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Re: covariance matrix, correlation matrix, decomposition
Posted:
Nov 24, 1999 4:04 AM


Michael Pronath wrote:
> It is often necessary to decompose a given covariance matrix C > (resp. correlation matrix R) as C=G*G'. Cholesky decomposition is > commonly used here. > But there are some others possible. Starting from the eigenvalue > decomposition 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' = C > > G, G1, and G2 have different properties (G is triangular, G1'*G1 is > diagonal, and G2 is symmetric), so one of them may be preferable over > the others in some cases. Note that all three of them can be used to > generate random numbers with a given covariance matrix, and all three > of them generate exactly the same distribution. > > For example, when putting a grid into a space of normal distributed > parameters: 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 transformed > grid 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 statistical > parameters are physical quantities, and you have capacitances (1e12) > as well as donations (1e20). Stability of the decomposition could be > an issue here and give an advantage to one of them.
What about an SVD of G (not C !) Just computing C (not even decomposing) makes the condition number worse. Say, you have the SVD of G : G = U S V' where U and V are orthogonal
matrices and S has nontrivial elements only on its diagonal. Then G*G' = U S S' U' where SS' has the diagonal (s_{ii}^2) and zeros elsewhere. You can extract all information about C from U and S and it should be much more.
 Helmut Jarausch Lehrstuhl fuer Numerische Mathematik Institute of Technology, RWTH Aachen D 52056 Aachen, Germany



