Fitting an Object

Because adding constants "helps" data fit a theory, there is an old joke: "Five constants?? You can fit an elephant with five constants!" In a 1975 article James Wei tested this and found that there is a least squares Fourier sine series that will fit these coordinates

x(t)=(Subscript[a, o])+(∑_i)(Subscript[a, i]) sin (((i t π)/36))
y(t)=(Subscript[b, o])+(∑_i)(Subscript[b, i]) sin (((i t π)/36))

that requires a minimum of 30 terms in the Fourier expansion. This Demonstration shows that you can get fairly close with 15 (five more if you want your elephant to have an eye).

The larger question addresses how far you should go to "back into" a theory from data. The plot of the data shows that it has little correlation. Indeed, Pearson's correlation coefficient for this data is very small. Yet with adequate computing power a set of equations can be found. The message becomes: if you want this elephant to have feathers, Mathematica can include them in the fit. Have you have found a theory in your data? No. Is your client impressed? Perhaps, if your client is a cartoon company.

THINGS TO TRY

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DETAILS

More information is available in chapter eight of Private Real Estate Investment and at mathestate.com.

J. Wei, "Least Square Fitting of an Object," CHEMTECH, 5(2), 1975 pp. 128–129.

Contributed by: Roger J. Brown

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