
# compute all the local Lcross functions
L <- localLcross(X)
# plot all the local Lcross functions against r
plot(L, main="local Lcross functions for amacrine", legend=FALSE)
# plot only the local L function for point number 7
plot(L, iso007 ~ r)
# compute the values of L(r) for r = 0.1 metres
L12 <- localLcross(X, rvalue=0.1)
```

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