Multi variable parameter estimation from data set

Hello,

I have a set of data from numerical simulation (36 simulations), I want to obtained a generalize empirical relation from the data.

Normally I can use the curve fitting toolbox to find the relation using a specific model equation for one variable, but

My data set is multi variable function that depends on three variables,

The model equation is of the from:

I = a/(b +x);

with a,b =f (Ct,Ia,and R)

0.64 < Ct < 0.98,

0.05 < Ia < 0.2

0.2 < R < 0.6

The empirical relation I want to finnaly obtained can be expressed as

where A, alpha, beta and zeta are constant

[Torsten]

And what are the fitting parameters ? A, alpha1, alpha2, beta1, beta2, zeta1, zeta2 and x ?

[Kabir Shariff]

A, alpha1, alpha2, beta1, beta2, zeta1, zeta2 are constant values that I want to obtained after the solution,

x is the independent variable

[Torsten]

So again please tell us what the parameters are you want to obtain from the fitting, for which parameters you have fixed values and for which parameters you have the 36 data points.

[Kabir Shariff]

I would like to obtain all the constant in the expression,

The function I depends on Ct, Ia and R. the dependence is usually a power relation(in form of alpha,beta,zeta) which should be a constant value. The vatiables Ct, Ia and R are inequatity contrained in the function.

Therefore, I would like to obtained the constant (alpha, beta,zeta) for the contrained values.

Thank you

[Torsten]

And you have 36 data for I and x ? And all other parameters are unknown ?

[Kabir Shariff]

No sir, I have a set of 36 numerical result

here is the data for clarification.

for K = [1 2 3] % This call file with different Ct values
    
    if K == 1
        Ct = 0.64;  % the actual Ct used in the simulation
    elseif K == 2
        Ct = 0.89;
    else 
        Ct = 0.98;
    end
        
    
    for DH = [20 40 60] % DH is R as mention above; It calls a file with selected Ct
        
        for p = [5 10 15 20] % p is Ia, sorry for the different names!!
            
            
            filename = sprintf('IaddK%dDH%dp%d.csv',K,DH,p); % file name in the form (IaddK2DH40p10.csv)
            
            file = importdata(filename);
            
            tag = file.data;
            
            x = tag(:,1);   Iobs = tag(:,2)*0.01;
            
            Ia = 0.01*p; R = 0.01*DH; % convering percentage 
            
            figure
            plot(x,Iobs)
            
        end
    end
end

For each of the 'filename', is a column for x and I(I_obs is the numerical data)

I want to develop a generalize model for all the sets of data, respecting the constraints.

I have tried to use Nonlinear Least-Squares, Problem-Based

https://fr.mathworks.com/help/optim/ug/nonlinear-least-squares-problem-based-basics.html

but it gives the parameters for each data, I want to generalize it please,

Thank you very much

Answer 1

[Torsten]

Ok.

Form one big matrix M with the data of all your Excel sheets.

First  column: x
Second column: I
Third  column: Ct corresponding to x and I
Fourth column: Ia corresponding to x and I
Fifth  column: R corresponding to x and I

Then you can use the following code:

M = your big matrix
% Initial values for the parameters 
% You might choose better guesses if you know better
A_0 = 1.0; 
alpha1_0 = 0.81;
alpha2_0 = 0.81;
beta1_0 = 0.125;
beta2_0 = 0.125;
zeta1_0 = 0.4;
zeta2_0 = 0.4;
x0 = [A_0;alpha1_0;alpha2_0;beta1_0;beta2_0;zeta1_0;zeta2_0];
lb = [-Inf,0.64;0.64;0.05;0.05;0.2;0.2];
ub = [Inf;0.98;0.98;0.2;0.2;0.6;0.6];
fun = @(p) M(:,2) - (p(1)*M(:,3).^p(2).*M(:,4).^p(4).*M(:,5).^p(6))./...
                    (p(1)*M(:,3).^p(3).*M(:,4).^p(5).*M(:,5).^p(7) - M(:,1));
x = lsqnonlin(fun,x0,lb,ub];
A = x(1)
alpha1 = x(2)
alpha2 = x(3)
beta1 = x(4)
beta2 = x(5)
zeta1 = x(6)
zeta2 = x(7)

[Kabir Shariff]

Hello, I was able to get a model from your explanation. Although the model gives almost the same parameters as defined in the 'ub'. And the accuracy of the model is not encouraging. Maybe I should use another model equation to have better results.

Thanks