Multiple curve fitting for parameter identification: multi-objective optimization
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I have 4 plots of x (eps) and y (sigmaexp) and I would like to best-fit them in the equation given by sigmanum2. The criteria to best-fit is to minimize sum of squares i.e. least sum of squares defined by function fitness. I have done this successfully for 1 plot by using fminsearch i.e. 1 objective function. Since I have 4 plots, I have 4 objective functions (or 4 values in y i.e. [F1; F2; F3; F4]) which must be minimized simultaneously. How can I do this or which function must be used for multiobjective optimization? Please help. I have done research from my end in the relevant forums, however I am not able to do it. Here is my code.
s = table2array(ExpDataMatlab); %excel data I have imported. Attached for your reference.
eps = s(:,1); % represents x
sigmaexp = s(:,2); % represent y
epsdot1 = s(:,3);
A = 332.8668; %constant
B = 128.6184; %constant
n = 0.4234; %constant
%x = [2000, 10.2234];
%y = fitness(x,A,B,n,eps,epsdot1,sigmaexp); %I used this to check if my function defined at the
%bottom works properly or not
ObjFcn = @(x)fitness(x,A,B,n,eps,epsdot1,sigmaexp);
x0 = [12000 4];
options = optimset('Display','iter','PlotFcns',@optimplotfval,'TolX',1e-10,'TolFun',1e-10,'Algorithm','interior-point');
[sol, fval, exitflag, output] = fminsearch(ObjFcn,x0,options);
C = sol(1); P = sol(2);
sigmanum4 = (A+B*(eps.^n)).*(1+(epsdot1./C).^(1/P)); %deduced value after optimization
plot(s(1:99,1),s(1:99,2),'+m');hold on;
plot(s(1:99,1),sigmanum4(1:99,1),'m');
hold off;
function y = fitness(x,A,B,n,eps,epsdot1,sigmaexp)
sigmanum2 = (A+B*(eps.^n)).*(1+(epsdot1./x(1)).^(1/x(2)));
F1 = sum((sigmaexp(1:99,1)-sigmanum2(1:99,1)).^2);%values for 982 /s % represent residual sum of squares
%F2 = sum((sigmaexp(100:206,1)-sigmanum2(100:206,1)).^2);%values for 2000 /s
%F3 = sum((sigmaexp(207:296,1)-sigmanum2(207:296,1)).^2);%values for 2900 /s
%F4 = sum((sigmaexp(297:368,1)-sigmanum2(297:368,1)).^2);%values for 4000 /s
%y = [F1; F2; F3; F4]; % for multiobjective minimization
y = [F1]; %for single objective minimization
%y = (0.01042*F1+0.00962*F2+0.01111*F3+0.01389*F4); %weighted residual sum of squares (Milani 2008)
end
Thank you.
采纳的回答
Star Strider
2021-4-3
I am not certain what you want to do.
To fit different parameter sets to different parts of the data, this works:
T1 = readtable('ExpDataMatlab.xlsx', 'VariableNamingRule','preserve');
eps = T1{:,1}; % represents x
sigmaexp = T1{:,2}; % represent y
epsdot1 = T1{:,3};
A = 332.8668; %constant
B = 128.6184; %constant
n = 0.4234; %constant
[Ueps,idx] = unique(epsdot1); % Find ‘Breaks’ In Data
didx = diff([idx; numel(eps)+1]); % Find Lengths Of Data Segments
idxmtx = [idx idx+didx-1]; % Matrix Of Index Segments
objfcn = @(x,idx) (A+B*(eps(idx).^n)).*(1+(epsdot1(idx)./x(1)).^(1/x(2))); % Objective Function
fitness = @(x,idx) norm(sigmaexp(idx) - objfcn(x,idx)); % Fitness Function
x0 = [12000 4];
options = optimset('Display','iter','PlotFcns',@optimplotfval,'TolX',1e-10,'TolFun',1e-10,'Algorithm','interior-point');
for k = 1:size(idxmtx,1) % Data Segment Parameter Estimation Loop
idxrng = idxmtx(k,1):idxmtx(k,2);
[sol(k,:), fval(k), exitflag(k), output{k}] = fminsearch(@(x)fitness(x,idxrng),x0,options);
end
figure
for k = 1:size(idxmtx,1) % Plotting Loop
subplot(4,1,k)
idxrng = idxmtx(k,1):idxmtx(k,2);
plot(eps(idxrng), [sigmaexp(idxrng) epsdot1(idxrng)])
hold on
plot(eps(idxrng), objfcn(sol(k,:),idxrng), '--g')
hold off
grid
ylim([0 5000])
end
It is straightforward to fit the entire data set to one set of parameters, if that is what you want to do.
.
7 个评论
Venkatesh Madhukar Deshpande
2021-4-4
Venkatesh Madhukar Deshpande
2021-4-4
编辑：Venkatesh Madhukar Deshpande
2021-4-4
To obtain single set of values for x, I removed these lines
[Ueps,idx] = unique(epsdot1);
didx = diff([idx; numel(eps)+1]);
idxmtx = [idx idx+didx-1];
and edited some code as shown below
objfcn = @(x) (A+B*(eps.^n)).*(1+(epsdot1./x(1)).^(1/x(2)));
fitness = @(x) norm(sigmaexp - objfcn(x));
Using this, I think it is optimizing for all 4 curves and giving me a single set of x. Is this correct?
The fitted curves and experimental curves are shown below. Code used is
C = sol(1); P = sol(2);
sigmanum4 = (A+B*(eps.^n)).*(1+(epsdot1./C).^(1/P));
plot(s(1:99,1),s(1:99,2),'+m');hold on;
plot(s(1:99,1),sigmanum4(1:99,1),'m');
plot(s(100:206,1),s(100:206,2),'.b');
plot(s(100:206,1),sigmanum4(100:206,1),'b');
plot(s(207:296,1),s(207:296,2),'dk');
plot(s(207:296,1),sigmanum4(207:296,1),'k');
plot(s(297:368,1),s(297:368,2),'og');
plot(s(297:368,1),sigmanum4(297:368,1),'g');
legend('Exp (982 /s)','Curve-fit (982 /s)','Exp (2000 /s)','Curve-fit (2000 /s)','Exp (2900 /s)','Curve-fit(2900 /s)','Exp (4000 /s)','Curve-fit (4000 /s)','Location','northwest');
hold off;
1. Is there a way to improve this fit like by using lsqcurvefit or some other function? For some reason, the fitted curves are not able to catch up with slopes of the experimental curve 2. If I want to provide lower and upper bound for searching then how can I do it?
Thank you
Star Strider
2021-4-4
编辑：Star Strider
2021-4-5
My pleasure!
As I mentioned, I was not certain what result you wanted.
Try this:
T1 = readtable('ExpDataMatlab.xlsx', 'VariableNamingRule','preserve');
eps = T1{:,1}; % represents x
sigmaexp = T1{:,2}; % represent y
epsdot1 = T1{:,3};
A = 332.8668; %constant
B = 128.6184; %constant
n = 0.4234; %constant
[Ueps,idx] = unique(epsdot1); % Find ‘Breaks’ In Data
didx = diff([idx; numel(eps)+1]); % Find Lengths Of Data Segments
idxmtx = [idx idx+didx-1]; % Matrix Of Index Segments
objfcn1 = @(x,idx) (A+B*(eps(idx).^n)).*(1+(epsdot1(idx)./x(1)).^(1/x(2))); % Objective Function
objfcn2 = @(x) (A+B*(eps.^n)).*(1+(epsdot1./x(1)).^(1/x(2))); % Objective Function
fitness = @(x) norm(sigmaexp - objfcn2(x)); % Fitness Function
x0 = [12000 4];
options = optimset('Display','iter','PlotFcns',@optimplotfval,'TolX',1e-10,'TolFun',1e-10,'Algorithm','interior-point');
[sol, fval, exitflag, output] = fminsearch(@(x)fitness(x),x0,options);
figure
hold on
for k = 1:size(idxmtx,1) % Plotting Loop
idxrng = idxmtx(k,1):idxmtx(k,2);
hp = plot(eps(idxrng), sigmaexp(idxrng),'.');
plot(eps(idxrng), objfcn1(sol,idxrng), 'Color',hp.Color, 'LineWidth',1.5)
grid
ylim([350 650])
end
hold off
grid
xlabel('Iteration')
yllabel('Function Value')
legend('Exp (982 /s)','Curve-fit (982 /s)','Exp (2000 /s)','Curve-fit (2000 /s)','Exp (2900 /s)','Curve-fit(2900 /s)','Exp (4000 /s)','Curve-fit (4000 /s)','Location','northwest')
producing:
Here ‘objfcn1’ fits each section of the data, and ‘objfcn2’ estimates the parameters for the entire data set. This is a bit more compact.
.
Venkatesh Madhukar Deshpande
2021-4-5
Star Strider
2021-4-5
As always, my pleasure!
Hi there. I am jumping in on your conversation since I am looking forward to solve a problem too. Just like @Venkatesh Madhukar Deshpande I have to find the model parameters of a theoretical model using experimentally obtained data. As I can see in the figure posted, the fit obtained is good enough which I am also getting in my case however I am worried about matching the first and the last points in the data. Is there a way in which I can bound the algorithm to start from eactly first point and end at the last point so that the error between the predicted and the experimental values is minimum?
Fixing a point in a fit is to the cost of one fit parameter. E.g. if you want to fit to a function y = a + bx and you want to fix y(0) = 0 and y(1) = 1, it follows that a = 0 and b = 1 - thus from the 2 fit parameters a and b, 2 - 2 = 0 are left after the boundary values of y were fixed.
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