pcfcross

Multitype pair correlation function (cross-type)

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

Usage

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

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 $$ p(r) = \lambda_i \lambda_j g_{i,j}(r) \,{\rm d}x \, {\rm d}y $$ 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 \(j\).

• 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) 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 \(c\).

The companion function pcfdot computes the corresponding analogue of Kdot.

See Also

Mark connection function markconnect.

Multitype pair correlation pcfdot, pcfmulti.

Pair correlation pcf,pcf.ppp.

Kcross

Aliases

• pcfcross

Examples

# NOT RUN {
 data(amacrine)
 p <- pcfcross(amacrine, "off", "on")
 p <- pcfcross(amacrine, "off", "on", stoyan=0.1)
 plot(p)
# }