Cross Validated Bandwidth Selection for Pair Correlation Function
Uses composite likelihood or generalized least squares cross-validation to select a smoothing bandwidth for the kernel estimation of pair correlation function.
bw.pcf(X, rmax=NULL, lambda=NULL, divisor="r",
kernel="epanechnikov", nr=10000, bias.correct=TRUE,
cv.method=c("compLik", "leastSQ"), simple=TRUE, srange=NULL,
..., verbose=FALSE, warn=TRUE)X |
A point pattern (object of class |
rmax |
Numeric. Maximum value of the spatial lag distance r for which g(r) should be evaluated. |
lambda |
Optional.
Values of the estimated intensity function.
A vector giving the intensity values
at the points of the pattern |
divisor |
Choice of divisor in the estimation formula:
either |
kernel |
Choice of smoothing kernel, passed to |
nr |
Integer. Number of subintervals for discretization of [0, rmax] to use in computing numerical integrals. |
bias.correct |
Logical. Whether to use bias corrected version of the kernel estimate. See Details. |
cv.method |
Choice of cross validation method: either
|
simple |
Logical. Whether to use simple removal of spatial lag distances. See Details. |
srange |
Optional. Numeric vector of length 2 giving the range of bandwidth values that should be searched to find the optimum bandwidth. |
... |
|
verbose |
Logical value indicating whether to print progress reports during the optimization procedure. |
warn |
Logical. If |
With cv.method="leastSQ", the bandwidth
h is chosen to minimise an unbiased
estimate of the integrated mean-square error criterion
M(h) defined in equation (4) in Guan (2007a).
The code implements the fast algorithm of Jalilian and Waagepetersen
(2018).
With cv.method="compLik", the bandwidth
h is chosen to maximise a likelihood
cross-validation criterion CV(h) defined in
equation (6) of Guan (2007b).
M(b) = \int_{0}^{rmax} \hat{g}^2(r;b) r dr - ∑_{u,v}
The result is a numerical value giving the selected bandwidth.
A numerical value giving the selected bandwidth.
The result also belongs to the class "bw.optim"
which can be plotted.
The bandwidth bw returned by bw.pcf
is the standard deviation of the smoothing kernel,
following the standard convention in R.
As mentioned in the documentation for
density.default and pcf.ppp,
this differs from other definitions of bandwidth that can be
found in the literature. The scale parameter
h, which is called the bandwidth in some literature,
is defined differently.
For example for the Epanechnikov kernel, h is the half-width
of the kernel, and bw=h/sqrt(5).
Rasmus Waagepetersen and Abdollah Jalilian. Adapted for spatstat by Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
Guan, Y. (2007a). A composite likelihood cross-validation approach in selecting bandwidth for the estimation of the pair correlation function. Scandinavian Journal of Statistics, 34(2), 336–346.
Guan, Y. (2007b). A least-squares cross-validation bandwidth selection approach in pair correlation function estimations. Statistics & Probability Letters, 77(18), 1722–1729.
Jalilian, A. and Waagepetersen, R. (2018) Fast bandwidth selection for estimation of the pair correlation function. Journal of Statistical Computation and Simulation, 88(10), 2001–2011. https://www.tandfonline.com/doi/full/10.1080/00949655.2018.1428606
b <- bw.pcf(redwood) plot(pcf(redwood, bw=b))
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