I have my doubts about polynomial fitting - at least over order 2 as well as fourier analysis because of the implication of more than a very rough cyclicity, but if Jane wants to go into greater depth, I would be interested.
OTOH, there are many techniques in FD that show promise for application, such as methods used in the analysis of boundary layer flow and the transition from turbulence to smooth flow and back again, such as scaling laws, etc, and the issue of short term vs. long term predictability. We are having some luck with the analysis of the 2nd derivative (acceleration) & 3rd (jerk) as useful classifiers for sharp transients, but it's still very much up in the air...