Re: Problem with Fit
- To: mathgroup at smc.vnet.net
- Subject: [mg17363] Re: [mg17304] Problem with Fit
- From: David Withoff <withoff>
- Date: Mon, 3 May 1999 01:46:00 -0400
- Sender: owner-wri-mathgroup at wolfram.com
> I have experienced what appears to be an underflow or overflow problem
> when using Fit. I fitted a quadratic {1,x,x^2} to a set of 26 data
> points where the x-value was in the range 0 to 25000 and the y-value
> was in the range 0 to 40; the results were as expected. I then tried
> fitting a cubic {1,x,x^2,x^3} and the results were much worse, with
> a larger RMS error than the quadratic. Trying higher powers such as
> {1,x,x^2,x^4} and {1,x,x^2,x^5} gave increasingly poor fits.
>
> I then repeated the process but with all the x-values divided by 1000,
> i.e. in the range 0 to 25. This time everything worked correctly and
> the cubic fit was significantly better than the quadratic. There had
> been no warnings or error messages with the original values, the only
> indication of the problem being the very poor fit.
>
> Is this a known problem? Is it reasonable to expect Mathematica to be
> able to issue a warning that something is amiss? The version I am using
> is Mathematica 2.2 for SPARC; is there a later one for this platform?
>
> Richard.
See the NumericalMath`PolynomialFit` package, especially the documentation
for this package, which discusses the underlying mathematical issues and
what to do about it. The general problem of polynomial fitting is also
occasionally used in numerical analysis classes to illustrate numerical
instability. The traditional solution to the numerical stability problem
is to scale the data (as you mentioned), which usually solves most of
the problem, and to use orthogonal polynomials, especially Chebyshev
polynomials, rather than powers of the independent variable.
The functions in the Statistics`LinearRegression` package will generate
a warning message in these examples to warn you of the numerical problem,
but for polynomials the recommended solution is the one used in the
NumericalMath`PolynomialFit` package. The Fit function is a bare-bones
linear regression function, and isn't designed to do any of the
diagnostics that would detect this problem.
The same problem can in principle come up when fitting any set of
functions, not just polynomials, that aren't numerically linearly
independent over the range of the data, but in practice the most common
examples involve polynomial fitting, so that is the context in which
this is usually discussed.
Dave Withoff
Wolfram Research