       Re: Unstable solutions to NonlinearFit

• To: mathgroup at smc.vnet.net
• Subject: [mg32492] Re: Unstable solutions to NonlinearFit
• From: "John Doty" <jpd at w-d.org>
• Date: Wed, 23 Jan 2002 01:00:15 -0500 (EST)
• References: <a2b4lp\$ljd\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <a2b4lp\$ljd\$1 at smc.vnet.net>, ashcroft at remss.com  wrote:

> 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.

It's not the range of data values that matters as much as the range of
measurement error.

> 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
>
> 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.)

For this type of measurement, the errors are generally proportional to the
quantity measured, except perhaps at the low end. You're not measuring
watts to nanowatt accuracy, are you? In this case, it makes sense to fit
to the log of the data.

The unweighted least squares procedure yields the most likely parameter
values under the assumption that the measurement errors are normally
distributed with equal variances. The nice thing about least squares is
that it's not terribly sensitive to violations of these assumptions, but
measurement variances that differ by 16 orders of magnitude strain it a
bit.

Alternatively, you can apply weights that are inversely proportional to
your estimates of the measurement variances. With your huge dynamic range,
however, this may still be numerically unstable in a calculation with only
16 digits of precision (depends on the details of the code in the fitting
package).

--
| John Doty		"You can't confuse me, that's my job."
| Home: jpd at w-d.org
| Work: jpd at space.mit.edu

```

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