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volume of ellipsoid

7 messages · Yuanzhi Li, Carl Witthoft, yuanzhi +2 more

Yuanzhi Li
# 
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-?)% of  
the probability in the distribution based on Mahalanobis distance:

?:(x-?)'?^(-1)(x-?)??2p(?)

where x and ? 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
Carl Witthoft
# 
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

            

      
      
      
    

        

      
      
    

        
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      yuanzhi
    

    

      
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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. 



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      yuanzhi
    

    

      
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Carl Witthoft 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.
thanks!



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      Michael Friendly
    

    

      
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On 11/14/2013 9:35 AM, yuanzhi wrote:

            

      
      
      
    

        
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.

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      David Winsemius
    

    

      
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On Nov 15, 2013, at 8:16 AM, Michael Friendly wrote:

        

            

      
      
      
    

        
Thank you so much for that reference, Michael, as well as the  
programming supplements that you constructed and linked in that  
encyclopedic review.

            

      
      
      
    

        
Copying back the omitted text from the OP who probably is under the  
misapprehension that we are all using Nabble:

            

      
      
      
    

        
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.

            

      
      
      
    

  
    

      
      
    

  


  


      

    

    
2 days later

    

      
      

        


  

        

      yuanzhi
    

    

      
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Michael Friendly wrote

            

      
      
      
    

        

      
      
    

        

            

      
      
      
    

        

      
      
    

        
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)'?^(-1)(x-x0)-?2p(?) 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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