# pcfmulti

##### Marked pair correlation function

For a marked point pattern, estimate the multitype pair correlation function using kernel methods.

- Keywords
- spatial, nonparametric

##### Usage

```
pcfmulti(X, I, J, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("translate", "Ripley"),
divisor = c("r", "d"),
Iname = "points satisfying condition I",
Jname = "points satisfying condition J")
```

##### Arguments

- X
- The observed point pattern, from which an estimate of the cross-type pair correlation function $g_{ij}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor).
- I
- Subset index specifying the points of
`X`

from which distances are measured. - J
- Subset index specifying the points in
`X`

to which distances are measured. - ...
- Ignored.
- r
- Vector of values for the argument $r$ at which $g(r)$ should be evaluated. There is a sensible default.
- kernel
- Choice of smoothing kernel,
passed to
`density.default`

. - bw
- Bandwidth for smoothing kernel,
passed to
`density.default`

. - stoyan
- Coefficient for default bandwidth rule.
- correction
- Choice of edge correction.
- divisor
- Choice of divisor in the estimation formula:
either
`"r"`

(the default) or`"d"`

. - Iname,Jname
- Optional. Character strings describing the members of
the subsets
`I`

and`J`

.

##### Details

This is a generalisation of `pcfcross`

to arbitrary collections of points.

The algorithm measures the distance from each data point
in subset `I`

to each data point in subset `J`

,
excluding identical pairs of points. The distances are
kernel-smoothed and renormalised to form a pair correlation
function.

- If
`divisor="r"`

(the default), then the multitype counterpart of the standard kernel estimator (Stoyan and Stoyan, 1994, pages 284--285) is used. By default, the recommendations of Stoyan and Stoyan (1994) are followed exactly. - If
`divisor="d"`

then a modified estimator is used: the contribution from an interpoint distance$d_{ij}$to the estimate of$g(r)$is divided by$d_{ij}$instead of dividing by$r$. This usually improves the bias of the estimator when$r$is close to zero.

There is also a choice of spatial edge corrections
(which are needed to avoid bias due to edge effects
associated with the boundary of the spatial window):
`correction="translate"`

is the Ohser-Stoyan translation
correction, and `correction="isotropic"`

or `"Ripley"`

is Ripley's isotropic correction.

The arguments `I`

and `J`

specify two subsets of the
point pattern `X`

. They may be any type of subset indices, for example,
logical vectors of length equal to `npoints(X)`

,
or integer vectors with entries in the range 1 to
`npoints(X)`

, or negative integer vectors.

Alternatively, `I`

and `J`

may be **functions**
that will be applied to the point pattern `X`

to obtain
index vectors. If `I`

is a function, then evaluating
`I(X)`

should yield a valid subset index. This option
is useful when generating simulation envelopes using
`envelope`

.

The choice of smoothing kernel is controlled by the
argument `kernel`

which is passed to `density`

.
The default is the Epanechnikov kernel.

The bandwidth of the smoothing kernel can be controlled by the
argument `bw`

. Its precise interpretation
is explained in the documentation for `density.default`

.
For the Epanechnikov kernel with support $[-h,h]$,
the argument `bw`

is equivalent to $h/\sqrt{5}$.

If `bw`

is not specified, the default bandwidth
is determined by Stoyan's rule of thumb (Stoyan and Stoyan, 1994, page
285) applied to the points of type `j`

. That is,
$h = c/\sqrt{\lambda}$,
where $\lambda$ is the (estimated) intensity of the
point process of type `j`

,
and $c$ is a constant in the range from 0.1 to 0.2.
The argument `stoyan`

determines the value of $c$.

##### Value

- An object of class
`"fv"`

.

##### See Also

##### Examples

```
adult <- (marks(longleaf) >= 30)
juvenile <- !adult
p <- pcfmulti(longleaf, adult, juvenile)
```

*Documentation reproduced from package spatstat, version 1.41-1, License: GPL (>= 2)*