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Science Forum Index » Statistics - Math Forum » how to marginalise gaussian distribution
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| shaobo hou |
 Posted: Mon Jan 01, 2007 10:59 pm |
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Given a D dimensional gaussian distribution with mean M and covariance matrix C, I want to compute its projection/marginalisation (I not sure what to call it) in q dimensional space, where q < D. The projection will have mean M' and covariance C'.
Initially I thought I can do this by simply projecting M onto the q dimensional hyperplane to produce M' and if the axes of the hyperlane is simply a subset of the original space, I can just select the appropriate entries of C to create C', e.g. If D = 3 and I want the gaussian to be projected onto the first and third axes,
C_11, C_12, C_13;
C = C_21, C_22, C_23;
C_31, C_32, C_33;
then
C'= C_11, C_13;
C_31, C_33;
However, I not sure how this can be done if the axes of the hyperplane is not a subset of the axes of the original space, i.e. if the axes of the hyperplane is an arbitrary set of orthogonal axes. Should I rotate the covariance matrix so that the hyperplane becomes parallel or perpendicular to the axes of the original space, so that the previous method can be applied? But how can this rotation be computed and applied?
thanks for any help
Shaobo |
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| shaobo hou |
| Posted: Mon Jan 01, 2007 11:03 pm |
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| oops, It doesn't seem to like the white spaces I put before some of the lines. I hope people can understand it, I was simply trying to represent C as a 3D matrix and C' as 2D submatrix of C. |
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| Stephen J. Herschkorn |
| Posted: Tue Jan 02, 2007 1:55 pm |
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Projection is a linear transformation. If X has multivariariate
normal distributurion with mean vector m and covaraince matrix C,
and A is a matrix of appropriate dimension, then A X has
multivariate normal distribution with mean vector A m and covariance
matrix A C A', where the single quote denotes transpose.
--
Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan |
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