pcfdot.inhom: Inhomogeneous Multitype Pair Correlation Function (Type-i-To-Any-Type)

Description

Estimates the inhomogeneous multitype pair correlation function (from type $i$ to any type) for a multitype point pattern.

Usage

pcfdot.inhom(X, i, lambdaI = NULL, lambdadot = 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 multitype pair correlation function $g_{i\bullet}(r)$ has been estimated
theo: vector of values equal to 1, the theoretical value of $g_{i\bullet}(r)$ for the Poisson process
trans: vector of values of $g_{i\bullet}(r)$ estimated by translation correction
iso: vector of values of $g_{i\bullet}(r)$ estimated by Ripley isotropic correction

as required.

Arguments

X: The observed point pattern, from which an estimate of the inhomogeneous multitype 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).
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.
lambdadot: Optional. Values of the estimated intensity function of the point pattern X. A numeric vector, pixel image or function(x,y).
r: Vector of values for the argument $r$ at which $g_{i\bullet}(r)$ 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 lambdadot is estimated by kernel smoothing.

Author

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

Details

The inhomogeneous multitype (type $i$ to any type) pair correlation function $g_{i\bullet}(r)$ is a summary of the dependence between different 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 a point of type $i$ at location $x$ and another point of any type at location $y$, where $x$ and $y$ are separated by a distance $r$, is equal to $$ p(r) = \lambda_i(x) lambda(y) g(r) \,{\rm d}x \, {\rm d}y $$ where $\lambda_i$ is the intensity function of the process of points of type $i$, and where $\lambda$ is the intensity function of the points of all types. For a multitype Poisson point process, this probability is $p(r) = \lambda_i(x) \lambda(y)$ so $g_{i\bullet}(r) = 1$.

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

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

Examples

Run this code

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

Run the code above in your browser using DataLab