Thread View: sci.math.stat
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Started by Soren Hein
Thu, 17 Dec 1992 15:54
Double decorrelation
Author: Soren Hein
Date: Thu, 17 Dec 1992 15:54
Date: Thu, 17 Dec 1992 15:54
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Here is one of those problems that appears to be simple, but
apparently isn't:
Given a matrix X (M by N) of stochastic variables, find two matrices
A (M by M) and B (N by N) such that for the transformed matrix
Y = A X B,
we have
E[Y Y^T] = I_M
E[Y^T Y] = I_N
where E[.] denotes expectation, ^T denotes transposition, and I_M and
I_N are identity matrices of size M by M and N by N, respectively.
/Soren
shein@robotics.berkeley.edu
(Needless to say, this is not a homework problem etc.)
Re: Double decorrelation
Author: Pierre Asselin
Date: Sat, 19 Dec 1992 18:10
Date: Sat, 19 Dec 1992 18:10
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In <1gq7rgINNcqk@agate.berkeley.edu> shein@nima.berkeley.edu (Soren Hein) writes: >Given a matrix X (M by N) of stochastic variables, find two matrices >A (M by M) and B (N by N) such that for the transformed matrix > Y = A X B, >we have > E[Y Y^T] = I_M > E[Y^T Y] = I_N I'm not sure it can be done. If you can settle for diagonal matrices instead of identities, a singular value decomposition (essentially) will do the job. I assume you know the covariance matrix of your Xij's. 1) Restrict A and B to orthogonal matrices. Then, Y Y^T = A X X^T A^T independent of B, and Y^T Y = B^T X^T X B independent of A. 2) Form the deterministic matrix E[X X^T] and diagonalize. The eigenvectors are the columns of A. 3) Form the deterministic matrix E[X^T X] and diagonalize. The eigenvectors are the rows of B. -- --Pierre Asselin, Santa Barbara, California pa@verano.sba.ca.us
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