B How can we accurately model data using curve fitting techniques?

[houlahound]

While using a piece of jewelry ie a fine chain hanging against a piece of graph paper then placing an origin, axes and scale on the paper I collected the position of the chain as a set of coordinates (x,y).

Entering the data into curve fitting software a perfect fit for a polynomial degree 2 was obtained.

Also fitted the data to a catenary made of exponentials.

My question is what does it mean to say the theoretical curve to model some data is this when you can adjust any polynomial within reason to get the same fit.

 

[jedishrfu]

We know that when you hang a chain you get a catenary. We also know that you can approximate a catenary within a given domain using a polynomial.

You were approaching the problem from the measurement end to get your polynomial and that's where you have to stop. You have no theory to explain the polynomial fit or why you came up with a polynomial. This happens a lot in science where the data is neatly described by a polynomial but there's no theory to explain it.

In one example, Cornell Univ folks had developed an AI that could discern the equations of motion from data about a compound pendulum and the equations were spot on. However later a biology team used the same program to analyze some cell data and once again they got a equation that was spot on but they couldn't publish because they couldn't explain the equation from theory.

In contrast, the catenary comes from analyzing the nature of the hanging chain and deriving the catenary from that analysis.

 

[houlahound]

thanks, so it's not so much a case of lack of rigour it is that a fitted curve gives no insight to what the key variables are?

 

[jedishrfu]

Here's that article I mentioned on the program that "understands" physics:

https://www.wired.com/2009/04/Newtonai/

Another example could be the evolution of the theory of gravitation from the cycles of Ptolemy to Kepler's laws to Newton's Law of Gravitation to Einstein's Theory of General Relativity. In each case, a better equation was developed to address the ever more precise collected data.

Here's Feynman's lecture on this evolution:

http://www.cornell.edu/video/richard-feynman-messenger-lecture-1-law-of-gravitation

 

[jedishrfu]

I found this related article in Wired about how students do experiments which you may find interesting:

https://www.wired.com/2016/09/might-gotten-little-carried-away-physics-time/

In this case, the students graph their data and fit it to a quadratic which is the correct answer and perhaps this makes students want to believe that everything works the same way.

However, the world intrudes and may make things locally in your limited experiment match a polynomial okay but in a bigger sense with theory we can find a better answer...