LINEAR REGRESSION

PLEASE ANSWER PART B. PART A IS INCLUDED FOR REFERENCE ONLY.

We investigate the solution of regression. For simplicity, we have only one feature x to predict y. Suppose we are given samples (x1, y), ..., (Xn, Yn). wo, Wi are parameters, and we are to find parameters that best fit the following relation: Wo + W1Xi = yi. (a) Centering. [5 pts] Let = 1 i=1 Xi and x'= xi . X'; are called centered since 2h-x' = 0. Let i be the values predicted by Li, wo, w1: i = wo + W1li. Show that can be predicted by X', as well. That is, there are parameters W, w such that i = wo + wix, and write w, w in terms of wo, w1, and , but not xi. (b) Loss function. [5 pts] In (a), we converted linear regression on xi with parameters wo, wi to linear regression on x; with w, w. Write the loss functions of the both. Specifically, let's assume J is the loss function of the former, and J' is of the latter