Let ${(x_i, y_i)}_{i=1}^n$ be a set of points in $R^2$ such that

Question

Let ${(x_i, y_i)}_{i=1}^n$ be a set of points in $R^2$ such that

$$\sum_{i=1}^n x_i = \sum_{i=1}^n y_i = 0.$$ Consider the values of $\alpha, \beta$ which minimize $$\sum_{i=1}^n |\alpha x_i + \beta - y_i|^2$$ (the linear least squares fit). Show that $\beta$ = 0.

Could someone please help me with this?

Answer 1

Hint:

$\displaystyle f(\alpha,\beta)=\sum_{i=1}^n (\alpha x_i + \beta - y_i)^2$ then \begin{cases} \dfrac{\partial f}{\partial\alpha}=0,\\ \dfrac{\partial f}{\partial\beta}=0. \end{cases}