# Thread: Write down the normal eq for estmiating the constant A and B

1. ## Write down the normal eq for estmiating the constant A and B

Question :

Write down the normal equations for estimating the constant A and B in the least-square fit of the model

$\displaystyle y = A + Bx,$

using n data points i.e. $\displaystyle (x,y),(x_2,y_2),.....,(x_n,y_n).$

2. take the sum of squares and differentiate wrt A and B.

3. ## Is this correct?

Originally Posted by matheagle
take the sum of squares and differentiate wrt A and B.

$\displaystyle y \ = \ A + Bx$

$\displaystyle q \ = \ \sum_{i} [y - (A+Bx)]^2$

Differentiating partuall wrt to A and B weget

$\displaystyle \frac{\partial q}{\partial A} \ = \ (-2)[y_i -(A + Bx_i)] \ = \ 0$

and
$\displaystyle \frac{\partial q}{\partial B} \ = \ (-2) x_i[y_i -(A + Bx_i)] \ = \ 0$

normal equation is

$\displaystyle \sum_{i} y_i \ = \ An + B \sum_{i} x_i$

$\displaystyle \sum_{i} x_i y_i \ = A \sum_{i} x_i + B \sum_{i} x_i ^2$

Is this correct??

4. Impressive, we got you to do some decent work.
The normal equations look good,
but you left out the sums in the partial derivatives
in the previous set of equations.

Originally Posted by zorro

$\displaystyle y_i \ = \ A + Bx_i$

$\displaystyle q \ = \ \sum_{i} [y_i - (A+Bx_i)]^2$

Differentiating wrt to A and B we get

$\displaystyle \frac{\partial q}{\partial A} \ = \ (-2)\sum_i[y_i -(A + Bx_i)] \ = \ 0$

and
$\displaystyle \frac{\partial q}{\partial B} \ = \ (-2)\sum_i x_i[y_i -(A + Bx_i)] \ = \ 0$

the normal equations are

$\displaystyle \sum_{i} y_i \ = \ An + B \sum_{i} x_i$

$\displaystyle \sum_{i} x_i y_i \ = A \sum_{i} x_i + B \sum_{i} x_i ^2$