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Old 08-27-2012, 01:08 PM
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magdon magdon is offline
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Join Date: Aug 2009
Location: Troy, NY, USA.
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Default Re: Recency weighted regression

Yes, there is a closed form solution which is obtained by taking the \alpha_t into the square:

E_{in}=\sum_{t}\alpha_t(\mathbf{w}\cdot\mathbf{x}_t-y_t)^2=\sum_{t}(\mathbf{w}\cdot\mathbf{x}_t\sqrt{\alpha_t}-y_t\sqrt{\alpha_t})^2

This is exactly an unscaled linear regression problem where you have rescaled each data point (\mathbf{x}_t,y_t) by \sqrt{\alpha_t}. So, after you rescale your data in this way, you can just run your old regression algorithm without the weightings.

Quote:
Originally Posted by itooam View Post
Thank you for all your help it has been really appreciated. I have one final question, do you know if there is a closed form solution to

E_{in}=\sum_{t}\alpha_t(\mathbf{w}\cdot\mathbf{x}_t-y_t)^2

(assuming \alpha is a vector with the same number of rows as x?)

i.e., the closed form solution as used for linear regression and regularization - copied from lecture notes is this:

W_{reg} = (Z^{T} Z+\lambda I)^{-1}Z^Ty

I am not sure where \alpha would end up in the above, the derivation is beyond me mathematically?
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