How to improve the fitting result

Hellow, there!

I am doing a fitting using quite involved custom equation. The definition of the function is as below:

function bif = bifit(A,bix,a,x,e,k,T)
    fun = @(bix,b,a,x,e,k,T) exp(-b.^2./(2.*a.^2)).*heaviside(bix-x-b-e).*exp(-(bix-x-b-e)./(k.*T));
    bif = integral(@(b) fun(bix,b,a,x,e,k,T),-1,1,'ArrayValued',1).*A;
end

Also, the code for fitting is as below:

ft = fittype('bifit(A,bix,a,x,e,k,T)','problem',{'x','k','T'},'independent','e'...
                                     ,'coefficients',{'A','bix','a'});


fo = fitoptions(ft);
fo.StartPoint = [129400 5.98470 0.01219];
fo.Lower = [120000 5.90000 0.00800];
fo.Upper = [300000 6.10000 0.13000];

bi = fit(data_x,data_y,ft,fo,'problem',{3.0169,8.61773*10^-5,161})
bi_coeff = coeffvalues(bi)
plot(bi,data_x,data_y,'-'),legend('exp','bi','location','Northeast')

'data_x' & 'data_y'(429x1 double respectively) are both column vector type data imported from excel file.

'e'(429x1 double) is a number type array(or matrix) data also imported from excel file.

The problem is that the result of the fitting returns the plot as shown below, which clearly does not fit the data:

The fitting should be as shown below:

The coeffvalues() returns the coefficient values as below, but it should be 129406 5.98480 0.01220 for the fit to look like the plot right above.

bi_coeff =

   1.0e+05 *

    1.2018    0.0001    0.0000

Thus, I was thinking to get a better fitting, I might need to extend the significant digit, or to bounds the x-axis value.

Any advice to improve the result of the fitting? Thanks a lot in advance!

[Wonkyung Choi]

I've discarded the data points that is less relevent to the fitting by hand from excel file.

Its plot looks like above. However, it is still less accurate and also has the problem that I have to discard the data point manually for each set of data. I am looking for more automatic and efficient way to fit. Any help would be deeply appreciated!

Answer 1

[Walter Roberson]

My tests indicate that over that range of constraints, unless data is discarded, then your best fit will always be with A = 120000 (the lower boundary of A).

I am getting comparatively small differences in fit around that boundary though. Basically all the minima I am finding are straight lines with slightly different offsets and angles.

Correction: I got a 0.05% improvement (2 parts in 10000) at 127570, 6.08408, 0.0780185. Was still a straight line.

[Walter Roberson]

format long g
data = readmatrix('t=70.xlsx');
data_x = data(:,1);
data_y = data(:,2);

e = readmatrix('e t=70.xlsx');

x = 3.0169; k = 8.61773*10^-5; T = 161;

[sx, sxidx] = sort(data_x);
sy = data_y(sxidx);

%function bif = bifit(A,bix,a,x,e,k,T)

obj = @(Abixa) bifit(Abixa(1), Abixa(2), Abixa(3), x, sx, k, T);

fo.StartPoint = [129400 5.98470 0.01219];
fo.Lower = [120000 5.90000 0.00800];
fo.Upper = [300000 6.10000 0.13000];

residue = @(Abixa) sum((obj(Abixa) - sy).^2);

options = optimoptions('fmincon');
options.Display = 'iter';

Samp = 100;
StartPoint = [fo.StartPoint; rand(Samp,3).*(fo.Upper - fo.Lower) + fo.Lower];

fprintf('Seeding with %d locations, please wait.\n', size(StartPoint,1));

trial_residues = arrayfun(@(row) residue(StartPoint(row,:)), (1:size(StartPoint,1)).');
[bestresidue, bestidx] = min(trial_residues);
bestStartPoint = StartPoint(bestidx,:);

fprintf('random selection had residue %g at point [%g,%g,%g]\n', bestresidue, bestStartPoint);

fprintf('Starting fmincon from best seed, please wait\n');

[best_Abixa, fval] = fmincon(residue, bestStartPoint, [], [], [], [], fo.Lower, fo.Upper, [], options);

fprintf('fmincon from there had residue %g at point [%g,%g,%g]\n', fval, best_Abixa);

ypred = obj(best_Abixa);
plot(sx, sy, 'k', sx, ypred, 'b');
legend('exp','bi','location','Northeast')

[Wonkyung Choi]

Ohh, I see. Thank you so much for examining the fit bounds.

I should do the fitting by hand in this case, thenXP. Again I really appreciate your help!

Answer 2

[Matt J]

Discard the data corresponding to data_x>2.98. Clearly, you do not want that to participate in the fit.

[Wonkyung Choi]

I know it's a kind of silly quesiton, but what kind of func should I use if I want to discard part of the data? It's because I also want to plot the data which I do not need to fit as shown above. Or are you suggesting that I discard the data in the excel from the first place(and not within the code)? I have lots of data to fit, so if I discard it in the excel from the first place, then it's too cumbersome.(plot the data, read out the line where I should discard the data, and then run the code for every single set of data.)

[Matt J]

Discard the data from the fit only

keep=data_x<2.98;

bi = fit(data_x(keep),data_y(keep),ft,fo,'problem',{3.0169,8.61773*10^-5,161});

plot(bi,data_x,data_y,'-'),legend('exp','bi','location','Northeast')