[R] density estimation on 2-D bounded domain

[Jason Turner]

On Mon, Jul 29, 2002 at 03:38:40PM +0200, Hans Muller wrote:
> density estimation on a 2 dimensional bounded domain
> ---------------------------------------------------------------------
> I am currently trying to estimate the probability
> density (PD) of  cancers within the breast using
...
> Of course a practical PD must be limited by the curve of the breast
> outline.
> I don't have a clue after perusing R documentation.
> 
> !! Is there a way to calculate with R statistics
> !! (with sm.density, or otherwise)
> !!  such a density with an imposed boundary ??

Bowman and Azalini [1], who wrote the sm package  suggest 
a two techniques: 

1) variable trasformation, from a bounded space to unbounded
(e.g. take logarithms of positive-only values) and,

2) a modified kernel, which has the effect of down-weighting
tails.  The function nnbr() in the sm library is used for this
by a nearest-neighbors method.

They cite Wand and Jones [2] for a more detailed
examination of each, but I haven't got a copy handy,
so you'll have to take their word for it.

Technique (1) will likely have to have an approximate
boundary (hemisphere, circle, parabola or similar - not 
familiar with mamography data format). 

(2) will also be approximate, in that the emperical density can 
still potentially cross boundaries.  The tails are reduced, but
not guaranteed to be eliminated at boundaries.

Cheers

Jason

[1] 
@book{
BowmanAzzalini1997,
author = "Adrian W. Bowman and Adelchi Azzalini",
title = "Applied Smoothing Techniques for Data Analysis",
publisher = "Oxford University Press",
year = 1997,
address = Oxford,
}

[2] 
@book{
WandJones1995,
author = "M. P. Wand and M. C. Jones",
title = "Kernel Smoothing",
publisher = "Chapman and Hall",
year = 1995,
address = London,
}
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