R: Multitype pair correlation function (i-to-any)

pcfdot {spatstat} | R Documentation

## Multitype pair correlation function (i-to-any)

### Description

Calculates an estimate of the multitype pair correlation function (from points of type `i` to points of any type) for a multitype point pattern.

### Usage

      pcfdot(X, i, ..., r = NULL,
             kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
             correction = c("isotropic", "Ripley", "translate"),
             divisor = c("r", "d"))

### Arguments

`X` | The observed point pattern, from which an estimate of the dot-type pair correlation function gdot[i](r) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor).
`i` | The type (mark value) of the points in `X` from which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of `marks(X)`.
`...` | 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; see Details.
`correction` | Choice of edge correction.
`divisor` | Choice of divisor in the estimation formula: either `"r"` (the default) or `"d"`. See Details.

### Details

This is a generalisation of the pair correlation function `pcf` to multitype point patterns.

For two locations x and y separated by a nonzero distance r, the probability p(r) of finding a point of type i at location x and a point of any type at location y is

p(r) = lambda[i] * lambda * gdot[i](r) dx dy

where lambda is the intensity of all points, and lambda[i] is the intensity of the points of type i. For a completely random Poisson marked point process, p(r) = lambda[i] * lambda so gdot[i](r) = 1.

For a stationary multitype point process, the type-`i`-to-any-type pair correlation function between marks i and j is formally defined as

g(r) = Kdot[i]'(r)/ ( 2 * pi * r)

where Kdot[i]'(r) is the derivative of the type-`i`-to-any-type K function Kdot[i](r). of the point process. See `Kdot` for information about Kdot[i](r).

The command `pcfdot` computes a kernel estimate of the multitype pair correlation function from points of type i to points of any type.

- 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 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). That is, h = c/sqrt(lambda), where lambda is the (estimated) intensity of the unmarked point process, and c is a constant in the range from 0.1 to 0.2. The argument `stoyan` determines the value of c.

The companion function `pcfcross` computes the corresponding analogue of `Kcross`.

### Value

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 gdot[i] has been estimated
`theo` | the theoretical value gdot[i](r) = r for independent marks.

together with columns named `"border"`, `"bord.modif"`, `"iso"` and/or `"trans"`, according to the selected edge corrections. These columns contain estimates of the function g[i,j] obtained by the edge corrections named.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

### See Also

Mark connection function `markconnect`.

Multitype pair correlation `pcfcross`, `pcfmulti`.

Pair correlation `pcf`,`pcf.ppp`.

`Kdot`

### Examples

     data(amacrine)
     p <- pcfdot(amacrine, "on")
     p <- pcfdot(amacrine, "on", stoyan=0.1)
     plot(p)

[Package spatstat version 1.54-0 Index]