Data Fitting with Hyperbolic and Lorentzian Function

Hi everyone,

I'm trying to fit my data (current through a device) at different temperatures with two models. The first one looks like this:

ft = fittype( 'a*cosh( b*(c-x) )^-2', 'independent', 'x', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Algorithm = 'Levenberg-Marquardt';
opts.Robust = 'Bisquare';

while the second is specified as follow (it's a Lorentzian function):

ft = fittype( 'a / (1 + ((x-b) / c)^2)', 'independent', 'x', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Robust = 'LAR';

I'm plotting the various coefficients (a, b, c) as the T increases.

I was wondering if someone could clarify me how those coeffiecients are calculated. I think the first one gets the full width of the data while the second gets the FWHM (the shape of the first function is similar to the second, but differentiates on the tail parts). This is quite relevant to me because I need to compare the two models, but the coefficients that I get so far are hardly comparable.

Thanks in advance.

[Michael Soskind]

Hi Alessandro,

This is a bit more of a question for someone who models lineshapes for a living. These values, in fact, should not be particularly comparable, at least directly. As you do note, the behavior at the wings is different for the two techniques.

Therefore, it doesnt make sense to compare these without comparing which has the lower mean square error, or some metric of choice, compared to the data after fitting.

Best,

Michael

[Alessandro Lodi]

Thanks Michal for your reply.

From a physical point of view, the model I'm studying should observe a transistion between the two lineshapes above certain T because the physics underlying it changes.

I'm aware that the problem of comparying directly those two lineshapes it's intrinsically incorrect, but was wondering if you'd have any suggestions on which statisical parameters could be valuable to extract from the fit object - I've already got the coefficient of determination and the RMSE for each fitting at each temperature.

Best,

Alessandro

[Michael Soskind]

Hi Alessandro,

All of the statements you make are correct and very important to note. The temperature does in fact lead to underlying changes in the physics, particularly when looking at this topic from a spectroscopic perspective. It is a bit difficult to say whether other statistical tehcniques are necessary. From my knowledge, the most commonly used statistical parameters to extract from the fit would be the RMSE.

Something that you seem to indicate is that you are looking for something to compare these different coefficients. There are other types of norms that could be used, rather than square error, you could do an alternatively weighted error. Typically, RMSE is what I have seen when using different fitting models. Maybe something useful is the attached file. This goes through some different fit types, and shows that RMSE is the most common method for evaluating fit.

I hope this helps,

Michael