Computes adaptive smoothing bandwidths for a spatial point pattern, according to the inverse-square-root rule of Abramson (1982).
# S3 method for ppp
bw.abram(X, h0,
...,
at=c("points", "pixels"),
hp = h0, pilot = NULL, trim=5, smoother=density.ppp)Either a numeric vector of length npoints(X)
giving the Abramson bandwidth for each point
(when at = "points", the default),
or the entire pixel image
of the Abramson bandwidths over the relevant spatial domain
(when at = "pixels").
A point pattern (object of class "ppp")
for which the variable bandwidths should be computed.
A scalar value giving the global smoothing bandwidth
in the same units as the coordinates of X.
The default is h0=bw.ppl(X).
Additional arguments passed to
as.im to control the pixel resolution,
or passed to density.ppp or smoother
to control the type of smoothing, when computing the pilot estimate.
Character string (partially matched) specifying whether
to compute bandwidth values at the points of X
(at="points", the default) or to compute bandwidths
at every pixel in a fine pixel grid (at="pixels").
Optional. A scalar pilot bandwidth, used for estimation
of the pilot density if required. Ignored if pilot is a pixel image
(object of class "im"); see below.
Optional. Specification of a pilot density
(possibly unnormalised).
If pilot=NULL the pilot density
is computed by applying fixed-bandwidth density estimation to
X using bandwidth hp.
If pilot is a point pattern, the pilot density is
is computed using a fixed-bandwidth estimate based
on pilot and hp.
If pilot is a pixel image (object of class "im"),
this is taken to be the (possibly unnormalised)
pilot density, and hp is ignored.
A trimming value required to curb excessively large bandwidths. See Details. The default is sensible in most cases.
Smoother for the pilot.
A function or character string, specifying the function
to be used to compute the pilot estimate when
pilot is NULL or is a point pattern.
Tilman M. Davies. Adapted by Adrian Baddeley Adrian.Baddeley@curtin.edu.au.
This function computes adaptive smoothing bandwidths using the methods of Abramson (1982) and Hall and Marron (1988).
The function bw.abram is generic. The function
bw.abram.ppp documented here is the method
for spatial point patterns (objects of class "ppp").
If at="points" (the default) a smoothing bandwidth is
computed for each point in the pattern X. Alternatively if
at="pixels" a smoothing bandwidth is computed for
each spatial location in a pixel grid.
Under the Abramson-Hall-Marron rule, the bandwidth at location \(u\) is
$$
h(u) = \mbox{\texttt{h0}}
* \mbox{min}[ \frac{\tilde{f}(u)^{-1/2}}{\gamma}, \mbox{\texttt{trim}} ]
$$
where \(\tilde{f}(u)\) is a pilot estimate of the spatially varying
probability density. The variable bandwidths are rescaled by \(\gamma\), the
geometric mean of the \(\tilde{f}(u)^{-1/2}\) terms evaluated at the
data; this allows the global bandwidth h0 to be considered on
the same scale as a corresponding fixed bandwidth. The trimming value
trim has the same interpretation as the required `clipping' of
the pilot density at some small nominal value (see Hall and Marron,
1988), to necessarily prevent extreme bandwidths (which
can occur at very isolated observations).
The pilot density or intensity is determined as follows:
If pilot is a pixel image, this is taken as the
pilot density or intensity.
If pilot is NULL, then the pilot intensity is
computed as a fixed-bandwidth kernel
intensity estimate using density.ppp applied to
the data pattern X using
the pilot bandwidth hp.
If pilot is a different point pattern on the same spatial
domain as X, then the pilot intensity is
computed as a fixed-bandwidth kernel
intensity estimate using density.ppp applied to
pilot using the pilot bandwidth hp.
In each case the pilot density or intensity is renormalised to become a probability density, and then the Abramson rule is applied.
Instead of calculating the pilot as a fixed-bandwidth density
estimate, the user can specify another density estimation procedure
using the argument smoother. This should be either a function
or the character string name of a function. It will replace
density.ppp as the function used to calculate the
pilot estimate. The pilot estimate will be computed as
smoother(X, sigma=hp, ...) if pilot is NULL,
or smoother(pilot, sigma=hp, ...) if pilot is a point
pattern. If smoother does not recognise the argument name
sigma for the smoothing bandwidth, then hp is effectively
ignored, as shown in the Examples.
Abramson, I. (1982) On bandwidth variation in kernel estimates --- a square root law. Annals of Statistics, 10(4), 1217-1223.
Davies, T.M. and Baddeley, A. (2018) Fast computation of spatially adaptive kernel estimates. Statistics and Computing, 28(4), 937-956.
Davies, T.M., Marshall, J.C., and Hazelton, M.L. (2018) Tutorial on kernel estimation of continuous spatial and spatiotemporal relative risk. Statistics in Medicine, 37(7), 1191-1221.
Hall, P. and Marron, J.S. (1988) Variable window width kernel density estimates of probability densities. Probability Theory and Related Fields, 80, 37-49.
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York.
bw.abram
# 'ch' just 58 laryngeal cancer cases
ch <- split(chorley)[[1]]
h <- bw.abram(ch,h0=1,hp=0.7)
length(h)
summary(h)
if(interactive()) hist(h)
# calculate pilot based on all 1036 observations
h.pool <- bw.abram(ch,h0=1,hp=0.7,pilot=chorley)
length(h.pool)
summary(h.pool)
if(interactive()) hist(h.pool)
# get full image used for 'h' above
him <- bw.abram(ch,h0=1,hp=0.7,at="pixels")
plot(him);points(ch,col="grey")
# use Voronoi-Dirichlet pilot ('hp' is ignored)
hvo <- bw.abram(ch, h0=1, smoother=densityVoronoi)
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