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

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ORESTE SAINT-JEAN le 28 Mar 2023
Commenté : Mathieu NOE le 29 Mar 2023
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?
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### Réponse acceptée

Davide Masiello le 29 Mar 2023
Modifié(e) : Torsten le 29 Mar 2023
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
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ORESTE SAINT-JEAN le 29 Mar 2023
Déplacé(e) : Star Strider le 29 Mar 2023
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.
Mathieu NOE le 29 Mar 2023
I feel better now :)

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