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A simple one-parameter equation that can be used to reproduce any plot


Pretty neat stuff.  The full short paper PDF is linked on the page below.

In a very surprising paper Steven Piantadosi shows that a simple function of one parameter (θ) can fit any collection of ordered pairs {Xi,Yi} to arbitrary precision. In other words, the same simple function can fit any scatter plot exactly, just by choosing the right θ. The intuition comes from chaos theory. We know from chaos theory that simple functions can produce seemingly random, chaotic behavior and that tiny changes in initial conditions can quickly result in entirely different outcomes (the butterfly effect). What Piantadosi shows is that the space traversed in these functions by changing θ is so thick that you can reverse the procedure to find a function that fits any scatter plot.

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This is just another formulation of the old nonsense about an infinite number of monkeys typing away forever and producing the complete works of Shakespeare.  It has nothing to do with Chaos Theory.

Hmm I'm not sure I understand (the paper looks way too complex for me to even try reading it) so I'll probably say something stupid, but… if the "single parameter" can be as precise/long as we want it to be then,  yeah, of course, you can describe any bitmap picture with a single parameter (which would happen to be the whole file's binary contents written as a single number) ?_?

This quote from the page was interesting:

the paper also tells us that Occam’s Razor is wrong. Overfitting is possible with just one parameter and so models with fewer parameters are not necessarily preferable even if they fit the data as well or better than models with more parameters.

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