[R] volume of ellipsoid

yuanzhi yuanzhi.li at usherbrooke.ca
Sun Nov 17 18:11:28 CET 2013

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 ...
> http://www.datavis.ca/papers/ellipses-STS402.pdf
> for the properties of ellipsoids and calculation of (hyper)volumes
> based on a spectral decomposition.
> 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.
> -- 
> Michael Friendly     Email: friendly AT yorku DOT ca
> Professor, Psychology Dept. & Chair, Quantitative Methods
> York University      Voice: 416 736-2100 x66249 Fax: 416 736-5814
> 4700 Keele Street    Web:   http://www.datavis.ca
> Toronto, ONT  M3J 1P3 CANADA
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Thank you for your reply. I think it is a good idea to calculate the volume
in Monte-Carlo method. So the the problem change to be have to quantify
whether a point is inside or outside of a region. Yes, it is easy to
determine a point inside or outside a ellipsoid by seeing whether
(x-x0)'Σ^(-1)(x-x0)-χ2p(α) is greater or smaller than zero, where x0 the
point we want to test. but how can we test whether a point in other shapes
of region(like the bounding box(R^p) as you said or the smallest convex
hull(R^p) of X)? Finally, I aslo want to know how calculate the hypervolume
of the bounding box (when P>3)? Thank you again.

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