Supplement 2.1: Point Clouds and Linear Regression Lines (2/3)

Best-fit straight lines passing through the origin

Our next step is to differentiate $S (a, b)$ with respect to a.

For simplicity, we consider a point cloud with a centroid $\bar{P} (\bar{x}', \bar{y}')$ in the origin of an $(x', y')$ coordinate system. Since the centroid is a point on the regression line, one has b=0 and

S (a) = \frac{1}{n} \sum_{i = 1}^{n} {(y_{i}^{'} - a x_{i}^{'})}^{2}

Differentiated: $S' (a) = \frac{2}{n} (a \sum_{i = 1}^{n} x_{i}^{2}' - \sum_{i = 1}^{n} x_{i}^{'} y_{i}^{'})$

and setting $S' (a) = 0$ yields:

a = \frac{\sum_{i = 1}^{n} x_{i}^{'} y_{i}^{'}}{\sum_{i = 1}^{n} x_{i}^{2}'}

where $\sum_{i = 1}^{n} x_{i}^{2}' \neq 0$ is a necessary condition.

We consider n points

P (x_{1}^{'}, x_{1}^{'}), ... P (x_{n}^{'}, x_{n}^{'})

, their centroid

\bar{P} (\bar{x}', \bar{y}')

lying in the origin of the coordinate system. Then the best-fit line

y' = a x'

is passing through the origin and has the slope:

a = \frac{\sum_{i = 1}^{n} x_{i}^{'} y_{i}^{'}}{\sum_{i = 1}^{n} x_{i}^{2}'}

Best-fit straight lines with intercept b

We set $x_{i}^{'} = x_{i} - \bar{x}$ and $y_{i}^{'} = y_{i} - \bar{y}$ which corresponds to a shift of coordinates, the new (x,y) coordinate axes being shifted by $(\bar{x}, \bar{y})$ with respect to the $(x', y')$ coordinates used in the left column.