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First of all I would not recommend the $M\cdot t$; it is fragile to the choice of $t_0$, isn't it? Nevertheless your specification seems to be close to the one of a linear regression (if $\epsilon$ is Gaussian and you metric is the Mahalanobis' one): just organize your dataset as a nice matrix and perform a linear regression.


You need to adjust your correlation matrix such that it becomes positive definite. There is an R routine that will do this for you - link. Or, if you want to do it yourself, i believe the general method is to do an eigen value decomposition, set any negative eigenvalues to zero, and then reconstruct the original matrix. If you're going to go down this ...

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