pcfglobal: (cross) pair correlation functions with a global intensity reweighting

Description

Compute \(g_\textrm{global}\) or \(c_\textrm{global}\)

Usage

pcfglobal(X, lambda=NULL, ..., sigma=bw.CvL(X), r=NULL, rmax=NULL,
    kernel="epanechnikov", bw=NULL, stoyan=0.15, normtol=.005, ratio=FALSE,
    discrete.lambda=FALSE, divisor=c("r", "d"),
    leaveoneout=TRUE, interpolate=TRUE, interpolate.fac=10, exp_prs=NULL,
    interpolate.maxdx=diameter(as.owin(X))/100, dump=FALSE)
pcfcross.global(X,Y, lambdaX=NULL, lambdaY=NULL, ...,
    sigma=bw.CvL(X), r=NULL, rmax=NULL, kernel="epanechnikov", bw=NULL,
    stoyan=0.15, normtol=.005, ratio=FALSE, discrete.lambda=FALSE,
    divisor=c("r", "d"), analytical=NULL, interpolate=TRUE,
    interpolate.fac=10, exp_prs=NULL,
    interpolate.maxdx=diameter(as.owin(X))/100, dump=FALSE)

Value

The return value is an object of class fv, just as for pcf

and pcfinhom. The object contains columns r, theo, and

global, corresponding respectively to the argument \(r\), the theoretical values of \(g(r)\) for a Poisson process, and \(g_\mathrm{global}(r)\).

Arguments

X, Y: point process of type ppp, on which to evaluate the (cross) \(K\)-function
lambda, lambdaX, lambdaY: intensity function estimates corresponding to X and Y. If omitted, intensity functions will be computed using density.ppp or densityfun.ppp (see discrete.lambda below)
...: extra args passed to density.ppp or densityfun.ppp, if applicable.
sigma: Bandwidth value to use for kernel-based intensity estimation, intensity functions and exp_prs are not provided by the user.
r: Values of \(r\) to evaluate \(K(r)\) at. If omitted, a sensible default is chosen, using the same conventions as Kest and Kinhom.
rmax: Maximum \(r\) to evaluate \(K(r)\) at. rmax is used to generate values for r, if omitted. If missing, a sensible default is chosen.
kernel: Kernel type for smoothing of pcf.
bw: Kernel bandwidth for smoothing of pcf.
stoyan: Coefficient for Stoyan's bandwidth selection rule. See pcf.ppp.
normtol: A tolerance to use for expectedPairs or expectedCrossPairs when computing monte-carlo estimates of the normalizing factor \(\gamma\). Expressed as a maximum fractional standard error.
ratio: If TRUE, assemble numerator and denominator of pcf estimator separately.
divisor: Whether to use the evaluation distance ("r") or the distance between points ("d") to normalize the contribution of each point pair.
analytical: If TRUE, use Diggle-Jones weights
discrete.lambda: If TRUE, and intensity function(s) are not supplied, estimate intensities by interpolating the values on a discrete lattice (using interp.im and density.ppp), instead of exactly (using densityfun.ppp).
interpolate: If TRUE, evaluate the expectedCrossPairs on a lattice and interpolate, rather than at the exact displacements observed in the pattern.
interpolate.fac: If interpolate, the lattice spacing will be sigma/interpolate.fac.
leaveoneout: Use the leave-one-out estimator for \(\gamma\). See Shaw et al 2020 for details.
exp_prs: A function that returns values for \(\gamma_\textrm{iso}(r)\). If \(\gamma\) is known explicitly, or the same calculation is being used for several point patterns, it can be much faster to compute it once and provide the function as exp_prs, since the computation of \(\gamma\) is usually the slowest part.
interpolate.maxdx: Upper bound on allowable lattice spacing for interpolation.
dump: For debugging purposes, include computed values of \(\gamma\) with the output, as attrs.

Author

Thomas Shaw <shawtr@umich.edu>

References

T Shaw, J Møller, R Waagepetersen. 2020. “Globally Intensity-Reweighted Estimators for \(K\)- and pair correlation functions”. arXiv:2004.00527 [stat.ME].

Examples

Run this code

rho <- funxy(function(x,y) 80*(1+x), owin())
X <- rpoispp(rho)
g <- pcfglobal(X)
#plot(g)

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