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Fit of multiple data sets with error

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Daniele Sonaglioni
Daniele Sonaglioni on 31 Mar 2021
Hi everybody,
i want to make a fit in Matlab of my experimental data with a known function. I have a code that is able to perform the fit of the data but it does not account for the error on my y-data. This fit is a bit particular because i want to fit three data set with the same function and thus the output fitted parameters are unique for the three cases. Each y-data set corresponds to a different temperature and my fitting function takes into account this fact.
My fitting function is this one:
where . The fitting parameters are .
Now my problem is: how can i make a fit of my data considering the experiemental error too?
Below i show the code with the data and the error associated. If possible, i want that the output parameters have their error.
Kb=8.26e-23;
m=17246;
yfcn = @(b,x) exp(-(x(:,2).*3.*Kb.*x(:,1)/m).^2) .* (1-2*b(1).*b(2).*(1 - sin(x(:,2).*b(3))./(x(:,2).*b(3))));
x=[0.5215 0.7756 1.2679 1.4701 1.6702 1.8680 2.0633 2.2693 2.4584 2.6442 2.8264 3.0046 3.0890 3.2611 3.4287 3.5917 3.7497 3.9309 4.0774 4.2183 4.3535 4.4827 4.5427 4.6628];
y1=[1.0290 1.0025 1.0158 1.0068 0.9705 0.9646 0.9596 0.9499 0.9811 0.9669 0.9519 0.9573 0.8989 0.9315 0.9618 0.9260 1.0481 0.9245 0.9733 0.8830 1.0203 0.9851 0.9314 0.9204];
y1_err=[0.0637 0.0520 0.0322 0.0239 0.0291 0.0232 0.0405 0.0228 0.0347 0.0364 0.0424 0.0298 0.0531 0.0252 0.0313 0.0230 0.0428 0.0255 0.0375 0.0278 0.0602 0.0467 0.0736 0.0836];
y2=[0.9773 1.0156 0.9362 1.0103 0.9414 0.9167 0.9257 0.9225 0.9127 0.9230 0.9380 0.8899 0.8047 0.9339 0.9012 0.9079 0.9497 0.8449 0.8837 0.8250 0.8738 0.8602 0.9106 0.8656];
y2_err=[0.0616 0.0523 0.0303 0.0239 0.0284 0.0223 0.0394 0.0223 0.0329 0.0351 0.0418 0.0283 0.0493 0.0252 0.0298 0.0226 0.0398 0.0239 0.0350 0.0265 0.0541 0.0425 0.0722 0.0801];
y3=[1.0433 0.9711 0.9913 0.9902 0.9427 0.9146 0.8849 0.9010 0.8876 0.9175 0.9329 0.8639 0.6970 0.8260 0.8675 0.8568 0.8748 0.8156 0.8041 0.7679 0.8333 0.8443 0.8336 0.8344];
y3_err=[0.0638 0.0504 0.0311 0.0232 0.0280 0.0219 0.0375 0.0215 0.0317 0.0343 0.0410 0.0272 0.0445 0.0227 0.0283 0.0213 0.0369 0.0228 0.0323 0.0248 0.0512 0.0405 0.0664 0.0766];
T1=120; %temperature reffered to y1
T2=140; %temperature reffered to y2
T3=160; %temperature reffered to y3
T1v = T1*ones(size(x));
T2v = T2*ones(size(x));
T3v = T3*ones(size(x));
xm = x(:)*ones(1,3);
ym = [y1(:) y2(:), y3(:)];
Tm = [T1v(:) T2v(:) T3v(:)];
xv = xm(:);
yv = ym(:);
Tv = Tm(:);
xTm = [Tv xv];
B0 = rand(3,1); % Use Appropriate Initial Estimates
B = fminsearch(@(b) norm(yv - yfcn(b,xTm)), B0); % Estimate Parameters
figure
for k = 1:3
idx = (1:numel(x))+numel(x)*(k-1);
subplot(3,1,k)
plot(x, ym(:,k), '.')
hold on
plot(x, yfcn(B,xTm(idx,:)), '-r')
hold off
grid
ylabel('Substance [Units]')
title(sprintf('y_{%d}, T = %d', k,xTm(idx(1),1)))
ylim([min(yv) max(yv)])
end
xlabel('x')
sgtitle(sprintf('$y=e^{-(\\frac{3xK_BT}{m})^2} (1-2%.3f\\ %.3f (1-\\frac{sin(x\\ %.3f)}{x\\ %.3f}))$',B,B(3)), 'Interpreter','latex')

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