Linear Regression Example

Let’s assume that we are given data with x values and their labels y.

x	Y
1	1
3	2
4	4
6	4
8	5
9	7
11	8
14	9

Now to find the line that fits best with the least square method, we simply need to find the parameters like slope and the intercept of the line i.e m and c.

Let equation of line be          y = bx + a

Where b is the slope and a is intercept which is given by

\[a = \frac{(\Sigma y)(\Sigma x^2) - (\Sigma x)(\Sigma xy)}{n(\Sigma x^2)-(\Sigma x)^2}\]

\[b = \frac{n\Sigma xy - (\Sigma x)(\Sigma y)}{n(\Sigma x^2)-(\Sigma x)^2}\]

Then

x	y	$x^2$	xy	$y^2$
1	1	1	1	1
3	2	9	6	4
4	4	16	16	16
6	4	36	24	16
8	5	64	40	25
9	7	81	63	49
11	8	121	88	64
14	9	196	126	81
Sum = 56	Sum = 40	Sum  = 524	Sum = 364	Sum = 256

Now 

$a = \frac{(40 \times 524) - (56 \times 364)}{(8 \times 524) - (56)^2 } = 0.545$

$b =\frac{(8\times 364)-(56\times 40)}{(8\times 524)-(56)^2} = 0.636 $

So equation of the line will be  \[y =0.636x + 0.545\]