Computes adaptive smoothing bandwidths according to the inverse-square-root rule of Abramson (1982).

```
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 `im`

age
of the Abramson bandwidths over the relevant spatial domain
(when `at = "pixels"`

).

- 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.

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).

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.

```
# '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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