[tex] y_i=A{x_i}^b [/tex](adsbygoogle = window.adsbygoogle || []).push({});

When I solve for A two different ways I am getting different answers..so somewhere I'm doing something wrong. If someone could point out where I would be grateful :).

Using logs:

[tex] y_i=A{x_i}^b [/tex]

[tex] ln(y_i)=ln(A)+b*ln(x_i) [/tex]

[tex] ln(y_i)-(ln(A)+b*ln(x_i))=r_i [/tex] for least squares we want to minimize: [tex] S=\sum_i^{n}{r_i}^2 [/tex] which means the gradient has to be zero. I only care about finding A right now so I only have to deal with the partial with respect to A:

[tex] \frac{\partial S}{\partial A}\sum_i^{n}{r_i}^2 =2 \sum_i^{n}{r_i}\frac{\partial r_i}{\partial A}= 2\sum_i^{n}(ln(y_i)-ln(A)-b*ln(x_i))\frac{1}{A}=0 [/tex]

The numerator inside the sum has to be zero, and we can ignore the 2, so:

[tex]\sum_i^{n}(ln(y_i)-ln(A)-b*ln(x_i))=0[/tex]

[tex]\sum_i^{n}ln(y_i)-n*ln(A)-b\sum_i^{n}ln(x_i)=0[/tex]

[tex] ln(A)=\frac{\sum_i^{n}ln(y_i)-b\sum_i^{n}ln(x_i)}{n} [/tex]

(this is the derivation that I think is correct).

But when i solve for A without taking the logs of each side first I get something else:

[tex] y_i - A{x_i}^b = r_i [/tex]

[tex] \frac{\partial S}{\partial A}\sum_i^{n}{r_i}^2 =2 \sum_i^{n}{r_i}\frac{\partial r_i}{\partial A}= 2\sum_i^{n}(y_i - A{x_i}^b)*-{x_i}^b=0 [/tex]

[tex]\sum_i^{n}(-{x_i}{y_i} + A{x_i}^{2b})=0 [/tex]

[tex]-\sum_i^{n}{x_i}{y_i}+A\sum_i^{n}{x_i}^{2b}=0 [/tex]

[tex] A=\frac{\sum_i^{n}{x_i}{y_i}}{\sum_i^{n}{x_i}^{2b}} [/tex]

And if you take the ln of it to compare it with what we got before you get:

[tex] ln(A)= ln(\frac{\sum_i^{n}{x_i}{y_i}}{\sum_i^{n}{x_i}^{2b}}) =ln(\sum_i^{n}{x_i}{y_i})-ln(\sum_i^{n}{x_i}^{2b}) [/tex]

Which is not the same as:

[tex] ln(A)=\frac{\sum_i^{n}ln(y_i)-b\sum_i^{n}ln(x_i)}{n} [/tex]

as far as I can tell...

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Solving for least square coefficients of power law

**Physics Forums | Science Articles, Homework Help, Discussion**