Line of Best Fit Example

Example

1. With points $\{(1,2),(2,2),(3,4)\}$
2. We get $A=\left[\begin{array}{cc}1& 1\\ 1& 2\\ 1& 3\end{array}\right]$
3. We get $x=\left[\begin{array}{c}a\\ b\end{array}\right]$
4. We get $c=\left[\begin{array}{c}2\\ 2\\ 4\end{array}\right]$
5. Then, compute $(A^{T}A)x=(A^{T}c)$

1 & 1 & 1\\ 1 & 2 & 3\\ \end{array}\right] \left[\begin{array}{cc} 1&1\\ 1 & 2\\ 1 & 3\\ \end{array}\right] \left[\begin{array}{cc} a\\ b\\ \end{array}\right] = \left[\begin{array}{ccc} 1 & 1 & 1\\ 1 & 2 & 3\\ \end{array}\right] \left[\begin{array}{cc} 2\\ 2 \\ 4 \\ \end{array}\right]

7. Compute later, and we get to REF and later $x=(\frac{2}{3},1)$
8. So, line of best fit is $y=bx+a⟹y=x+\frac{2}{3}$