The issue seems to be that the "best" fit involves a and b set to their lower bounds and NonlinearModelFit just isn't reaching those lower values. But because it then appears obvious that the lower bounds for those two parameters are the values that maximize the likelihood, setting those two parameters to their lower bounds is necessary.

t0 = Rationalize[time[[1]], 0];
q0 = Rationalize[heatflow[[1]], 0];
data = Rationalize[Thread[{time, heatflow}], 0];
model = (q0 - (a + b (t - t0)))*Exp[-(t - t0)/tau] + (a + b*(t - t0)) /. b -> 11 10^-10 /.  a -> 10^-6;
fit = NonlinearModelFit[data, {model, {10 < tau < 1200}}, {{tau, 658}}, t];
fit["BestFitParameters"]
(* {tau -> 660.9437814041655`} *)
mseMathematica = Mean[(data[[All, 2]] - (model /. t -> data[[All, 1]] /.fit["BestFitParameters"]))^2]
(* 3.0509905090105965`*^-10 *)
mseMatlab = Mean[(data[[All, 2]] - (model /. t -> data[[All, 1]] /. {a -> 0.0000010000000222047, 
        b -> -1.09997779553949 10^-9, tau -> 657.977318176476}))^2]
(* 3.0950005923166116`*^-10 *)

One obtains slightly better fits (using the mean square error) with Mathematica than Matlab.

As Neil Signer mentioned if you want better fits you need a different model or loosen up on the parameter restrictions. Also, $R^2$ is not a great measure of goodness of fit especially if the values are near 1 as in this case. Consider using fit["EstimatedVariance"] or the square root of that value instead.

Dear Rohit,

That "step" is on purpose as I picked two separate time segments for fitting the model.

Best regards,

Roland

Dear Neil,

Thank you very much for your response. Sorry about the txt file. I actually only created it to avoid uploading excess data from my actual file and extra code. It is therefore not linked to any of my issues. Interestingly, my code somehow worked with the txt as written at first. Anyway, I changed it to csv by now. Additionally, I realized that in Matlab I used slightly different data points. However, all the problems remain. The adjusted csv file is attached, now with the exact same datapoints as were used in Matlab.

The constraints in the fit are actually needed. I have rough estimates where the 3 parameters should be, and anything outside the constraints does not make sense from a physical point of view, especially if the sign is opposite. Also changing the model is not the way to go, as it was derived from the equations corresponding to the physical processes in the experiment.

The standard fit may match very well, however, I need it to be even better. Even with the new csv file, the Matlab fit was far better. Matlab gave: a = 0.0000010000000222047, b = -1.09997779553949E-09, and tau = 657.977318176476. Rsquared was 0.9999992.

Rsquared in Mathematica is 0.9995 when using all the options as I did at first, and a bit lower without them.

I think the fit could reach the same values as Matlab if Mathematica would obey the constraints, which it currently does not for b during the iterations.

Best regards,

Roland

Attachments:

Raw_heatflow_data.csv