nonlinear data fitting for a system of ODE using lsqcurvefit (finding unknown parameters)

I have a set of experimental data in staedy state condtion(without time dependent). This experiment has been done at 4 different concentration including 3 compounds. the values of concentration at inlet(x=0) and outlet(x=1) are known. i used ODE15s to solve the differenial equations and find the values at outlet and then compare them with real values to find the unknown kinitic parameters(i followed https://www.mathworks.com/matlabcentral/answers/43439-monod-kinetics-and-curve-fitting#comment_89455). but when i ran the code i got some error as below: Not enough input arguments.

Error in Untitled10>objfun (line 25)
[tSol,YSol]=ode15s(@diffeq,x,Z);

Error in lsqcurvefit (line 202)
            initVals.F = feval(funfcn_x_xdata{3},xCurrent,XDATA,varargin{:});

Error in Untitled10 (line 20)
[kfit,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@objfun,K,Cin,yy)

Caused by:
    Failure in initial objective function evaluation. LSQCURVEFIT cannot continue.

[Moji]

    clc
  clear all
  global K
  Y0=[0  0  0
  293.3  2.141445291  3.688379481
  462.8  2.536287882  4.200913729
  966.2  2.352145581  6.525317012
  1920  2.454642817  11.40142937
  ];
  K=[1 2 1];
  dt=0.001;
  x=[0:dt:0.02]';
  Cin=[0;200;500;1000;2000];
  yy=[0  0  0
  208.0886316  33.822843  78.56152111
  376.5730526  31.98339108  168.7154254
  802.9630526  29.05423171  204.8119954
  1770  35.44630378  295.906807
  ];
  [kfit,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@objfun,K,Cin,yy)

function [C]=objfun(K,Y0,x)
for i=1:5
Z=Y0(i,:);
[tSol,YSol]=ode15s(@diffeq,x,Z);
C(i,:)=YSol(end,:)
end
end

function dYdt = diffeq(x,Y)
global K
k1=K(1);
k2=K(2);
k3=K(3);
dadt=-k1*Y(1)/(1+k1*Y(1));
dbdt=k1*Y(1)/(1+k1*Y(1))-k2*Y(2);
dcdt=k2*Y(2)-k3*Y(3);
dYdt=[dadt;dbdt;dcdt];
end

[Star Strider]

@Mojtaba Malayeri — Please post the image you included with your previous (now deleted) Question.

It is essential to understanding what you want to do.

[Moji]

@Star Strider that image was my experimental data, it is included in my code now.

Answer 1

[Matt J]

[kfit,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@(K) objfun(K,Y0,x),K,Cin,yy)

[Moji]

thank you for your response, but i got error again:

Error using Untitled10>@(K)objfun(K,x,Y0)
Too many input arguments.

Error in lsqcurvefit (line 202)
            initVals.F = feval(funfcn_x_xdata{3},xCurrent,XDATA,varargin{:});

Error in Untitled10 (line 20)
[kfit,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@(K) objfun(K,x,Y0),K,Cin,yy)

Caused by:
    Failure in initial objective function evaluation. LSQCURVEFIT cannot continue.

[Matt J]

Your objective function doesn't seem to have any dependence on Cin. It should look like this

yy_predicted = objfun(K,Cin_data)

[Moji]

becaue this problem is steady-stat, i dont have data at different x between x=0:1 (i have only data of inlet and outlet). but it is done at different concentration; so Y0 is my inlet and yy is my outlet. each number in C_in related to a row of matrix Y0. I hope I have explained it well

[Matt J]

Then, shouldn't you have this?

 lsqcurvefit(@(a,b) objfun(a,b,x),  K,Y0,yy)

[Moji]

sorry, what are a and b here? actually, Y0 should be replaced instead of Cin but it is a vector 5*3. i got erorr

[Matt J]

They are the names of inputs to fun(a,b) where fun is the two-argument anonymous function created as follows,

   fun = @(a,b) objfun(a,b,x) ;

[Moji]

i replaced this function in main file: [kfit,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@(K,Y0) objfun(K,Y0,x), K,Y0,yy) but i got following message and can not be calculated: Optimization completed because the size of the gradient at the initial point is less than the default value of the optimality tolerance.

[Matt J]

This runs for me without error messages:

    Y0=[0  0  0
    293.3  2.141445291  3.688379481
    462.8  2.536287882  4.200913729
    966.2  2.352145581  6.525317012
    1920  2.454642817  11.40142937
    ];
    K0=[1 2 1];             %<---Change
    dt=0.001;                %<---Change
    x=[0:dt:0.02]';
    %Cin=[0;200;500;1000;2000];     %<---Change
    yy=[0  0  0
    208.0886316  33.822843  78.56152111
    376.5730526  31.98339108  168.7154254
    802.9630526  29.05423171  204.8119954
    1770  35.44630378  295.906807
    ];
    [kfit,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=...
         lsqcurvefit(@(a,b) objfun(a,b,x),K0,Y0,yy)  %<---Change

function [C]=objfun(K,Y0,x)
for i=1:5
Z=Y0(i,:);
[tSol,YSol]=ode15s(@(a,b)diffeq(a,b,K) ,x,Z); %<---Change
C(i,:)=YSol(end,:);
end
end

function dYdt = diffeq(x,Y,K)
%global K      <---Change
k1=K(1);
k2=K(2);
k3=K(3);
dadt=-k1*Y(1)/(1+k1*Y(1));
dbdt=k1*Y(1)/(1+k1*Y(1))-k2*Y(2);
dcdt=k2*Y(2)-k3*Y(3);
dYdt=[dadt;dbdt;dcdt];
end

[Moji]

Yes, it works now. thank you so much. i really appriciate. just my fitting value are negative. how can i add possitive constrain for unknown fitting parameters??

[Matt J]

You're welcome, but please Accept-click the answer if it solved your problem. To add positivity constraints:

lowerbounds=[0;0;0];
... = lsqcurvefit(@(a,b) objfun(a,b,x),K0,Y0,yy,lowerbounds);

[Moji]

i replace a lowerbound as follow, but i one of my parameter is negative: [kfit,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=... lsqcurvefit(@(a,b) objfun(a,b,x),K0,Y0,yy,[0;0;0]) another issue is that Rsdnrm =7.1019e+04 that is very large error

[Matt J]

You need to make lowerbounds a vector, like in my example.

[Moji]

yes, sorry i made mistake. but Rsdnrm = 2.1754e+05 is reasonable??

[Matt J]

Who knows? If you re-run as below, it will solve the exact same optimization problem, but give you a much smaller Rsdnrm,

 [kfit,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=...
   lsqcurvefit(@(a,b) objfun(a,b,x)/1e5,kfit,Y0,yy/1e5,[0;0;0])

[Moji]

because i have to use these parameters in another model to find the exact result. so it is very important to find the accurate values for these parameters.

[Matt J]

How well is it working on simulated Y0,yy data (for which you know the true K)?

[Moji]

the error is very large. i even change the initial gusses for fitting parameters, but the erorr doesn't change.

[Moji]

i checked it, but there are big differenc between them: simulated=[0 0 0 293.2800001 0.000216393 5.849608311 462.78 0.000216393 6.756985175 966.18 0.000216393 8.897246179 1919.98 0.000216393 13.87585578]

real value=[0 0 0 208.0886316 33.822843 78.56152111 376.5730526 31.98339108 168.7154254 802.9630526 29.05423171 204.8119954 1770 35.44630378 295.906807]

[Moji]

i changed every parameters, but the erorr value is fixed at 2.17e+05. i don't know where this issue come from!!!!!!!!!

[Matt J]

Make sure you study the material at Optimizing a Simulation or Differential Equation.

[Moji]

you mean i have to change my optimization solver instead of lsqcurvefit??

[Matt J]

No. The link I gave you gives advice applicable to all solvers.

[Moji]

Thanks for the suggestion. i implimened all case of options, but the issue still persists. very large erorr.