Let $ f(L) $ be a scalar function of the lower triangular Cholesky decomposition of the covariance matrix $ \Sigma $ such as $ \Sigma = LL' $. Let's assume that we know the first and second derivative of $ f(L) $ wrt to $L{_i}{_j} $: $$ dL=\frac{\text{d } f(L)}{\text{d vech}(L)} $$ $$ hL=\frac{\text{d}^2 f(L)}{\text{d vech}(L)^2} $$ where $ dL $ is a vector of the same size of $ vech(L) $ and $ hL $ is the corresponding hessian matrix.

To pass from the Cholesky factor to the (lower triangular) covariance matrix, Iain Murray has found an interesting formula for the first derivative: $$ \text{tril}(d\Sigma) = \Phi(L^{-T}(P+P^T)L^{-1}) \text{ where } P=\Phi(L^TdL) $$ and $ \Phi() $ takes the lower-triangular part of a matrix and halves its diagonal.

What I would like to find is the formula for $ h\Sigma $ as a function of $ dL $ and $ hL $.

A similar question has been asked here for the first derivative, and Iain Murray has written an interesting paper on the question of first derivatives.



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