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This question already has an answer here:

I have read many articles about PLS, but I could not understand the mathematical description yet. I know that it is quite similar to principal component regression (PCR), except that it takes into account the direction of the response variable.

Could you please provide for me a simple mathematical explanation e.g the formula and coefficient estimates form?

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marked as duplicate by Silverfish, ttnphns, gung, Sycorax, John Aug 7 '16 at 20:54

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ We already have a thread on Theory behind partial least squares regression. I think this question largely duplicates that one. If that doesn't adequately answer your question, perhaps you can edit this to make clear what is distinctive about your request? $\endgroup$ – Silverfish Aug 7 '16 at 19:40