R: Expected pairs in an inhomogeneous poisson process

expectedPairs {globalKinhom}

R Documentation

Expected pairs in an inhomogeneous poisson process

Description

Compute the expected number of pairs at a given displacement h in a poisson process with a given intensity function. This corresponds to the integrals \gamma of Shaw et al. 2020. The various functions correspond to the univariate and bivariate versions of the anisotropic or isotropic versions of \gamma. The final two options (expectedPairs_kernloo and expectedPairs_iso_kernloo), provide implementations of the leave-out kernel estimates of \gamma: \bar \gamma(h) and \bar \gamma ^\mathrm{iso}(r). In those cases, the point pattern X itself is passed to the routine, rather than the (true or estimated) intensities rho etc. The estimators for \bar \gamma(h) are only applicable to univariate processes. See Shaw et al, 2020 for details.

Usage

expectedPairs(rho, hx, hy=NULL, method=c("mc", "lattice"),
                tol=.005, dx=diff(as.owin(rho)$xrange)/200, maxeval=1e6,
                maxsamp=5e3)

expectedCrossPairs(rho1, rho2=NULL, hx, hy=NULL, method=c("mc", "lattice"),
                tol=.005, dx=diff(as.owin(rho1)$xrange)/200, maxeval=1e6,
                maxsamp=5e3)

expectedPairs_iso(rho, r, tol=.001, maxeval=1e6, maxsamp=5e3)

expectedCrossPairs_iso(rho1, rho2=NULL, r, tol=.001, maxeval=1e6, maxsamp=5e3)

expectedPairs_kernloo(X, hx,hy, sigma=bw.CvL, tol=.005, maxeval=1e6,
                            maxsamp=5e3, leaveoneout=TRUE)

expectedPairs_iso_kernloo(X, r, sigma=bw.CvL, tol=.001, maxeval=1e6,
                                    maxsamp=5e3, leaveoneout=TRUE)

Arguments

`rho1`, `rho2`, `rho`	Intensity functions, either of class `im` or `funxy`. This may be produced by `density.ppp` or `densityfun.ppp`, or provided by a fitted intensity model.
`X`	Point pattern of class `ppp` with the points of the pattern for which `\bar \gamma` is to be estimated.
`hx`, `hy`	For expectedPairs and expectedCrossPairs (i.e. `\gamma(h)`), the displacements `h \in \textrm{R}^2` to evaluate `\gamma` at. These can be in any format supported by xy.coords.
`r`	For the isotropic versions `\gamma^\mathrm{iso}(r)`, the separations `r` at which `\gamma^\mathrm{iso}` is to be evaluated.
`method`	Either mc (the default) or lattice. Compute integral using monte-carlo or on a lattice.
`tol`	A tolerance for how precise the integral should be. This is compared to a standard error for the mc estimate.
`sigma`	Smoothing bandwidth for direct kernel-based estimators `\bar \gamma`.
`leaveoneout`	Use leave-out estimators. This should generally be true except for the purpose of evaluating the bias of the standard estimators. See Shaw et al 2020 for details.
`maxeval`	Maximum number of evaluations of rho per iteration. Prevents memory-related crashes that can occur.
`maxsamp`	Maximum number of monte carlo samples per iteration. If this is too large, you may do more work than required to achieve tol.
`dx`	if method=="lattice", a lattice spacing for the computation. defaults to .01.

Value

The return value is a numeric vector with length equal to the number of displacements h passed

Author(s)

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].