Matlab fit to three dimensions function

Hi

I got three input vectors: x1, x2, x3 and output vectory y. How do i fit to my custom function?

y=a0+a1*x1+a2*x2+a3*x3+a11*x1^2+a22*x2^2+a33*x3^2+a12*x1*x2+a13*x1*x3+a23*x2*x3

Obviously I want to get parameters a1, a2, ...

For two variables (x1, x2) i can do it like:

f = fit([x1, x2], y, 'poly33');

But I'm struggling to do that for the function above.

Any help appreciated.

[the cyclist]

Do you also have the Statistics and Machine Learning Toolbox available, or only the Curve Fitting Toolbox?

[standy]

Yes, i got this toolbox.

Answer 1

[Bruno Luong]

n = length(y);
A = [ones(n,1) x1(:) x2(:) x3(:) x1(:).^2 x2(:).^2 x3(:).^2 x1(:).*x2(:) x1(:).*x3(:) x2(:).*x3(:)] \ y(:)

Answer 2

[the cyclist]

I would do it like this:

% Set the random number generator seed, for reproducibility
rng default

% Create some random data
N = 1000;
x1 = randn(N,1);
x2 = randn(N,1);
x3 = randn(N,1);

% Create a response variable with known coefficients, and some noise
y = 2 +  3*x1      +  5*x2     + 7*x3 ...
       + 11*x1.^2  + 13*x2.^2  + 17*x3.^2 ...
       + 19*x1.*x2 + 23*x1.*x3 + 29*x2.*x3 ...
       + 31*randn(N,1);
   
% Fit a quadratic model
mdl = fitlm([x1 x2 x3],y,'quadratic')

% % The above is equivalent to the following model, written out in full Wilkinson notation
% mdl = fitlm([x1,x2,x3],y,'y ~ x1 + x2 + x3 + x1^2 + x2^2 + x3^2 + x1:x2 + x1:x3 + x2:x3');

Almost all of this code is me creating the data, to illustrate everything. Since you have the data already, you should only need

mdl = fitlm([x1 x2 x3],y,'quadratic')

The resulting model object, mdl, has methods for lots of information about the model fit.

[standy]

How does 'quadratic' fit to my equasion?

[the cyclist]

Unless I misunderstood, you want all possible linear terms and squared terms. That's a quadratic polynomial. Notice that right underneath, in some commented-out code, I show the full notation if I had written out all terms.

[standy]

Ahhh

Makes sense now. I got the output like:

Linear regression model:
    y ~ 1 + x1*x2 + x1*x3 + x2*x3 + x1^2 + x2^2 + x3^2

Estimated Coefficients:
                    Estimate         SE         tStat      pValue 
                   __________    __________    ________    _______

    (Intercept)        1.4267        1.1233      1.2701    0.23281
    x1              -0.015338      0.024769    -0.61922    0.54962
    x2               -0.90665        3.3137    -0.27361    0.78995
    x3                      0             0         NaN        NaN
    x1:x2           0.0031125      0.012077     0.25772    0.80185
    x1:x3                   0             0         NaN        NaN
    x2:x3               0.765        3.0553     0.25038    0.80736
    x1^2           6.8008e-05    0.00011935     0.56982    0.58137
    x2^2                    0             0         NaN        NaN
    x3^2                    0             0         NaN        NaN
    

Does it mean that a0=1, a1=-0.015338 etc?

[the cyclist]

Yes