[R] volume of ellipsoid
dwinsemius at comcast.net
Fri Nov 15 17:15:52 CET 2013
On Nov 15, 2013, at 8:16 AM, Michael Friendly wrote:
> On 11/14/2013 9:35 AM, yuanzhi wrote:
>> Hi, Carl Witthoft
>> yes, it looks like a mathematical question. I will try based on your
>> suggestion to calculate the volume of the intersection. But I still
>> want to
>> know whether there are some functions in R which can calculate the
>> volume of
>> an ellipsoid(area for p=2, hypervolume for p>3) containing X, just
>> like the
>> "convhulln" function in "geometry" package which can calculate the
>> volume of
>> convex hull containing X.
> See the Appendix A.2 in my paper on Elliptical Insights ...
Thank you so much for that reference, Michael, as well as the
programming supplements that you constructed and linked in that
> for the properties of ellipsoids and calculation of (hyper)volumes
> based on a spectral decomposition.
Copying back the omitted text from the OP who probably is under the
misapprehension that we are all using Nabble:
>>> But the problem is how to calculate the volume of intersection
>>> between 2, 3 or more ellipsoids. Are there some functions which
>>> can calculate the volume of intersection between two region or
>>> functions which directly calculate the volume of a union of two
>>> region(the region here is ellipsoid). OR yo you have any good
>>> ideas solving this problem in R? Thank you all in advance!
> The intersection of general ellipsoids is mathematically extremely
> complex. You can approximate it by acceptance sampling -- finding the
> proportion of random points in R^p in the bounding box of the
> ellipsoids which are contained in both.
So that reduces the problem to defining a function that returns TRUE
for a point when it is in the interior of an ellipse. So reading your
section on statistical ellipsoids, I think an intersection test for
n=2 could require that the squared Mahalanobis distance from the
centroids be less than the c_1^2 and c_2^2 values for the two
ellipsoids under consideration.
> Michael Friendly Email: friendly AT yorku DOT ca
David Winsemius, MD
Alameda, CA, USA
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