Normal Equations

**joatmon** · December 5th 2012, 12:29 PM

Show that the best least squares fit to a set of measurements $y_1,...y_m$ by a horizontal line y = $\lambda$ (where $\lambda$ is a constant) is their average:

$\lambda = \frac{y_1 + ... + y_m}{m}$

Other than that I need to use normal equations and that this is a problem concerning orthogonality, I am completely stuck and have no idea how to proceed. Can anyone help? Thanks.

**HallsofIvy** · December 5th 2012, 12:56 PM

. That's pretty straight forward if you use the definition isn't it? You want to find the value $\lambda$ such that $\sqrt{\sum_{i=1}^m (y_i- \lambda)^2}$ is a minimum. And because the square root function is is increasing that's the same as finding $\lambda$ such that $\sum_{i=1}^m (y_i- \lambda)^2$ is a minimum.

So take the derivative with respect to $\lambda$ and set it equal to 0: $2\sum_{i=1}^m \lambda- y_i= 0$ . That's the same as $\sum_{i=1}^m \lambda- \sum_{i= 1}^m y_i= m\lambda- \sum_{i=1}^m y_i= 0$ . Solve that for $\lambda$ .

**joatmon** · December 5th 2012, 12:57 PM

Thank you, but no derivatives. I have to approach this as a linear algebra problem using normal equations.

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