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)

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.

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 intensity values at all locations, or a function(x,y) which can be evaluated to give the intensity value at any location.
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: This argument is for internal use only.
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.

Author

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz

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.

Examples

Run this code

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

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