Least Squares Method

this is an exceptionally optional note, added for completeness of discussion in other notes

Imagine you have a bunch of values of $x$ and $y$ which are related linearly, i.e. you want to find parameters $m$ and $b$ that would make $mx+b$ as close to $y$ as possible.

You could do so using least squares method. In fact, this is exactly how Excel finds trend line for linear equations. Let’s denote individual data points $(x_i,y_i)$, and there are $n$ such points. You can find optimal $m$ and $b$ as follows:

$$m=\frac{n{\sum }_{i=1}^n{x}_i{y}_i-\left({\sum }_{i=1}^n{x}_i\right)\left({\sum }_{i=1}^n{y}_i\right)}{n{\sum }_{i=1}^n{x}_i^2-{\left({\sum }_{i=1}^n{x}_i\right)}^2}$$

and

$$b=\frac{1}{n}\left(\sum _{i=1}^n{y}_i-m\times \sum _{i=1}^n{x}_i\right)$$