spatstat (version 1.30-0)

pcfcross.inhom: Inhomogeneous Multitype Pair Correlation Function (Cross-Type)

Description

Estimates the inhomogeneous cross-type pair correlation function for a multitype point pattern.

Usage

pcfcross.inhom(X, i, j, lambdaI = NULL, lambdaJ = NULL, ...,
               r = NULL, breaks = NULL,
               kernel="epanechnikov", bw=NULL, stoyan=0.15,
               correction = c("isotropic", "Ripley", "translate"),
               sigma = NULL, varcov = NULL)

Arguments

X

The observed point pattern, from which an estimate of the inhomogeneous 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

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).

j

The type (mark value) of the points in X to which distances are measured. A character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

lambdaI

Optional. Values of the estimated intensity function of the points of type i. Either a vector giving the intensity values at the points of type i, a pixel image (object of class "im") giving the i

lambdaJ

Optional. Values of the estimated intensity function of the points of type j. A numeric vector, pixel image or function(x,y).

r

Vector of values for the argument $r$ at which $g_{ij}(r)$ should be evaluated. There is a sensible default.

breaks

Optional. An alternative to the argument r. Not normally invoked by the user.

kernel

Choice of smoothing kernel, passed to density.default.

bw

Bandwidth for smoothing kernel, passed to density.default.

...

Other arguments passed to the kernel density estimation function density.default.

stoyan

Bandwidth coefficient; see Details.

correction

Choice of edge correction.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambdaI or lambdaJ is estimated by kernel smoothing.

Value

• A function value table (object of class "fv"). Essentially a data frame containing the variables
• r
  the vector of values of the argument $r$ at which the inhomogeneous cross-type pair correlation function $g_{ij}(r)$ has been estimated
• theo
  vector of values equal to 1, the theoretical value of $g_{ij}(r)$ for the Poisson process
• trans
  vector of values of $g_{ij}(r)$ estimated by translation correction
• iso
  vector of values of $g_{ij}(r)$ estimated by Ripley isotropic correction
• as required.

Details

The inhomogeneous cross-type pair correlation function $g_{ij}(r)$ is a summary of the dependence between two types of points in a multitype spatial point process that does not have a uniform density of points.

The best intuitive interpretation is the following: the probability $p(r)$ of finding two points, of types $i$ and $j$ respectively, at locations $x$ and $y$ separated by a distance $r$ is equal to $$p(r) = \lambda_i(x) lambda_j(y) g(r) \,{\rm d}x \, {\rm d}y$$ where $\lambda_i$ is the intensity function of the process of points of type $i$. For a multitype Poisson point process, this probability is $p(r) = \lambda_i(x) \lambda_j(y)$ so $g_{ij}(r) = 1$.

The command pcfcross.inhom estimates the inhomogeneous pair correlation using a modified version of the algorithm in pcf.ppp.

If the arguments lambdaI and lambdaJ are missing or null, they are estimated from X by kernel smoothing using a leave-one-out estimator.

See Also

pcf.ppp, pcfinhom, pcfcross, pcfdot.inhom

Examples

data(amacrine)
  plot(pcfcross.inhom(amacrine, "on", "off", stoyan=0.1),
       legendpos="bottom")