How to find the best fit to a set of data?

I have discrete regular grid of a-b points and their corresponding “c” values and I interpolate it further to get a smooth curve. Now from interpolation data, I further want to create a polynomial equation for curve fitting. How to fit 3D plot in polynomial?

I try to do this in MATLAB. I used Surface fitting toolbox in MATLAB (r2010a) to curve fit 3 dimensional data. But, how does one find a formula that fits a set of data to the best advantage in MATLAB. Any advise?

This is just a small portion of my data.

a = [ 0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001,0.011, 0.011, 0.011, 0.011, 0.011, 0.011, 0.011, 0.011, 0.011, 0.011];

b = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];

c = [ -.304860225, .170315374, .343019354, .370114906, .373180536, .36719579, .363397853, .363417755, .366962504, .379710865, -.304860225, .170315374, .343019354, .370114906, .373180536, .36719579, .363397853, .363417755, .366962504, .379710865];

Thanks in advance.

[James Phillips]

I fitted your 3D data subset to the 3D equation

z = a*exp(b*x+c*y*y) + Offset

And got what I consider a fairly good fit:

Degrees of freedom (error): 16
Degrees of freedom (regression): 3
Chi-squared: 0.000421944557298
R-squared: 0.999490260845
R-squared adjusted: 0.999394684754
Model F-statistic: 10457.5343591
Model F-statistic p-value: 1.11022302463e-16
Model log-likelihood: 79.2849184051
AIC: -7.52849184051
BIC: -7.3293453858
Root Mean Squared Error (RMSE): 0.00459317187409
a = -1.0098597038687287E+00
       std err squared: 1.23700E-04
       t-stat: -9.07981E+01
       p-stat: 0.00000E+00
       95% confidence intervals: [-1.03344E+00, -9.86282E-01]
b = 4.4388958377603527E-01
       std err squared: 1.06567E+00
       t-stat: 4.29996E-01
       p-stat: 6.72934E-01
       95% confidence intervals: [-1.74451E+00, 2.63229E+00]
c = -4.0672678853444422E-01
       std err squared: 4.04870E-05
       t-stat: -6.39212E+01
       p-stat: 0.00000E+00
       95% confidence intervals: [-4.20216E-01, -3.93238E-01]
Offset = 3.6932328705567163E-01
       std err squared: 1.78766E-06
       t-stat: 2.76226E+02
       p-stat: 0.00000E+00
       95% confidence intervals: [3.66489E-01, 3.72158E-01]

Coefficient Covariance Matrix
[ 4.69065343e+00 2.45739605e+02 1.73568438e+00 1.49019190e-02]
[ 2.45739605e+02 4.04097132e+04 -9.72087773e-02 -5.35271954e-02]
[ 1.73568438 -0.09720878 1.53525294 0.11068472]
[ 0.01490192 -0.0535272 0.11068472 0.0677873 ]

If you would like details including how I plotted the 3D data and 3D equation, please email me directly.

James Phillips [e-mail]

[James Phillips]

So that you can visually inspect the resulting fitted surface, I also made a 3D animation of the fitted surface and data rotating in 3-space:

http://zunzun.com/images/rotation.gif

The animated GIF is 6.6 Meg.

James

[Syeda]

Please check your mailbox! Thankyou for replying

Answer 1

[Image Analyst]

First of all, you have 2D data. You do not have 3D data. You have 2 independent variables, which can each be an index in a matrix, and for those you have one value. This is a 2D array, not 3D because you do not have 3 independent variables. Sure you can plot it with surf() and it looks kind of 2.5D-ish since it's a perspective projection of a surface displayed on your flat 2D monitor, but that's still 2D not 3D.

That said, you can use John D'Errico's polyfitn(): http://www.mathworks.com/matlabcentral/fileexchange/34765-polyfitn That's what I use to fit a nice smooth polynomial to a non-uniform background image. This will give me a nice smooth background image free from any video noise since it's a model, not actual noisy image data.

By the way, there does not appear to be a "Surface Fitting Toolbox." If you're actually using the Curve Fitting Toolbox, then please add that the the product list to the bottom right of your original question.

[Syeda]

I tried John D'Errico's polyfitn()

p = polyfitn([a(:),b(:)],c(:),3) % generates the coefficients of third degree polynomial

polyn2sym(p)

(This results may not be accurate, as it is just a small portion of my data)

Anyways, it gives me an equation.

Now, I want to plot the real data and the curve we created to fit the data. I want plot to compare fit with original data. Any advice?