I have a large weighted least squares system with millions of observations. I can't store the least squares matrix in memory, so I only store the normal equations matrix:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

A^T W A

[/tex]

where [itex]A[/itex] is dense and n-by-p and [itex]W = diag(w_1,...,w_n)[/itex]. It takes averylong time to build this normal equations matrix due to the large number of observations. However, I also want to iteratively re-weight the observations to reduce the effect of outliers, and so at the next iteration, the weight matrix will be the product of the original weights [itex]W[/itex] with some new weights [itex]V = diag(v_1,...,v_n)[/itex]. Therefore I need to construct the matrix:

[tex]

A^T W' A

[/tex]

where [itex]W' = diag(w_1 v_1, w_2 v_2, ..., w_n v_n)[/itex].

Does anyone know of a slick (fast) way to determine the matrix [itex]A^T W' A[/itex] given the matrix [itex]A^T W A[/itex] and the new weights [itex]V[/itex]? I am hoping I don't need to start over from scratch to build [itex]A^T W' A[/itex] for each iteration, since it takes so long.

I did search the literature and didn't find much on this topic.

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# Large weighted least squares system

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