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 ...
http://www.datavis.ca/papers/ellipses-STS402.pdf
Thank you so much for that reference, Michael, as well as the
programming supplements that you constructed and linked in that
encyclopedic review.
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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