hat matrices about mgwr HOT 4 CLOSED

Wei, just tagging you here for tracking. Don't worry about it, & keep plugging that dissert 😄

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My working notebook on the subject. Not primetime, by far, but anyone interested in digging in can see how I started.

I'll also be updating that notebook as I keep working.

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I also tried the math above and did not get luck.
e.g. Using a simple two covariates example:

f1 = X1B1 = S1(y - f2)
f2 = X2B2 = S2(y - f1)
=> y_hat = f1+f2 = S1(y - f2) + S2(y - f1) = Sy
=> S1y - S1f2 + S2y - S2f1 = Sy
=> (S1+S2)y - (S1f2 + S2f1) = Sy

Instead of taking inverse of y (as a vector), I wonder if taking partial derivative of y makes sense which can also leave S alone on the right hand side. But this gives another problem that need to solve the partial derivative of the second term (S1f2+S2f1), which is a function of y. I am thinking if we could find another equation/relationship among f1,f2 and y rather than equations above, the second term might be solvable. Just some random thoughts.

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Mathematical details to come, but I think we can close this based on Hanchen's recent efforts.

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