Estimates the inhomogeneous multitype pair correlation function (from type \(i\) to any type) for a multitype point pattern.
pcfdot.inhom(X, i, lambdaI = NULL, lambdadot = NULL, ...,
r = NULL, breaks = NULL,
kernel="epanechnikov", bw=NULL, adjust.bw=1, stoyan=0.15,
correction = c("isotropic", "Ripley", "translate"),
sigma = NULL, adjust.sigma = 1, varcov = NULL)A function value table (object of class "fv").
Essentially a data frame containing the variables
the vector of values of the argument \(r\) at which the inhomogeneous multitype pair correlation function \(g_{i\bullet}(r)\) has been estimated
vector of values equal to 1, the theoretical value of \(g_{i\bullet}(r)\) for the Poisson process
vector of values of \(g_{i\bullet}(r)\) estimated by translation correction
vector of values of \(g_{i\bullet}(r)\) estimated by Ripley isotropic correction
as required.
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).
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).
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.
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.
This argument is for internal use only.
Choice of one-dimensional smoothing kernel,
passed to density.default.
Bandwidth for one-dimensional smoothing kernel,
passed to density.default.
Numeric value. bw will be multiplied by this value.
Other arguments passed to the one-dimensional kernel density estimation
function density.default.
Bandwidth coefficient; see Details.
Choice of edge correction.
Optional arguments passed to density.ppp
to control the smoothing bandwidth, when lambdaI and/or
lambdadot is estimated by spatial kernel smoothing.
Numeric value. sigma will be multiplied by this value.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz
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.
The arguments bw and adjust.bw control the
degree of one-dimensional smoothing of the estimate of pair correlation.
If the arguments lambdaI and/or lambdadot are missing or
null, they will be estimated from X by spatial kernel
smoothing using a leave-one-out estimator,
computed by density.ppp.
The arguments sigma, varcov
and adjust.sigma control the degree of spatial smoothing.
pcf.ppp,
pcfinhom,
pcfdot,
pcfcross.inhom
plot(pcfdot.inhom(amacrine, "on", stoyan=0.1), legendpos="bottom")
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