Finding the original weights after data normalisation

Question

Suppose we have vectors $x_1$ and $x_2$, each has ($n$) samples. Both $x_1$ and $x_2$ are my independent variables.

Suppose we also have a vector $y$ which has ($n$) samples and is y my dependent variable.

I would like to perform a linear regression in the form: $$y=b_0 + b_1 x_1 + b_2 x_2$$

however, before performing the regression, all my variables ($y,x_1,x_2$) were normalised such that their mean is zero and std. deviation is $1$.

After the regression, we get $a_0, a_1$ and $a_2$ which are the weights. How can I get the original weights ($b_0,b_1,b_2$) given the weights obtained from the normalised data ($a_0,a_1$ and $a_2$)?

Answer 1

Suppose the original vectors are $y, x_1, x_2$ and we have

$$\hat{y}=\frac{y-\mu_y}{\sigma_y},\hat{x}_i=\frac{x_i-\mu_{x_i}}{\sigma_{x_i}}$$

and we have

$$\hat{y}=a_0+a_1\hat{x}_1+a_2\hat{x}_2$$

then we have

$$\frac{y-\mu_y}{\sigma_y}=a_0+\sum_{i=1}^2a_i\frac{x_i-\mu_{x_i}}{\sigma_{x_i}}$$

Hence, $$b_0=\mu_y+\sigma_y\left(a_0-\sum_{i=1}^2 \frac{a_i\mu_{x_i}}{\sigma_{x_i}}\right)$$ For $i\ge 1$, $$b_i=\frac{\sigma_ya_i}{\sigma_{x_i}}$$