Chuck, John:
If we know already, a priori, that the data is from a "smooth function",
that means (moving from left to right say), the extended line or the
extended parabola from the last two or last three points respectively is
always a very good predictor of the next point, then I would suggest
some form of interpolation. You can get very bad behavior from least
squares polynomial fitting. If the data is very noisy, then it will
not meet the smoothness assumption and some kind of least squares
polynomial fitting will be better than interpolation so long as the
underlying signal is well matched to the polynomial degree. If it is a
signal with very poor signal to noise ratio (for example), these fitting
algorithms are very problematic. Then we need to talk about
understanding the dynamics that produce the underlying signal so that we
can have a predictor that we can "correct" with the noisy observations.
This is like Kalman Filtering/Smoothing. If the underlying dynamics is
very nonlinear, or not well approximated locally by lines, as well as
the observation of the signal, then welcome to nonlinear filtering and
the theory of infinite dimensional functions spaces of the Hilbert type
and stochastic driven parabolic partial differential equations. I
needed 4.5 years to get a Ph.D. to understand the latter and that was
the first time I could write down a real phase locked loop with
nonlinear observation (sinusoidal phase detector) and understand the
mechanics.
JUST SAY NO. Doing all of these kinds of approximations, predictors,
etc. in the real world, DSP type, control type, etc. is an art form in
many cases based on some science or assumed knowledge. Chuck's question
is too wide without further specifics to give it a one answer fits all.
Bob
John Aldridge wrote:
cswiger wrote:
This is for the mathematicians out there - what is a simple
working algorithm for creating a function model to fit an
arbitrary number of data points.
You could try a least squares fitted polynomial
http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
has a description of how it's done.
--
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