Minimize the sum of squared errors between the experimental and predicted data in order to calculate two parameters

In my research work, I use a model and I want to minimize the sum of squared errors between the experimental and predicted data in order to calculate two parameters.

The experimental data are:

u exp: [0.709; 0.773 ;0.823 ;0.849 ;0.884 ;0.927 ;0.981 ;1.026 ;1.054 ;1.053 ;1.048;1.039] ;

observed at z=[ 0.006;0.012;0.018;0.024;0.03;0.046;0.069;0.091;0.122;0.137;0.152;0.162];

The equation of the model that I use is:

u model=0.1073*((log(0.13/z)-1/3*(1-(z/0.13)^3)+2*a*(1+(b)^0.5)*cos(11.89*z)); and I want to calculate the parameters “a” et “b” by minimizing the sum of squared errors between “u exp” and “u model”.

Someone here can help me please?

Thank you already for your help!

Answer 1

[Davide Masiello]

You can use MatLab's fmincon.

z     = [0.006;0.012;0.018;0.024;0.03;0.046;0.069;0.091;0.122;0.137;0.152;0.162];
u_exp = [0.709;0.773;0.823;0.849;0.884;0.927;0.981;1.026;1.054;1.053;1.048;1.039];
u_mod = @(P) 0.1073*(log(0.13./z)-1/3*(1-(z/0.13).^3)+2*P(1).*(1+P(2).^0.5).*cos(11.89*z));
sum_sq_err = @(P) sum((u_exp-u_mod(P)).^2);

P = fmincon(sum_sq_err,[0.1,0.1]);

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

a = P(1)

a = 2.0158

b = P(2)

b = 0.3185

hold on
plot(z,u_exp,'o')
plot(z,u_mod(P))
hold off
grid on

[Mathieu NOE]

If you don't have the optimisation toolbox, you can use fminsearch

z     = [0.006;0.012;0.018;0.024;0.03;0.046;0.069;0.091;0.122;0.137;0.152;0.162];
u_exp = [0.709;0.773;0.823;0.849;0.884;0.927;0.981;1.026;1.054;1.053;1.048;1.039];
u_mod = @(P) 0.1073*(log(0.13./z)-1/3*(1-(z/0.13).^3)+2*P(1).*(1+P(2).^0.5).*cos(11.89*z));
sum_sq_err = @(P) sum((u_exp-u_mod(P)).^2);

P = fminsearch(sum_sq_err,[0.1,0.1])

P = 1×2

1.8969 0.4387

hold on
plot(z,u_exp,'o')
plot(z,u_mod(P))
hold off
grid on

[ORESTE SAINT-JEAN]

Thanks a lot for your help Davide and Mathieu !

I haven't really had optimization toolboxes, but after installing it, the algorithm works fine. Now, I can adapt easily the equations of others functions that I use to continue my work.

Thank you !

[ORESTE SAINT-JEAN]

z = [0.006;0.012;0.018;0.024;0.03;0.046;0.069;0.091;0.122;0.137;0.152;0.162];

u_exp = [0.345;0.281;0.231;0.205;0.17;0.127;0.073;0.028;0.00;0.0010;0.0060;0.015];

u_mod = @(P) 0.1073*((log(0.132./z)-(1/3)*(1-(z/0.132).^3)+2*P(2).*(1+(P(1)).^0.5).*(cos(pi*z/0.264)).^2));

sum_sq_err = @(P) sum((u_exp-u_mod(P)).^2);

P = fminsearch(sum_sq_err,[0.01,0.01])

hold on

plot(z,u_exp,'o')

plot(z,u_mod(P))

hold off

grid on

P =

0.0506 0.2198

With the god data, everything it's ok.

THANK YOU FOR YOUR HELPS!!

[Mathieu NOE]

I feel better now :)