Maximum Likelihood Matrix Decomposition – Difference Between Cholesky Decomposition and Log-Cholesky Decomposition

cholesky decompositiondlmmatrixmatrix decompositionmaximum likelihood

Is there any difference between a Cholesky decomposition and a log-cholesky decomposition? If yes, what is the difference?

In the paper "An R package for dynamic linear models" by Giovanni Petris ( he refers to the paper "Unconstrained Parametrizations of Variance-Covariance Matrices" by Pinheiro and Bates (1996)).

In this case since the model is not a standard one, we use the general create dlm to define a build which we subsequently use to find the MLEs of the model parameters. IN order to avoid an optimization problem with complicated constraints, we parametrize V in terms of the elements of its log-Cholesky decomposition

I know "Cholesky decomposition" $L L^T$

$A = \begin{bmatrix}a_{11} & a_{21} & a_{31} \\a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\end{bmatrix} = \begin{bmatrix}l_{11} & 0 & 0 \\l_{21} & l_{22} & 0\\ l_{31} & l_{32} & l_{33}\end{bmatrix} \begin{bmatrix}l_{11} & l_{21} & l_{31} \\0 & l_{22} & l_{23}\\ 0 & 0 & l_{33}\end{bmatrix} \qquad , $

but I do not know "Log-Cholesky decomposition".

Best Answer

I think it's less confusing to call it the log-Cholesky paramet(e)rization rather than the log-Cholesky decomposition (i.e., the "decomposition" part doesn't change ...)

From Pinheiro's thesis (1994, UW Madison) - I think it has the same information as the paper you cite:

6.1.2 Log-Cholesky Parametrization If one requires the diagonal elements of $\boldsymbol L$ in the Cholesky factorization to be positive then $\boldsymbol L$ is unique. In order to avoid constrained estimation, one can use the logarithms of the diagonal elements of $\boldsymbol L$. We call this parametrization the log-Cholesky parametrization. It inherits the good computational properties of the Cholesky parametrization, but has the advantage of being uniquely defined.

In other words, in your notation it would be:

\begin{bmatrix} \log(l_{11}) & 0 & 0 \\ l_{21} & \log(l_{22}) & 0\\ l_{31} & l_{32} & \log(l_{33})\end{bmatrix}

For what it's worth, when defining a parameter vector for a model you also need to define an order in which the matrix is unpacked; for example, in lme4 the log-Cholesky lower triangle is unpacked in column-first order, i.e. $\theta_1 = \log(l_{11})$, $\theta_2=l_{21}$, $\theta_3=l_{31}$, $\theta_4=\log(l_{22})$, ...

Related Question