cov2theta function

Compute a Packed Representation of a Covariance Matrix

Transform covariances matrices to a packed representation or compute the inverse transformation.

cov2theta(Sigma)
theta2cov(theta)

Arguments

• Sigma: an $n$-by-$n$ real, symmetric positive definite matrix. Only the upper triangle is seen .
• theta: a numeric vector of length $n*(n+1)/2$ whose first $n$ elements are positive.

Returns

A vector like theta (cov2theta) or a matrix like Sigma (theta2cov); see Details .

Details

An $n$-by-$n$ real, symmetric, positive definite matrix $\Sigma$ can be factorized as

The upper triangular Cholesky factor $R$ can be written as

where $R1$ is a unit upper triangular matrix and $D = diag(diag(R1' * R1))$

and $Ds = diag(diag(Sigma))$

are diagonal matrices.

cov2theta takes $\Sigma$ and returns the vector $\theta$

of length $n*(n+1)/2$ containing the log diagonal entries of $D_{\sigma}$ followed by (in column-major order) the strictly upper triangular entries of $R_{1}$. theta2cov computes the inverse transformation.