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In fitting any curve algorithms use a score to calculate how good the fit is, for example there is least square (the sum of the square of the difference between the fit and the data) where the lower score means a good fit and higher score means a bad fit. What other functions can I use for this "score" of fit?

I understand that this question is pretty broad and can potentially have different answers depending on the nature of the problem at hand. So if someone could just point out a document or a research paper or a link that would be great.

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  • $\begingroup$ It's mostly implicitly assumed, but their is a smoothness criteria. You could fit some very wiggly curve which went near all the data points. This would have a low sum of squares error but would not be a good fit. You can of course replace square with absolute value or 4th power if you wanted to penalise large errors. $\endgroup$ – user121049 Jul 3 '17 at 11:39
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Another criterion (though by no means the only other one) is to minimize the maximum absolute deviation between the curve and the data points. (You can search the phrase "minimize maximum deviation" for references and applications.)

In keeping with the comment by user121049, you can also add terms to the criterion function that penalize overfitting. See, for instance, https://en.wikipedia.org/wiki/Regularization_(mathematics).

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  • $\begingroup$ Thanks. Is there a book or a reference that you can recommend which is comprehensive of the known estimates of quality of fit? Would greatly appreciate. $\endgroup$ – Shaz Jul 4 '17 at 8:58
  • $\begingroup$ There is also the question of the uniqueness of the score. What I mean by uniqueness is that the best fit (the best score, either highest or lowest) should only correspond to a unique set of parameters involved. If more than one set of parameters can give you the best score then in my opinion that is not a very good chart to score the best fit. $\endgroup$ – Shaz Jul 4 '17 at 10:13
  • $\begingroup$ Sorry, I don't know any books that consider much of a variety of penalty functions. Regarding uniqueness, I understand the attraction of it, and any strictly convex function will guarantee uniqueness (so long as there are no constraints on the parameter values). $\endgroup$ – prubin Jul 4 '17 at 13:27
  • $\begingroup$ So I tried the idea of using higher powers instead of a square of the residual to score the fit. Something interesting happened: While for the problem at hand the least square method gives only one parameter with the best fit, minimizing fourth power of residual gives more than one parameter with best fit. Is that usual? $\endgroup$ – Shaz Jul 5 '17 at 6:05

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