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Re: covariance matrix, correlation matrix, decomposition
Posted:
Nov 24, 1999 4:04 AM
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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 180° 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 90°
> 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 (1e-12)
> 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
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