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Fit Any Scatter Plot June 7, 2018

A wonderful paper in the archives of the University of Rochester  shows how any random scatter plot can be fit to a curve with enough parameters, and thence a lower number of same is often thought to be a good measure of an expression’s fitness for use…until now. “The mathematician John von Neumann famously admonished that with four free parameters he could make an elephant, and with five he could make it wiggle its trunk…The aim of this short note is to show that, in fact, very simple, elementary models exist that are capable of fitting arbitrarily many points to an arbitrary precision using only a single real-valued parameter θ. This is not always due to severe pathologies—one such model, studied here, is infinitely continuously differentiable as a function of θ. The existence of this model has implications for statistical model comparison, and shows that great care must be taken in machine learning efforts to discover equations from data since some simple models can fit any data set arbitrarily well.”

Tall claim?  Nope.  The author, Steven T. Piantadosi, shows two examples of data points fitted with a simple equation

can be fit to any arbitrary set of data plots……like these:

Mind you, the parameter θ needs to be calculated precisely: ”Both use r = 8 and require hundreds to thousands of digits of precision in θ.”.

Gee whiz (and hilarity) aside, the paper demonstrates the fallacy of using unreasonable models for this sort of algorithmic from-data derivation to create meaning from what might be noise, or Joan Miro’s signature.