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October 18

14th-degree polynomial functions in the real world?

Epstein's Amendment to Moore's Law states that the doubling time for the most economical semiconductor density increases by 6 months every 10 years. According to Wolfram Alpha, this amounts to a polynomial growth function with an order of 13.8629. Are polynomial functions with orders that high seen frequently in the sciences? (Yes, I know it's a bit more pessimistic than what the historical data support.) If not, does this more likely mean that they don't occur in nature or society (and that therefore this model isn't plausible), or that they tend to be mistaken for exponential functions? Neon Merlin 18:38, 18 October 2011 (UTC)[reply]

It's an empirical formula, so you do what you have to to fit the data. In a situation like this, there's not some platonic ideal law of nature causing the rate of development of microchips that we're digging for. The trends are caused by all sorts of unpredictable and complicated interactions and the actual function describing the rate is not likely to be anything that can be nicely expressed. Even a 14 degree polynomial is much simpler than what's actually going on. The advantage to simpler models isn't that they are a priori any more accurate, it's that at some point more detail doesn't improve our predictive power since we're dealing with so much uncertainty. Only with very fundamental science (like in physics) do we often (and somewhat miraculously) find that the most accurate models are also the simplest. Rckrone (talk) 19:41, 18 October 2011 (UTC)[reply]

NB: This is a simple power function, not a polynomial with a bunch of intermediate terms that are free to float, so it's probably not much more prone to overfitting than an exponential function would be. Neon Merlin 02:44, 19 October 2011 (UTC)[reply]

Yes, this is a power law (well, nearly), not a polynomial. Power laws do tend to have smallish exponents, and I can't think of an example of another this large. The Lennard-Jones potential isn't far off, though. 130.88.73.65 (talk) 09:44, 20 October 2011 (UTC)[reply]