spatstat (version 1.14-10)

pcfcross: Multitype pair correlation function

Description

Calculates an estimate of the cross-type pair correlation function for a multitype point pattern.

Usage

pcfcross(X, i, 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

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.

...

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 $g_{i,j}$ has been estimated
• theo
  the theoretical value $g_{i,j}(r) = 1$ 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

The cross-type pair correlation function is a generalisation of the pair correlation function pcf to multitype point patterns.

For two locations $x$ and $y$ separated by a distance $r$, the probability $p(r)$ of finding a point of type $i$ at location $x$ and a point of type $j$ at location $y$ is $$p(r)={\lambda }_i{\lambda }_j{g}_{i,j}(r)\mathrm{d}x\mathrm{d}y$$ where $\lambda_i$ is the intensity of the points of type $i$. For a completely random Poisson marked point process, $p(r) = \lambda_i \lambda_j$ so $g_{i,j}(r) = 1$. Indeed for any marked point pattern in which the points of type i are independent of the points of type j, the theoretical value of the cross-type pair correlation is $g_{i,j}(r) = 1$. For a stationary multitype point process, the cross-type pair correlation function between marks $i$ and $j$ is formally defined as $$g_{i,j}(r)=\frac{K_{i,j}^{\prime }(r)}{2\pi r}$$ where $K_{i,j}^\prime$ is the derivative of the cross-type $K$ function $K_{i,j}(r)$. of the point process. See Kest for information about $K(r)$.

The command pcfcross computes a kernel estimate of the cross-type pair correlation function between marks $i$ and $j$. 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 cross-type pair correlation.

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

See Also

pcf, pcf.ppp, Kcross

Examples

data(amacrine)
 p <- pcfcross(amacrine, "off", "on")
 plot(p)