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Unstable solutions to NonlinearFit
*To*: mathgroup at smc.vnet.net
*Subject*: [mg32418] Unstable solutions to NonlinearFit
*From*: ashcroft at remss.com (Peter Ashcroft)
*Date*: Sat, 19 Jan 2002 01:17:18 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
I'm having trouble using the NonlinearFit function to
find the best fit for some data. I get answers, but
in some cases they appear nonsensical. Also, I notice
that the Confidence Intervals are *very* large for some
of the parameters.
The data in question describes antenna gain patterns, and
has a very large dynamic range. I suspect that this large
range of data values (from 10^-8 to 1) is the reason that
some of the parameters are estimated so badly.
Simply looking at a plot of the data on a logarithmic
scale suggests that it could be fit well by something of
the form: Exp[b r1^2] + Exp[c + d r1] + 10^f
(In other words, the data has an Exp[b r1^2] part for
small values of r1, a slowly diminishing Exp[c + d r1]
part for larger values of r1, and eventually plateaus
at some constant positive value.)
Here's an example where the fit doesn't turn out so well,
(as judged by a plot of the fit on a logarithmic scale,
and the confidence intervals that are very large).
NonlinearRegress[linearvpairs,
Exp[b r1^2] + Exp[c + d r1] +
10^f, {r1}, {{b, -1}, {c, -1}, {d, -1}, {f, -1}},
RegressionReport -> {BestFit, AsymptoticCorrelationMatrix,
ParameterCITable}]
BestFit ->
3.649189*10^-9 + E^(-16.209103 - 4.15162 r1) + E^(-4.106467 r1^2)
ParameterCITable-> Estimate Asymptotic SE CI
b -4.1064 0.0139 -4.13384,-4.0790
c -16.209 30903.73 -60595.3,60562.8
d -4.1516 236105.04 -462829.,462821.
f -8.4378 5643.3565 -11070.8,11053.9
Note that "b" has a tight cinfidence interval, but the others
are extremely large.
Here's another example that's even more pathological. I don't
know what subtlety of the data differentiates this case from the
one above. (All the data sets look fairly similar on visual
inspection.)
NonlinearRegress[linearhpairs,
Exp[b r1^2] + Exp[c + d r1] +
10^f, {r1}, {{b, -1}, {c, -1}, {d, -1}, {f, -1}},
RegressionReport -> {BestFit}]
BestFit ->
3.772468*10^-313 + E^(-3.974 r1^2) + E^(-22.626 + 1.698 r1)
Note the 10^-313! I couldn't even compute the confidence
intervals in this case.
I know that I could get much better behaved fits if I
fitted to the logarithm of the data rather than the
data itself, but I would prefer to fit to the data
directly if possible. (Reasoning that what I want to
minimize is the sum of the squared error in power
rather than the squared error in log of the power.)
Does anyone have any suggestions for how I might set
the NonlinearFit options in order to get more stable
results? Thanks.
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