Fit of multiple data sets with error

Question

Daniele Sonaglioni 2021-3-31

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/788799-fit-of-multiple-data-sets-with-error

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')