# [R] volume of ellipsoid

Carl Witthoft carl at witthoft.com
Thu Nov 14 13:25:26 CET 2013

```Well, given that an upvoted question on math.stackexchange got no answers,
I'd say you're asking a very difficult question.   Perhaps this paper
http://www.geometrictools.com/Documentation/IntersectionOfEllipsoids.pdf

will be of some help.   It's possible that you could do:
1) find the ellipse of intersection of the two ellipsoids
2) find some formula for the equivalent of the "spherical cap" volume
3) apply that formula to each ellipsoid, using the intersection ellipse as
the base of the caps.

All this, btw, is math,  not really related to R or to software programming
in general.

yuanzhi wrote
> Hi, everyone!
>
> I have a matrix X(n*p) which is n samples from a p-dimensional normal
> distribution. Now I want to make the ellipsoid containing (1-α)% of
> the probability in the distribution based on Mahalanobis distance:
>
> μ:(x-μ)'Σ^(-1)(x-μ)≤χ2p(α)
>
> where x and Σ is the mean and variance-covariance matrix of X. Are
> there some functions in R which can plot the ellipsoid and calculate
> the volume (area for p=2, hypervolume for p>3) of the ellipsoid? I
> only find the "ellipsoidhull" function in "cluster" package, but it
> deal with ellipsoid hull containing X, which obvious not the one I want.
>
> After that, a more difficult is the following. If we have a series of
> matrix X1, X2, X3,… Xm which follow different p-dimensional normal
> distributions. And we make m ellipsoids (E1, E2, … Em) for each matrix
> like the before. How can we calculate the volume of union of the m
> ellipsoids? One possible solution for this problem is the
> inclusion-exclusion principle:
>
> V(⋃Ei)(1≤i≤m)=
> V(Ei)(1≤i≤m)-∑V(Ei⋂Ej)(1≤i<j≤m)+∑V(Ei⋂Ej⋂Ek)(1≤i<j<k≤m)+(-1)^(m-1)V(E1⋂…⋂Em)
>
> 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!
>
> Yuanzhi
>
> ______________________________________________
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> and provide commented, minimal, self-contained, reproducible code.

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