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)

Arguments

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)

Number or character string identifying the type (mark value) of the points in X from which distances are measured.

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

lambdadot

Optional. Values of the estimated intensity function of the point pattern X. A numeric vector, pixel image or function(x,y).

Vector of values for the argument $r$ at which $g_{i\bullet}(r)$ 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.

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.

Value

A function value table (object of class "fv"). Essentially a data frame containing the variables
rthe vector of values of the argument $r$ at which the inhomogeneous multitype pair correlation function $g_{i\bullet}(r)$ has been estimated
theovector of values equal to 1, the theoretical value of $g_{i\bullet}(r)$ for the Poisson process
transvector of values of $g_{i\bullet}(r)$ estimated by translation correction
isovector of values of $g_{i\bullet}(r)$ estimated by Ripley isotropic correction
as required.

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

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

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