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So this is more a question to help me understand what is going on rather than application:

So in normal regression we have

$$ \mathbf{Y=X} \mathbf{\beta} + \varepsilon $$

Now the part that really confuses me:

A regression with orthogonal polynomial regression is just:

$$ \mathbf{Y= \phi(X)} \mathbf{W}+ \varepsilon $$

where $ \mathbf{\phi(X) = P^{T} X}$ and $W= \mathbf{P\beta}$

Where P is the orthonormal change of basis matrix that you can obtain in R using contri.poly()?These P transformations chosen usually appear to also centre the data? Also if I am correct, how exactly are these change of basis matrix determined?

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