## FANDOM

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Least squares is a form of approximation, used to make a prediction based on experimental data.

Given a set of points $(x_1, y_1) (x_2, y_2)...(x_n, y_n)$, and the deviation from the points $d=y_i - (ax_i + b)$ minimize $D=\sum_{i=1}^n (y_i - (ax_i+b))^2$.

Finding the minimum involves talking a look a the derivates.

• $\frac{\delta D}{\delta a}=\sum_{i=1}^n 2 (y_i - (ax_i + b)) (-x_i)$
• $\frac{\delta D}{\delta b}=\sum_{i=1}^n 2 (y_i - (ax_i + b)) (-1)$

Removing unnecessary 2 constant (solving to approach 0), and simplifying:

• $\frac{\delta D}{\delta a}=\sum_{i=1}^n (y_i - (ax_i + b)) (-x_i) = \sum (a x_i^2 + x_i b + - x_i y_i)$
• $\frac{\delta D}{\delta b}=\sum_{i=1}^n (y_i - (ax_i + b)) (-1) = \sum (a x_i + b - y_i)$

This can be re-written as a 2x2 linear system:

• $a \sum x_i^2 + b \sum x_i = \sum x_i y_i$
• $a \sum x_i + b n = \sum y$