Multi variable parameter estimation from data set

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Kabir Shariff
Kabir Shariff 2022-3-29
评论： Kabir Shariff 2022-3-30
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
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Torsten
Torsten 2022-3-29
编辑：Torsten 2022-3-29
And you have 36 data for I and x ? And all other parameters are unknown ?
Kabir Shariff
Kabir Shariff 2022-3-29
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

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Torsten
Torsten 2022-3-29
编辑：Torsten 2022-3-29
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)
  1 个评论
Kabir Shariff
Kabir Shariff 2022-3-30
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

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