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Is there a general approach to finding self-similar solutions; i.e. collapsing several functions onto a single function by some transformation?

I have data from some experiments, and the functions generated clearly appear to have some kind of self-similar solution. They are the result of varying a small number (4) of dimensionless parameters, and I'd like to find how (or if) I can collapse these onto a single function.

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I've always relied on physical insights and finding an appropriate nondimensionalization. If your data arises from a physical process where you know the equation or at least the geometry of the problem you may have some luck via the Buckingham pi theorem followed by trial and error. See also this paper:

Price, James F. "Dimensional Analysis of Models and Data Sets: Similarity Solutions and Scaling Analysis." (2006). PDF

Four is a non-trivial number of parameters when the functionality is unknown. In general you will need some additional insight to determine the (expected) functionality of the data and then you can perform some curve fitting.

"With four parameters I can fit an elephant, and with five I can make him wiggle his trunk." - John von Neumann

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  • $\begingroup$ Thanks much for the Price reference! I have some physical insights into appropriate nondimensionalization, but of course the devil is in the details. $\endgroup$
    – Aurelius
    Sep 15 '14 at 13:59

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