MATLAB: Fit data with dependent parameters

fit dependent

Hi,

There are two rows of data, x and y. I would like to fit y = f(x), where

f(x) = a*x^3 + b*x^2 + (2a+3b)*x,

i.e. parameters are not independent.

I tried to use the function "fittype", but it does not work (Licensing error: -101,147).

I would like to know if there is any other way to solve it.

Thank you!

Best Answer

Yours is a linear problem, however the easiest way to estimate the parameters is likely an unconstrained nonlinear solver, such as fminsearch:

x = ...;
y = ...;
objfcn = @(b,x) b(1).*x.^3 + b(2).*x.^2 + (2*b(1) + 3*b(2)).*x;
[B,resnorm] = fminsearch(@(b) norm(y - objfcn(b,x)), [1;1]);
xv = linspace(min(x), max(x));
figure
plot(x, y, 'pb')
hold on
plot(xv, objfcn(B,xv), '-r')
hold off
grid

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MATLAB: Fsolve interchangeable with nlinfit, but how

It’s not exactly interchangable, however it’s possible to use it to estimate the parameters.

Example —

x = linspace(0, 10, 10);
y = exp(x.*0.2)+randn(size(x))/10
objfcn = @(b,x) b(1) + b(2).*exp(x.*b(3));
B = fsolve(@(b) norm(y - objfcn(b,x)), rand(3,1));
figure
plot(x, y, 'p')
hold on
plot(x, objfcn(B,x), '-r')
hold off
grid
text(1.5,6,sprintf('y = %.3f + %.3f\\cdote^{%.3f\\cdotx}',B))

producing (with this set of random values):

Experiment with this approach with your own data.

MATLAB: FMINCON: Interior-Point algo -> Is there a way to pass extra parameters to the hessian function or allow the objective function to return 3 output variables (function value, gradient and hessian) when there are no nonlinear constraints

Also, even if my hessian could be computed solely using 'x', this approach would not be ideal - computing the hessian requires computing the objective function and gradient so it would basically double the computing effort at each iteration.

To address this issue, you can adapt the technique described here,

https://www.mathworks.com/help/optim/ug/objective-and-nonlinear-constraints-in-the-same-function.html

The link talks about using nested functions and externally scoped variables to share values between the objective and constraint functions, but the same technique applies if you want to share between the objective function code and the HessianFcn code (or any other code).

For you, this would probably look like the following:

function x=optimizationMain(DATA,x0,A,b,Ae,beq,lb,ub)
  state.x=[]; state.f=[];state.g=[];
   
  opts=optimoptions('fmincon','SpecifyObjectiveGradient',true,...
                              'HessianFcn',@myHessianFcn);
  
   x=fmincon(@objectiveWrapper,x0,A,b,Ae,beq,lb,ub,opts); %run optimization
    function varargout=objectiveWrapper(x)
        
        if ~isequal(x,state.x)
         [varargout{1:nargout}]=myfun(x,DATA);
          if nargout>1, 
             state.x=x; state.f=varargout{1}; state.g=varargout{2}; 
          end
        else
           varargout={state.f,state.g};
        end         
        
    end
    function H=MyHessianFcn(x, ~)
        
        if ~isequal(x,state.x)
            state.x=x;
            [state.f,state.g]=myfun(x,DATA);
        end
        H=.... some function of state and DATA
      
       
    end
end

Best Answer

Related Solutions

MATLAB: Fsolve interchangeable with nlinfit, but how

MATLAB: FMINCON: Interior-Point algo -> Is there a way to pass extra parameters to the hessian function or allow the objective function to return 3 output variables (function value, gradient and hessian) when there are no nonlinear constraints

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