Carl Witthoft wrote > 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 >> >> ______________________________________________ >> <email>R-help@
>> mailing list >> https://stat.ethz.ch/mailman/listinfo/r-help >> PLEASE do read the posting guide >> http://www.R-project.org/posting-guide.html >> and provide commented, minimal, self-contained, reproducible code. 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. thanks! -- View this message in context: http://r.789695.n4.nabble.com/volume-of-ellipsoid-tp4680409p4680446.html Sent from the R help mailing list archive at Nabble.com. ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.