```Date: Nov 23, 1999 12:11 PM
Author: Michael Pronath
Subject: covariance matrix, correlation matrix, decomposition

It is often necessary to decompose a given covariance matrix C(resp. correlation matrix R) as C=G*G'.  Cholesky decomposition iscommonly used here.But there are some others possible.  Starting from the eigenvaluedecomposition C=Q*L*Q', there is   G1 = Q*sqrt(L), as  G1*G1' = Q*sqrt(L)*sqrt(L)*Q'=Q*L*Q' = C  or   G2 = Q*sqrt(L)*Q', as G2*G2' = Q*sqrt(L)* Q'*Q *sqrt(L)*Q' =                                = Q*sqrt(L)*   1  *sqrt(L)*Q' = CG, G1, and G2 have different properties (G is triangular, G1'*G1 isdiagonal, and G2 is symmetric), so one of them may be preferable overthe others in some cases.  Note that all three of them can be used togenerate random numbers with a given covariance matrix, and all threeof them generate exactly the same distribution.For example, when putting a grid into a space of normal distributedparameters: The grid is generated in a "normed" space (zero mean,unity variance), and each grid point q is then transformed into the"real" space grid point p = p0 + G*q .  The shape of the transformedgrid depends on the choice of G:   1) Cholesky:      The grid is "sheared", angles between grid                   vertices vary largely between 0 and 180Â° and                   depend on the order of the parameters   2) G1:            The grid is scaled along its axes and rotated.                   Angles between grid vertices are all 90Â°   3) G2:            The grid looks like a rhombus.The condition of C may be extremely bad, e.g. if the statisticalparameters are physical quantities, and you have capacitances (1e-12)as well as donations (1e20).  Stability of the decomposition could bean issue here and give an advantage to one of them.I'd like to know if anybody has made some more profound analysis aboutthis, and the pro's and con's of the various methods.Michael Pronath--
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