## Least Squares fit to a linear equation

This code applies a Least Squares fit of the supplied data to the form y = mx + b. X is the observed input and Y is the observed output of the system that you are trying to fit into the standard linear equation form.

If you know that the system you are trying to fit is linear, this function can be a life saver! However, don't think it is a magic wand and can be used to approximate any system. It can be modified to fit any polynomial equation you may want to, but this code has been designed for only the linear case.

```
function [m, b] = ls_linear(x_observed, y_observed)
%
% Perform Least Squares curve-fitting techniques to solve the coefficients
% of the linear equation: y = m*x + b. Input and Output equation
% observations must be fed into the algorithm and a best fit equation will
% be calculated.
%
% Usage: [M,B] = ls_linear(observed_input, observed_output);
%
% observed_input is the x-vector observed data
% observed_output is the y-vector observed data
%
% Author: sparafucile17 06/04/03
%
if(length(x_observed) ~= length(y_observed))
error('ERROR: Both X and Y vectors must be the same size!');
end
%Calculate vector length once
vector_length = length(x_observed);
%
% Theta = [ x1 1 ]
% [ x2 1 ]
% [ x3 1 ]
% [ x4 1 ]
% [ ... ... ]
%
Theta = ones(vector_length, 2);
Theta(1:end, 1) = x_observed;
%
% Theta_0 = [ y1 ]
% [ y2 ]
% [ y3 ]
% [ y4 ]
% [ ... ]
%
Theta_0 = y_observed;
%
% Theta_LS = [ M ]
% [ B ]
%
Theta_LS = ((Theta' * Theta)^-1 ) * Theta' * Theta_0;
%Return the answer
m = Theta_LS(1);
b = Theta_LS(2);
```