covariance-matrices

Covariance matrices

Filed under: ADMB Tricks

Parameterization via the Cholesky factor (Incomplete example)

If you want to estimate the parameters of a covariance matrix S you must ensure that the resulting matrix is:

1. symmetric and

2. positive definite.

To achieve this you do not parameterize S directly, but rather its Cholesky factor L, i.e. S = LL', see <http/en.wikipedia.orwikCholesky_decomposition>

 

The following two step procedure is recommended:

1. Parameterize the correlation matrix C via the Cholesky factor as explained here Correlation matrix

2. Scale C with the standard deviations to obtained S.

Complete example given in C.tpl and C.dat.

Constrained covariance matrices

Sometimes you want elements in the C (or S) to be zero, say S(1,2) = 0, meaning the element 1 and 2 are uncorrelated. An example of how to achieve this is provided here:

constrained_cor.tpl

constrained_cor.dat