Getting into machine learning, have a reasonable background in statistics and understand the basic principles of linear algebra (matrix multiplication etc.) - but am having a damn hard time figuring out why reducing a polynomial regression works.
For example, say we have this function:
$y =$ $\beta_0$ + $\beta_1$$x_1$ + $\beta_2$$x_2$$^2$
From what I've seen on 4 videos and 6 articles, we can use the following substitutions:
- $x_2$ = 1
- $x_3$ = $x$
- $x_4$ = $x^2$
To create the following model: $y =$ $\beta_0$ + $\beta_1$$x_1$ + ($\beta_2$$x_2$ + $\beta_3$$x_3$ + $\beta_4$$x_4$)
And then, fine - we can solve that as a normal multiple linear regression, and all is well.
But why, why does this work? I really cannot find an explanation for this.