spatstat (version 1.31-3)

pcfdot: 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, ...)

Arguments

X

The observed point pattern, from which an estimate of the dot-type pair correlation function $g_{i\bullet}(r)$ 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).

...

Arguments passed to pcf.ppp.

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 $g_{i\bullet}$ has been estimated
• theo
  the theoretical value $g_{i\bullet}(r) = 1$ 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.

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 $$p(r) = \lambda_i \lambda g_{i\bullet}(r) \,{\rm d}x \, {\rm d}y$$ 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 $g_{i\bullet}(r) = 1$. For a stationary multitype point process, the type-i-to-any-type pair correlation function between marks $i$ and $j$ is formally defined as $$g_{i\bullet}(r) = \frac{K_{i\bullet}^\prime(r)}{2\pi r}$$ where $K_{i\bullet}^\prime$ is the derivative of the type-i-to-any-type $K$ function $K_{i\bullet}(r)$. of the point process. See Kdot for information about $K_{i\bullet}(r)$.

The command pcfdot computes a kernel estimate of the multitype pair correlation function from points of type $i$ to points of any type. It uses pcf.ppp to compute kernel estimates of the pair correlation functions for several unmarked point patterns, and uses the bilinear properties of second moments to obtain the multitype pair correlation.

See pcf.ppp for a list of arguments that control the kernel estimation.

The companion function pcfcross computes the corresponding analogue of Kcross.

See Also

Mark connection function markconnect.

Multitype pair correlation pcfcross. Pair correlation pcf,pcf.ppp. Kdot

Examples

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