# markconnect

##### Mark Connection Function

Estimate the marked connection function of a multitype point pattern.

- Keywords
- spatial, nonparametric

##### Usage

```
markconnect(X, i, j, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", ..., normalise=FALSE)
```

##### Arguments

- X
- The observed point pattern.
An object of class
`"ppp"`

or something acceptable to`as.ppp`

. - i
- Number or character string identifying the type (mark value)
of the points in
`X`

from which distances are measured. - j
- Number or character string identifying the type (mark value)
of the points in
`X`

to which distances are measured. - r
- numeric vector. The values of the argument $r$ at which the mark connection function $p_{ij}(r)$ should be evaluated. There is a sensible default.
- correction
- A character vector containing any selection of the
options
`"isotropic"`

,`"Ripley"`

or`"translate"`

. It specifies the edge correction(s) to be applied. - method
- A character vector indicating the user's choice of
density estimation technique to be used. Options are
`"density"`

,`"loess"`

,`"sm"`

and`"smrep"`

. - ...
- Arguments passed to the density estimation routine
(
`density`

,`loess`

or`sm.density`

) selected by`method`

. - normalise
- If
`TRUE`

, normalise the pair connection function by dividing it by $p_i p_j$, the estimated probability that randomly-selected points will have marks $i$ and $j$.

##### Details

The mark connection function $p_{ij}(r)$ of a multitype point process $X$ is a measure of the dependence between the types of two points of the process a distance $r$ apart.

Informally $p_{ij}(r)$ is defined as the conditional probability, given that there is a point of the process at a location $u$ and another point of the process at a location $v$ separated by a distance $||u-v|| = r$, that the first point is of type $i$ and the second point is of type $j$. See Stoyan and Stoyan (1994).

If the marks attached to the points of `X`

are independent
and identically distributed, then
$p_{ij}(r) \equiv p_i p_j$ where
$p_i$ denotes the probability that a point is of type
$i$. Values larger than this,
$p_{ij}(r) > p_i p_j$,
indicate positive association between the two types,
while smaller values indicate negative association.

The argument `X`

must be a point pattern (object of class
`"ppp"`

) or any data that are acceptable to `as.ppp`

.
It must be a multitype point pattern (a marked point pattern
with factor-valued marks).

The argument `r`

is the vector of values for the
distance $r$ at which $p_{ij}(r)$ is estimated.
There is a sensible default.

This algorithm assumes that `X`

can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in `X`

as `X$window`

)
may have arbitrary shape.

Biases due to edge effects are
treated in the same manner as in `Kest`

.
The edge corrections implemented here are
[object Object],[object Object]
Note that the estimator assumes the process is stationary (spatially
homogeneous).

The mark connection function is estimated using density estimation techniques. The user can choose between [object Object],[object Object],[object Object],[object Object]

##### Value

- An object of class
`"fv"`

(see`fv.object`

). Essentially a data frame containing numeric columns r the values of the argument $r$ at which the mark connection function $p_{ij}(r)$ has been estimated theo the theoretical value of $p_{ij}(r)$ when the marks attached to different points are independent - together with a column or columns named
`"iso"`

and/or`"trans"`

, according to the selected edge corrections. These columns contain estimates of the function $p_{ij}(r)$ obtained by the edge corrections named.

##### References

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

##### See Also

Multitype pair correlation `pcfcross`

and multitype K-functions `Kcross`

, `Kdot`

.

Use `alltypes`

to compute the mark connection functions
between all pairs of types.

Mark correlation `markcorr`

and
mark variogram `markvario`

for numeric-valued marks.

##### Examples

```
# Hughes' amacrine data
# Cells marked as 'on'/'off'
data(amacrine)
M <- markconnect(amacrine, "on", "off")
plot(M)
# Compute for all pairs of types at once
plot(alltypes(amacrine, markconnect))
```

*Documentation reproduced from package spatstat, version 1.20-2, License: GPL (>= 2)*