MATLAB: Line search algorithm: nonlinear least squares method for fitting experimental data to a model

Question

Hello, I have tried to fit my experimental data with a nonlinear model. But the error between the obtained curves is very high. I tried to solve it with lsqnonlin and lsqcurvefit but the exitflag that I received is 1, so i can not get better results with this model. I find then that in some articles, they used line search method. I am not familiar with these types of codes. Can you, please, help me adapting my code to line search algorithm? Otherwise, any ideas to get better fit.

Thank you in advance.

clc
close all
clear all
figure;
A=importdata('1.xlsx');
xdata = ...
 [A(:,1) A(:,1) A(:,1)];
ydata = ...
 [A(:,2) (A(:,3)) (A(:,4))];
%%


% Create the  model.
fun = @(x,xdata)[(x(1)*exp(-(x(2)*(357.15-x(4)))/(x(3)+357.15-x(4))))...
        ./(1.+(xdata(:,1).*(x(1)*exp(-(x(2)*(357.15-x(4)))./(x(3)+357.15-x(4))))./x(6)).^(1-x(5))),...
        (x(1)*exp(-(x(2)*(367.15-x(4)))./(x(3)+367.15-x(4))))...
        ./(1.+(xdata(:,1).*(x(1)*exp(-(x(2)*(367.15-x(4)))./(x(3)+367.15-x(4))))./x(6)).^(1-x(5))),...
        (x(1)*exp(-(x(2)*(377.15-x(4)))./(x(3)+377.15-x(4))))...
        ./(1.+(xdata(:,1).*(x(1)*exp(-(x(2)*(377.15-x(4)))./(x(3)+377.15-x(4))))./x(6)).^(1-x(5)))];
%%
% Fit the model using the starting point 
% x0 = [1e5;10;10;340;0.2;500];
x0 = [1e4;2;2;300;0.3;6];
options = optimoptions('lsqcurvefit','Algorithm','levenberg-marquardt','Display','iter','TolX', 1e-6, 'TolFun', 1e-6, 'MaxFunEvals', 40000, 'MaxIter', 400);
lb = [1 1 1 1 0 1];
ub = [1e17 100 100 1000 1 1e9];
x =  lsqcurvefit(fun,x0,[xdata(:,1) xdata(:,1) xdata(:,1)],[ydata(:,1) ydata(:,2) ydata(:,3)],lb,ub,options)
%%
% Plot the data and the fitted curve.
loglog(xdata(:,1),fun(x,xdata(:,1)))
 hold on
loglog(xdata(:,1),ydata(:,1),'ko')
hold on
loglog(xdata(:,1),ydata(:,2),'ro')
hold on
loglog(xdata(:,1),ydata(:,3),'go')
legend('Fitted1','Fitted2','Fitted3','Data1','Data2','Data3')
title('Data and Fitted Curve')

Answer 1

Hi, Kaouthar, the code changed to fellow, it ensures all parameters are postive, and the label of x_axis is your true xdata. However one thing should be noticed: the parameters are not unique for x1, x2 x3 and x4.

VarConstant c=[357.15,367.15,377.15];
SharedModel;
Parameter x(6)>0;
XAxis = xdata;
Variable xdata,y;
Function  y=(x1*exp(-(x2*(c-x4))/(x3+c-x4)))/(1+(xdata*(x1*exp(-(x2*(c-x4))/(x3+c-x4)))/x6)^(1-x5));
Data;
0.5,0.659,0.869,1.15,1.51,1.99,2.63,3.47,4.57,6.03,7.95,10.5,13.8,18.2,24.1,31.7,41.8,55.2,72.7,95.9,127,167,220,290,383,505,665,878,1160,1530,2010,2650,3500;
24.434,24.336,22.598,19.534,16.388,13.596,11.234,9.282,7.692,6.514,5.649,4.839,4.161,3.602,3.159,2.792,2.499,2.262,2.070,1.916,1.792,1.687,1.601,1.537,1.501,1.473,1.447,1.418,1.387,1.345,1.297,1.228,1.072;
Data;
0.5,0.659,0.869,1.15,1.51,1.99,2.63,3.47,4.57,6.03,7.95,10.5,13.8,18.2,24.1,31.7,41.8,55.2,72.7,95.9,127,167,220,290,383,505,665;
20.583,19.918,18.385,15.886,13.327,11.047,9.116,7.537,6.299,5.326,4.484,3.814,3.249,2.796,2.438,2.140,1.905,1.721,1.567,1.442,1.341,1.259,1.194,1.145,1.115,1.096,1.078;
Data;
0.5,0.659,0.869,1.15,1.51,1.99,2.63,3.47,4.57,6.03,7.95,10.5,13.8,18.2,24.1,31.7,41.8,55.2,72.7,95.9,127,167,220,290,383,505,665;
17.141,16.598,15.950,14.201,12.054,10.024,8.269,6.820,5.642,4.709,3.946,3.326,2.826,2.417,2.088,1.823,1.609,1.438,1.297,1.185,1.094,1.021,0.962,0.919,0.889,0.868,0.851;

Result 1:

Root of Mean Square Error (RMSE): 0.750624723639767

Sum of Squared Residual: 49.0190603893171

Correlation Coef. (R): 0.994414241612346

R-Square: 0.988859683921457

Adjusted R-Square: 0.988000769669663

Determination Coef. (DC): 0.986551039923055

F-Statistic: 342.846071379397

Parameter Best Estimate

-------------------- -------------

x1 50.7166624975047

x2 1.74233039846288

x3 61.0256533126286

x4 370.012581419946

x5 0.34018118177961

x6 14.1379270601172

Result 2:

Root of Mean Square Error (RMSE): 0.750624723639768

Sum of Squared Residual: 49.0190603893173

Correlation Coef. (R): 0.994414241654891

R-Square: 0.988859684006072

Adjusted R-Square: 0.988000769760861

Determination Coef. (DC): 0.986551039936596

F-Statistic: 342.846071879207

Parameter Best Estimate

-------------------- -------------

x1 67.2294239650589

x2 2.02418733773269

x3 52.5281917961558

x4 361.515106529892

x5 0.340181179660682

x6 14.1379272335838

Result 3:

Root of Mean Square Error (RMSE): 0.750624723639767

Sum of Squared Residual: 49.019060389317

Correlation Coef. (R): 0.994414241636979

R-Square: 0.988859683970447

Adjusted R-Square: 0.988000769722583

Determination Coef. (DC): 0.986551039933522

F-Statistic: 342.846071436732

Parameter Best Estimate

-------------------- -------------

x1 743.828260682761

x2 4.42788617741519

x3 24.0130073020205

x4 332.999932457755

x5 0.34018118195002

x6 14.1379269290681