explanation of least squares

Question

Can someone explain me how to start this problem: method of least squares fit,

$y=Bx$ to the following $n=6$ points $(3,4),(1,2),(5,4),(6,8),(3,6),(4,5)$

I have calculated $x$ mean and $y$ mean, and calculated $\large \frac{(x-xm)(y-ym)}{(x-xm)^{2}}$ but it doesn't give me a right answer

What is the answer you expect to get, and what answer are you getting? How did you do the calculations? — Jonathan Christensen, Jan 22 '13 at 22:50
Possibly useful explanation of least squares linear regression — MJD, Jan 22 '13 at 23:18

copper.hat · Accepted Answer · 2013-01-22T23:12:04.777

2

Presumably you are trying to find the $B$ that minimizes $f(B) = \sum_{k=1}^6 (y_k-Bx_k)^2$. This is a strictly convex quadratic, so it has a unique minimizer, and we can find the solution by differentiating and setting the derivative to zero.

We have $\frac{df(B)}{dB} = -2 \sum_{k=1}^6 x_k(y_k-Bx_k) = -2 ((\sum_{k=1}^6 x_ky_k)-B (\sum_{k=1}^6 x_k^2))$. Setting $\frac{df(B)}{dB} =0$ gives the equation $$\frac{df(B)}{dB} = -2 ((\sum_{k=1}^6 x_ky_k)-B (\sum_{k=1}^6 x_k^2)) = 0$$

and hence the minimizing $B$ is given by $B = \frac{\sum_{k=1}^6 x_ky_k}{\sum_{k=1}^6 x_k^2}$. In this particular case, the computation gives $B = \frac{5}{4}$.

edited Jan 22 '13 at 23:12

answered Jan 22 '13 at 22:56

copper.hat

178,207

have no idea how you got the last equation, but thats a correct value – Jack F Jan 22 '13 at 23:03
I have elaborated slightly; does this clarify it for you? – copper.hat Jan 22 '13 at 23:12
ooooo i see it. Thank you so much – Jack F Jan 22 '13 at 23:13

explanation of least squares

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