Ax + b Least Squares Minimization Standard Form

[cook11]

All -

Given a set of data {(xi, yi)| i = 1,2,...,m} and the regression equation f(x) = ax + b, I want to use the simplex method to minimize the equation Sigma [(yi - f(xi))/f(xi)]^2. However, I am stuck on how to initially organize the problem. I am not sure whether the equation, Sigma [(yi - f(xi))/f(xi)]^2, needs to be put into some sort of standard form or not. Also, I am having trouble comprehending how to turn the individual [(y - f(x))/f(x)]^2 equations into constraints.

The end goal for this is to turn it into a program. The simplex method should run sufficiently fast for the type of data I will be feeding the program. However, I first need to understand the logic before any code gets written.

Let me know if I'm not stating the problem clear enough. Thank you for helping.

 

[statdad]

You are trying to minimize

$$S(a,b) = \sum_{i=1}^m {\left(\frac{y_i - (a + bx_i)}{a+bx_i}\right)^2}$$

is this correct? I'm not sure what this will give you - certainly not a regression result.

I'm not aware of any way to use the simplex method - designed for linear optimization problems, for a problem that is so far from being linear. (L1 regression problems can be solved as linear programming problems, but that is the closest item I know of.)

 

[cook11]

statdad -

Correct. That's the right equation. It's the Minimum Squares Percent Error (MSPE) equation used in econometrics. I figured out there are a handful of algorithms/methods to minimize nonlinear equations. Variations of Newton's Method are fairly popular for this task. Thank you for your input!