bw.abram.ppp {spatstat.explore}

Abramson's Adaptive Bandwidths For Spatial Point Pattern

Description

Computes adaptive smoothing bandwidths for a spatial point pattern, according to the inverse-square-root rule of Abramson (1982).

Usage

## S3 method for class 'ppp'
bw.abram(X, h0, 
         ...,
         at=c("points", "pixels"),
         hp = h0, pilot = NULL, trim=5, smoother=density.ppp)

Arguments

X

A point pattern (object of class "ppp") for which the variable bandwidths should be computed.

h0

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.

at

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").

hp

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.

pilot

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.

trim

A trimming value required to curb excessively large bandwidths. See Details. The default is sensible in most cases.

smoother

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.

Details

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.

Value

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").

Author(s)

Tilman Davies [e-mail]. Adapted by Adrian Baddeley [e-mail].

References

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.

See Also

bw.abram

Examples

# '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)