MATLAB: Multivariate Regression Parameter Optimization

minimizationmultivariatemultivariate regressionoptimizationparameter optimizationregression

I am trying to find the values that will optimize parameters in an equation. Two measured values are related by an equation to equal a known value.
The equation is: y=a*(x^b)*(z^c)
x & z = measured values (vectors of length n)
a,b & c = the unknown free parameters (single values)
I also have known values Y that y should approximately equal. Therefore, I want to find the set of parameters a,b&c that minimizes the difference between Y and y (given all measured values).
What is the best way to do this in MATLAB?
Thanks!
-Andrew

Best Answer

NLINFIT is good.
NonLinearModel.fit is also good.
But since your problem involves fitting a surface with only two independent variables, it can be done very simply using the Curve Fitting Toolbox functions.
You can do it interactively using CFTOOL and then generate the MATLAB code automatically (recommended), or if you want to write the code out by hand yourself, you can do something along these lines:
x = rand(100,1);
z = rand(100,1);
atrue = 2.5;
btrue = 1.7;
ctrue = 1.2;
Y = atrue*(x.^btrue).*(z.^ctrue) + 0.05*randn(size(x)) ;
scatter3(x,z,Y);
hold all;
F = fittype('a*x^b*z^c','Independent',{'x' 'z'});
M = fit([x z],Y,F) % Or specify an initial guess: M = fit([x z],Y,F,'Start',[0 0 0])
plot(M)
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