Need help with a least squares method fitting.

I am doing a final project for a class and the first part is about fitting a 2nd and 6th order polynomials to some data using the least squares method. The catch is the general linear least squares method cannot be used, i.e. {y}=[Z]{a}+{e}.

clear all;

clc;

load('data')

x = [t]';

y = [r]';

N=[14400 sum(x) sum(x.^2);sum(x) sum(x.^2) sum(x.^3);sum(x.^2) sum(x.^3) sum(x.^4)];

q=[sum(y) sum(x.*y) sum(x.^2.*y)];

a=N/q

%Polyfit Check to compare

x=[t]';

y=[r]';

b=polyfit(x,y,2)

fprintf('The data is fit as y=(%.4f)+(%.4f)(x)+(%.4f)x^2\n\n',b(1),b(2),b(3));

And the results I get do not match up. 

a =

     1.0e+04 *

      0.0000
      0.0001
      1.5963

b =

     1.0e+03 *

     -0.0000    0.0001    8.0980

The data is fit as y=(-0.0000)+(0.1205)(x)+(8098.0146)x^2

I don't know exactly what I am doing wrong. The number 14400 is in there because the data is two 14401 X 1 doubles. I have also attached the data. Thanks in advance

Answer 1

[James Tursa]

The least squares equations you are trying to solve, using your variable names, are

N * a = q'

So do this instead (where the transpose is simply to turn q into a column vector):

a = N \ q';

Then you can see how they match up (the fliplr stuff is just to get the numbers to line up for output):

>> fliplr((N\q')')
ans =
  1.0e+003 *
  -0.000000012897721   0.000120497278334   8.098014646584689
>> polyfit(x,y,2)
ans =
  1.0e+003 *
  -0.000000012897721   0.000120497278334   8.098014646584003