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".


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. $\theta_1 = \log(l_{11})$, $\theta_2=l_{21}$, $\theta_3=l_{31}$, $\theta_4=\log(l_{22})$, ...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.