bw.diggle | R Documentation |
Uses cross-validation to select a smoothing bandwidth for the kernel estimation of point process intensity.
bw.diggle(X, ..., correction="good", hmax=NULL, nr=512, warn=TRUE)
X |
A point pattern (object of class |
... |
Ignored. |
correction |
Character string passed to |
hmax |
Numeric. Maximum value of bandwidth that should be considered. |
nr |
Integer. Number of steps in the distance value |
warn |
Logical. If |
This function selects an appropriate bandwidth sigma
for the kernel estimator of point process intensity
computed by density.ppp
.
The bandwidth \sigma
is chosen to
minimise the mean-square error criterion defined by Diggle (1985).
The algorithm uses the method of Berman and Diggle (1989) to
compute the quantity
M(\sigma) = \frac{\mbox{MSE}(\sigma)}{\lambda^2} - g(0)
as a function of bandwidth \sigma
,
where \mbox{MSE}(\sigma)
is the
mean squared error at bandwidth \sigma
,
while \lambda
is the mean intensity,
and g
is the pair correlation function.
See Diggle (2003, pages 115-118) for a summary of this method.
The result is a numerical value giving the selected bandwidth.
The result also belongs to the class "bw.optim"
which can be plotted to show the (rescaled) mean-square error
as a function of sigma
.
A single numerical value giving the selected bandwidth.
The result also belongs to the class "bw.optim"
(see bw.optim.object
)
which can be plotted to show the bandwidth selection criterion
as a function of sigma
.
The smoothing parameter sigma
returned by bw.diggle
(and displayed on the horizontal axis of the plot)
corresponds to h/2
, where h
is the smoothing
parameter described in Diggle (2003, pages 116-118) and
Berman and Diggle (1989).
In those references, the smoothing kernel
is the uniform density on the disc of radius h
. In
density.ppp
, the smoothing kernel is the
isotropic Gaussian density with standard deviation sigma
.
When replacing one kernel by another, the usual
practice is to adjust the bandwidths so that the kernels have equal
variance (cf. Diggle 2003, page 118). This implies that sigma = h/2
.
.
Berman, M. and Diggle, P. (1989) Estimating weighted integrals of the second-order intensity of a spatial point process. Journal of the Royal Statistical Society, series B 51, 81–92.
Diggle, P.J. (1985) A kernel method for smoothing point process data. Applied Statistics (Journal of the Royal Statistical Society, Series C) 34 (1985) 138–147.
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
density.ppp
,
bw.optim.object
.
Alternative methods:
bw.ppl
,
bw.scott
,
bw.CvL
,
bw.frac
.
attach(split(lansing))
b <- bw.diggle(hickory)
plot(b, ylim=c(-2, 0), main="Cross validation for hickories")
if(interactive()) {
plot(density(hickory, b))
}
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