On Thu, 10 Dec 2009, Keith Jewell wrote:
Hi All (especially Duncan and Baptiste),
Summary (of lengthy bits below):
I will have a convex hull in multiple (3 to 10) dimensions derived from many
(thousands) of points by geometry::convhulln.
I will need to categorise other 'test' points as inside/outside that convex
hull . e.g. given:
----------------------
require(geometry)
ps <- matrix(rnorm(4000),ncol=4) # 'calibration' set
phull <- convhulln(ps) # convex hull
xs <- matrix(rnorm(1200),ncol=4) # 'test' set
-----------------
How do I categorise each point (row) in xs as inside/outside(/on) phull???
There is tripack::in.convex.hull but that doesn't handle my dimensionality.
Thanks to Duncan Murdoch for the suggestion (just a few lines down,
previously made by Baptiste Auguie): of testing a single point thus:
i) add the test point to the set of points defining the convex hull,
ii) recalculate the convex hull
iii) if the test point is part of the new convex hull, then it was outside
the original
BUT I have many (thousands) of test points, so this would involve very many
convex hull calculations. My suggestion, immediately below, requires finding
the signs of perpendicular distances from each test point to each
multidimensional 'plane' defining the convex hull (NB: phull is a matrix in
which each row defines such a 'plane').
Many?
set.seed(1234)
ps <- matrix(rnorm(4000),ncol=4)
phull <- convhulln(ps)
xs <- matrix(rnorm(1200),ncol=4)
phull2 <- convhulln(rbind(ps,xs))
nrp <- nrow(ps)
nrx <- nrow(xs)
outside <- unique(phull2[phull2>nrp])-nrp
done <- FALSE
while(!done){
+ phull3 <- convhulln(rbind(ps,xs[-(outside),]))
+ also.outside <- (1:nrx)[-outside][unique(phull3[phull3>nrp])-nrp]
+ print(length(also.outside))
+ outside <- c(outside,also.outside)
+ done <- length(also.outside)==0
+ }
[1] 3
[1] 0
phull2 was evaluated once, phull3 twice.
Any point that is in the convex hull of rbind(ps,xs) is either in or
outside the convex hull of ps. Right? So, just recursively eliminate
points that are in the convex hull of the larger set.
Chuck
p.s. for
xs <- matrix(rnorm(120000),ncol=4)
it required about a dozen iterations
Baptiste has found a Matlab implementation
<http://www.mathworks.com/matlabcentral/fileexchange/10226-inhull> of (what
looks like) my algorithm. I don't speak Matlab, but this looks non-trivial
to code in R. I'll do it if I have to, but if it already exists it would be
nice. If I do have to code it, I'd really appreciate an expression in
algebra rather than Matlab!
Any pointers will be much appreciated,
Keith Jewell
"Duncan Murdoch" <murd...@stats.uwo.ca> wrote in message
news:4b20e1ea.3030...@stats.uwo.ca...
On 10/12/2009 5:15 AM, Keith Jewell wrote:
Hi,
Doing some more reading, I think the problem is easier because the hull
is convex. Then an algorithm for testing points might be:
a) Define the convex hull as a set of planes (simplexes).
[as returned by convhulln!!]
b) Define one point, i, known to be interior
[e.g. mean of all the points defining the hull]
c) If point x is
i) for every plane, on the same side as i; x is interior
ii) for every plane, on the same side as i or in the plane; x is in
the surface
iii) else x is exterior
That looks like it would work, but wouldn't it be easier to do the
following:
Compute the convex hull with the new point added. If the point is
exterior, the new point will be part of the hull. If interior, it won't
be. If it is on the boundary, it's probably unpredictable, but due to
rounding error, that's probably true even with a perfect algorithm.
I didn't notice that you said how your original polygon is defined, but if
it is defined as a convex hull or in terms of its vertices, the above
method would work. If it's defined some other way, it might be hard.
Duncan Murdoch
So now I need to find the directions of points from multidimensional
planes.Perhaps I can find the vectors of the perpendiculars from the
points to the planes (possibly extended) and test for
parallel/anti-parallel?
I feel that I'm in the right direction because this uses the structure of
a convex hull returned by convhulln. But, I still feel I'm re-inventing
the wheel. Surely this has been done before? Isn't a (the?) major purpose
of a convex hull to test other points for inclusion?
Perhaps when I get the geometry sorted this will be so easy I'll
understand why noone has pointed me to an existing R function, but
currently I feel I and Baptiste are wandering in the dark :-(
Any hints?
Thanks in advance,
Keith Jewell
-----------------------------------------------------------------
"baptiste auguie" <baptiste.aug...@googlemail.com> wrote in message
news:de4e29f50912040550m71fbffafnfa1ed6e0f4451...@mail.gmail.com...
Hi,
Yet another one of my very naive ideas on the subject: maybe you can
first evaluate the circumscribed and inscribed spheres of the base set
of points (maximum and minimum of their distances to the center of
gravity). Any points within a distance smaller than the infimum is
good, any point further than the supremum is not good. This should be
faster than the calculation of a convex hull for each point. Of course
the usefulness of this first test really depends on how aspherical is
your base convex hull.
I do hope to read a real answer from someone who knows this stuff!
HTH,
baptiste
2009/12/4 Keith Jewell <k.jew...@campden.co.uk>:
Hi,
I seek to identify those points in/outside a multidimensional convex
hull
(geometry::convhulln). Any suggestions?
Background just in case I'm going down a really wrong road:
Given an observed data set with one dependent/observed variable (Y) and
multiple (3 to 10) independent/design variables (X1, X2, ...) I want to
increase the number of points by interpolating. I'm using expand.grid(X)
to
generate the X points and locfit::predict.locfit to interpolate the Y
values. No problem so far.
BUT my observed X data set is very far from rectangular, so the set of
points produced by expand.grid includes many "extrapolations" which I'd
want to remove. I'm aware of the problems of defining extrapolation in
multiple dimensions and am minded to define it as "outside the convex
hull",
hence my question.
In searching r-project.org I came across a thread in which baptiste
auguie
said "one general way to do this would be to compute the convex hull
(?chull) of the augmented set of points and test if the point belongs to
it"; an approach I'd considered generalising to multiple points thus
(pseudo
R code)...
----------------
baseHull <- convhulln(baseSet)
augHull <- convhulln(augSet)
while (augHull != baseHull)
{ augSet <- augSet [-(augHull !%in% baseHull)]
augHull <- convhulln(augSet)
}
--------------------
... but this has that horrible loop including a convhulln!
In the (real, typical) test data set I'm using for development baseSet
is 5
columns by 2637 rows while baseHull has only 42 distinct points. If
augHull
has a similarly simple hull, then each time round the loop will remove
only
a few dozen points; I need to remove many thousands.
And (in my naivete) I think there must be a better way of testing for a
point inside a polygon, I'd have thought convhulln must need to do that
often!
Thanks in advance for any pointers,
Keith Jewell
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______________________________________________
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PLEASE do read the posting guide
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and provide commented, minimal, self-contained, reproducible code.
______________________________________________
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Charles C. Berry (858) 534-2098
Dept of Family/Preventive Medicine
E mailto:cbe...@tajo.ucsd.edu UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 92093-0901
______________________________________________
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
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