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I am working on fitting an equation with three fitting parameters A,B,C to a some sets of data (Y). The equation is

$$ Y=A*e^{-B/(X-C)} $$

I have multiple sets of Y and X from multiple samples. I first used nlinfit in MATLAB to fit A, B, and C in each sample and when I did this I found that the results for A and B across all data sets can be described perfectly ($R^2=.92$) as $B=C_1*ln(A)+C_2$ where C1 and C2 are constants. With nonlinear fitting I know that correlation between parameters is not necessarily a feature of the data itself, but I'm wondering if there is a way to check that. If A and B are actually this correlated across samples, it would have some important implications.

I have tried fitting this in a different way - a linearized form of the equation is

$$ ln(Y)=ln(A)-B/(X-C) $$

with the correct choice of C, a plot of 1/(X-C) vs ln(Y) is linear, so I did a simple routine in excel to vary C to maximize the $R^2$ of a line fit to ln(Y) vs 1/(X-C). Doing so gave me a better fit of the data (the final $R^2$ of each line fit is greater than .998), with generally the same A,B,C fits as the MATLAB. It also increased the correlation between A and B. I am confident there is no other C that could fit the data, as it is related to a temperature I can measure (though is not exactly that temperature - it varies between samples, but only by at most 50 C). As a result I don't see any way for there to be another A or B that fits the data either, but maybe I am wrong.

Does the correlation between A and B here still sound like a result of the data fitting or is it a real correlation? I can make physical arguments either way so any help you could give would be much appreciated!

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  • $\begingroup$ @giorgioluciano's comment is helpful. An alternative way to check the correlation between parameters is to plug the model into a random forests framework, replacing the standard decision tree with the model for which you want a parameter test. Then, save the parameter estimates from each iteration of the RF (note that RF, as outlined by Breiman, randomly resamples both objects and variables based on the bootstrap. You could also use a jacknife here). Based on this saved output, run your correlations. $\endgroup$
    – user78229
    Commented Nov 28, 2016 at 12:40

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It depends. First of all QQ-plot and all other diagnostic plots could help you. But most of all you should also try to fit your data with an indipendent data set. As an example. Your model really looks like an experiment of fitting kinetics data (probably because I'm writing myself code for Avrami modelling and non isothermal data) if you have the same results for two INDIPENDENT dataset and the model created with the first one can predict (and so you will use RMSE of the training set - test set) the results of the second yours. Two caveats are very important when you have to evaluate correlation. 1) Correlation does not means causality and 2) generally you investigate to find linear correlation If you need help with fitting chemical data I will be glad to help ;)

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  • $\begingroup$ close - its conductivity data in a polymer, thought the literature borrows a lot from kinetic fitting. if i were to find an independent dataset (I'm not sure what exactly qualifies as independent for my system), would i expect the same C1 and C2 to apply if this is just a data fitting anomaly? $\endgroup$
    – kddrsn
    Commented Nov 28, 2016 at 20:17
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Let's say you find a model with a good C1 and C2 while trying to minimize the error in fit. the only way to knw if they are a good approximation is to rerun the experiment and try to predict your new data using the old coefficient and compare the results. the two experiment need to be performed in say condition but in a different "batch" (two different days/week just to be sure that you are trying to avoid to reproduce a systematic error). I hope i understood your reply and if I can I will be glad to help more

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