fmincon returning results with large errors

I am using fmincon to do some fitting. The fitted parameters are A and B. I do 500 simulations for varying values of A (B is always 1). I do the fittings 500 times. In most cases, fitted A and B are accurate. In some cases where Y (and A) is very small (~0), B is ~2.5 times bigger.

How can I improve my fits? Why is B in the plot above doing these weird spikes? Example code where fitted B is not accurate.

global X Y deltaX
X = linspace(0,25,1000);
Y = [0,6.88363868053220e-06,2.61850086044514e-05,7.31760916473311e-05,0.000169703525107875,0.000335095365782447,0.000570492105509620,0.000848334324266810,0.00112111070650578,0.00134616301294618,0.00150637795687461,0.00161103712983892,0.00168157211368303,0.00173645598272875,0.00178586388577830,0.00183385079765603,0.00188133039133770,0.00192820866515512,0.00197531256730628,0.00202495694380148,0.00207931456044459,0.00213768017818540,0.00219701992890083,0.00225523654851226,0.00231059493361869,0.00236274290280846,0.00241225288555139,0.00245952660700232,0.00250526005739466,0.00254982927731439,0.00259379301196966,0.00263704176625230,0.00268006945437819,0.00272282025528020,0.00276533860178240,0.00280774203039247,0.00284990694888583,0.00289193218896916,0.00293375224159887,0.00297539229362766,0.00301687646156951,0.00305819262887141,0.00309933318385231,0.00314029812651220,0.00318108745685109,0.00322170117486898,0.00326213928056586,0.00330240177394174,0.00334248865499662,0.00338239992373049,0.00342213558014336,0.00346169562423523,0.00350108005600609,0.00354028887545594,0.00357932208258480,0.00361825212432555,0.00365701700530112,0.00369561562408123,0.00373404798066589,0.00377231407505510,0.00381041390724885,0.00384834747724715,0.00388611478505000,0.00392371583065739,0.00396115061406933,0.00399841913528582,0.00403552139430685,0.00407245739113243,0.00410922712576256,0.00414584722157688,0.00418236613869793,0.00421872887837678,0.00425493544061343,0.00429098582540786,0.00432688003276010,0.00436261806267012,0.00439819991513794,0.00443362559016356,0.00446889508774697,0.00450400840788817,0.00453896555058717,0.00457376651584397,0.00460841130365855,0.00464289991403093,0.00467726557986839,0.00471152168503671,0.00474563118934736,0.00477959409280035,0.00481341039539566,0.00484708009713329,0.00488060319801326,0.00491397969803555,0.00494720959720017,0.00498029289550712,0.00501322959295640,0.00504601968954801,0.00507866318528194,0.00511116008015821,0.00514351037417680,0.00517574009128327,0.00520784217947271,0.00523980694091487,0.00527163497746057,0.00530332689096067,0.00533488328326600,0.00536630475622739,0.00539759191169569,0.00542874535152174,0.00545976567755637,0.00549065349165042,0.00552140939565473,0.00555203399142014,0.00558252788079749,0.00561289166563762,0.00564312594779136,0.00567323132910956,0.00570320841144305,0.00573305779664267,0.00576278008655926,0.00579237588304366,0.00582184578794671,0.00585119040311925,0.00588041033041211,0.00590950617167613,0.00593847852876216,0.00596732800352102,0.00599605519780357,0.00602465518980816,0.00605312684019562,0.00608147672119711,0.00610970539106717,0.00613781340806028,0.00616580133043096,0.00619366971643374,0.00622141912432310,0.00624905011235357,0.00627656323877966,0.00630395906185587,0.00633123813983672,0.00635840103097672,0.00638544829353037,0.00641238048575220,0.00643919816589670,0.00646590189221839,0.00649249222297179,0.00651896971641139,0.00654533493079172,0.00657158842436727,0.00659773075539258,0.00662376248212213,0.00664968416281045,0.00667549635571204,0.00670119961908142,0.00672679451117310,0.00675228159024158,0.00677765748679504,0.00680291309244907,0.00682806075934081,0.00685310098461930,0.00687803426543361,0.00690286109893277,0.00692758198226583,0.00695219741258184,0.00697670788702984,0.00700111390275888,0.00702541595691801,0.00704961454665628,0.00707371016912273,0.00709770332146641,0.00712159450083636,0.00714538420438164,0.00716907292925129,0.00719266117259436,0.00721614943155989,0.00723953820329693,0.00726282798495453,0.00728601927368173,0.00730911256662759,0.00733210836094114,0.00735500715377145,0.00737780944226754,0.00740051572357848,0.00742312649485330,0.00744564204747113,0.00746804590573406,0.00749035393145055,0.00751256655973064,0.00753468422568437,0.00755670736442181,0.00757863641105298,0.00760047180068795,0.00762221396843676,0.00764386334940945,0.00766542037871608,0.00768688549146670,0.00770825912277135,0.00772954170774008,0.00775073368148294,0.00777183547910998,0.00779284753573124,0.00781377028645677,0.00783460416639663,0.00785534961066086,0.00787600705435950,0.00789657693260261,0.00791705968050023,0.00793745573316242,0.00795776552569921,0.00797798949322067,0.00799812807083683,0.00801818169365774,0.00803815079679346,0.00805802434759900,0.00807780975707944,0.00809751037005942,0.00811712656893272,0.00813665873609315,0.00815610725393450,0.00817547250485058,0.00819475487123519,0.00821395473548212,0.00823307247998517,0.00825210848713814,0.00827106313933482,0.00828993681896903,0.00830872990843455,0.00832744279012519,0.00834607584643474,0.00836462945975700,0.00838310401248577,0.00840149988701485,0.00841981746573804,0.00843805713104914,0.00845621926534194,0.00847430425101024,0.00849231247044785,0.00851024430604856,0.00852810014020616,0.00854588035531447,0.00856358533376727,0.00858120595119917,0.00859874351318242,0.00861620554506136,0.00863359239094699,0.00865090439495034,0.00866814190118241,0.00868530525375423,0.00870239479677679,0.00871941087436111,0.00873635383061821,0.00875322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deltaX = X(2);
A = 0.00111978493959457;
B = 1;

opts = optimset('TolX',1e-4,'TolFun',1e-4,'Diagnostics','on','MaxFunEvals',1000,'MaxIter',800,'Display','final','UseParallel','never');
low_bound = [0 0.0001];
up_bound = [5 5];

%% I do the fitting 3 times with varying starting paramaters A and B. Then I take the results with the smallest fitting error.
[fitted_params1,error1,exitflag1,output1] = fmincon(@fitting_function,[0 0.5],[],[],[],[],low_bound,up_bound,[],opts);
fitted_params1 % returns [0.0011 2.4454]
error1 % returns 1.1038e-06

% I GIVE THIS SOLVER THE ORIGINAL VALUES OF A AND B!!!
[fitted_params2,error2,exitflag2,output2] = fmincon(@fitting_function,[A B],[],[],[],[],low_bound,up_bound,[],opts); 
fitted_params2 % returns [0.0011 2.2865]
error2 % returns 9.9851e-07

[fitted_params3,error3,exitflag3,output3] = fmincon(@fitting_function,[0.5 1],[],[],[],[],low_bound,up_bound,[],opts);
fitted_params3  % returns [0.0011 2.4455]
error3 % returns 1.1039e-06

%% ...
% my code takes the the fitted_params of the fitting with the smallest
% error (in this case fitted_params2)

function Error = fitting_function(initial)
    A_fit=initial(1);
    B_fit=initial(2);
    global X Y deltaX

    Y_fit = A_fit.*conv(helper(X)./0.6,exp(-A_fit./B_fit.*X)).*deltaX;

    %Objective function
    Error=sum((Y_fit(2:length(X))-Y(2:end)).^2);
end

%Helper function
function y = helper(x)
    y = 0.809/(0.0563*sqrt(2*pi))*exp(-(x-0.17046).^2/(2*0.0563^2)) ...
       + 0.330/(0.132*sqrt(2*pi))*exp(-(x-0.365).^2/(2*0.132^2))  ...
       + 1.050*exp(-0.1685*x)./(1+exp(-38.078*(x-0.483)));
end

Results of Code:

A = 0.0011 AND A_fitted = 0.0011

B = 1 AND B_fitted = 2.2865 <= error is big in this case.

Answer 1

[Matt J]

Notice that your actual fit is fine, despite the discrepancy in B. This occurs because as Y and A become small, the influence of B on the shape of the curve diminishes and B_fitted = 2.2865 gives approximately the same curve as B=1. You have more than one solution, in other words.

[Tara]

Thanks Matt for your answer.

I realise I might have more than 1 solution for B. But in the second fitting i do, I start with the values of A and B that I used to generate the Y curves. Still, the fitting converges to another solution. Can I change the options for fmincon so that it converges to the original values of A and B?

If you look at the plot for B, most of my 500 fitting trials give B=1 except for that small chunck. I wonder why the fitting fails only for that small section.

[Matt J]

That "small chunk" is where the value of B has diminished influence over the shape of the curve (because A is small and everything is close to zero). When the curve is insensitive to the value of the parameter there is nothing you can do to ensure a certain result for that parameter.

Answer 2

[Matt J]

One thing I think would help is to pre-normalize Y prior to the fit so that it is never uniformly close to zero, e.g.,

Y=Y./sum(Y); %normalize

Similarly, reparametrize your model as

Y_fit = C_fit.*conv(helper(X)./0.6,exp(-D_fit.*X)).*deltaX;

After fitting the normalized Y, retrieve the original parameters from

A_fit=C_fit*sum(Y);
B_fit=A_fit/D_fit;

[Matt J]

I also recommend that you normalize your objective

    %Objective function
    Error=sum((Y_fit(2:length(X))-Y(2:end)).^2)/(numel(Y)-1);

You might also consider switching from fmincon to lsqcurvefit, since it appears that you don't have very complicated constraints.