Can I maximize the correlation between a linear combination of

Averi Mitchell

Averi Mitchell

Answered question

2022-06-16

Can I maximize the correlation between a linear combination of variables and some other variable?

Answer & Explanation

Zayden Wiley

Zayden Wiley

Beginner2022-06-17Added 21 answers

Step 1
Some observations when n = 1 w.o. intercept to guess what is going on.
Let's fix X our predictor and Y the target.
Than:
c o v ( α X , Y ) = α c o v ( X , Y )
is a linear function of α . Not much to be maximized here, just a change of scale changes the covariance in a linear (unbounded) fashion.
Instead:
c o r r ( α X , Y ) = c o r r ( X , Y )
is scale independent. Again, not much to be maximized here.
This shows that you are not looking at a sensible metric, I think.
LR instead would look at E [ ( α X Y ) 2 ] ( one may be more used to the form of this expression as the sum of the squared residuals ), that is quadratic in α and can therefore be minimized.

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