The SSM Procedure

Covariance Parameterization

The covariance matrices specified by the COV and COV1 options in the STATE statement must be positive semidefinite. When these matrices are of general form and are not user-specified, they are internally parameterized by their Cholesky root. Suppose that $\Sigma $, an $m \times m$ positive semidefinite matrix of rank $\Mathtext{r}$, is such a covariance matrix. Then, $\Sigma $ can always be written as

\[ \Sigma = R R^{'} \]

where the (generalized) Cholesky root, $\Mathtext{R}$, is an $m \times r$ lower triangular matrix with nonnegative diagonal elements (that is, $R[i,j] = 0 \; \mr{if} j > i$ and $R[i,i] \geq 0,\; 1 \leq i \leq r$). The SSM procedure parameterizes $\Sigma $ by the elements of its Cholesky root, which adds $r*(r+1)/2 + r*(m-r)$ elements to the parameter vector $\pmb {\theta }$.