Thank you for your comments. In reply to your comment about accessibility, I did modify my earlier reply to you with a remark stating that I would be happy to share the text of that original publication with anyone who contacts me ( buhlmann@umn.edu ). I stated in my earlier reply that the three different data sets that are fitted simultaneously represent different spectral features. For those who dig deep and look at that publication, they will see that these spectral features refer to NMR spectroscopy (nuclear magnetic resonance spectroscopy). For those not familiar with this technique, let me give an example that is different but would follow a similar logic. Imagine a purple compound reacted in a chemical reaction to give a green product. Three different data sets would be the intensity of absorption of red and blue light of the reacting compound, and the intensity of absorption of green light of the product. Averaging the different data sets would not make any sense, but as all three data sets represent the same reaction, the same set of ODEs. And why is it preferable to solve the ODEs and not just make a fit with any function that gives a nice fit. The reaction constant, kzero, cannot be obtained from a fit with any function that gives a nice fit. It can only be obtained from using a physically meaningful model, as one gets it from solving the ODEs. That reaction constant, kzero, is a very useful parameter as it can be used to predict the speed of reaction of experiments that have not been performed, for example with different concentrations of the reacting compound. Hope that helps.

Thank you for your interest. And I apologize if the notebook is not fully self-explanatory. It was really written initially assuming that everyone looking at it would have had a look at the peer-reviewed publication that is mentioned in the abstract. I would be happy to provide anyone who contacts me with the text of that publication ( buhlmann@umn.edu ). Let me give you a reply, which would probably even make more sense for someone who has had a look at that publication; but I will try to be as clear as I can. As you saw, there were multiple sets of y values. You cannot meaningfully average those, though. The data are all from one experiment, but the data are very different in character. As described in that publication, the data describe the reaction of one chemical reagent to a product reagent, and the different sets of data are spectral features of the reacting reagent and the product compound. Averaging these is not meaningful. However, each set of data reflects the same reaction kinetics, and therefore they all depend on the same reaction constant, which I labelled kzero. That really gives two different options. Either one can make a fit of each data set, which will give for three data sets three different values for kzero, and then one can apply usual statistics to determine an average and a confidence interval for kzero. The disadvantage of this method is that the confidence interval for this determination of kzero is then determined with a degree of freedom of 2, which does not reflect well that the original data contain many more data points. (Imagine for example what would happen if had only two data sets available and made to separate fits; you would get two kzero values, and with only one degree of freedom, the confidence interval for kzero would be huge, definitely not reflecting the large number of data points in the initial data.) The second option (which I chose) is to fit all data in one fit using the Kronecker Delta, which is really the main feature of this notebook. I think chemists underutilize the Kronecker Delta. The question then is how to get an estimate of the accuracy of the resulting kzero, a question for which I then used bootstrapping.

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