Why do fit results and parameter confidence intervals depend on magnitude of units? Why are confidence intervals nonsense on a good fit?

I have attached a short script which computes group delay vs. frequency from the measured phase of a log-periodic antenna, with the data file also attached. (Sorry tried to use the inline code entry box, but Firefox did not seem to play well with it).

The group delay is fitted to a 4-parameter function using "fit" and I plot the results and the parameter estimatest and confidence intervals.

The problem is this: first, if I use the original units, Hz for frequency and seconds for group delay, the fit will not properly converge to any reasonable values for the parameters. The resulting confidence intervals are also nonsensical.

This problem goes away when I convert to MHz and nanoseconds, which are units that give values falling in a numerical range that is commonly used for antennas. However, even though the fit is excellent by eye, the confidence intervals on the fitted parameters are nonsensical.

So my questions are: why does the fit depend on units and why are the confidence intervals crazy even with a good fit?

curve1 =

General model:

curve1(x) = a.*(x./b).^(-c) + d

Coefficients (with 95% confidence bounds):

a = 7.5e-08 (-0.001166, 0.001166)

b = 1.7e+08 (-2.643e+12, 2.644e+12)

c = 1 (0.9718, 1.028)

d = 1.5e-08 (1.415e-08, 1.585e-08)

curvem =

General model:

curvem(x) = a.*(x./b).^(-c) + d

Coefficients (with 95% confidence bounds):

a = 76.35 (-1.624e+06, 1.624e+06)

b = 171.7 (-2.506e+06, 2.506e+06)

c = 1.457 (1.456, 1.458)

d = 3.091e-08 (-0.01276, 0.01276)

Answer 1

[Torsten]

Your model is not adequate because it needs 3 parameters, not 4.

Write it as

curvem(x) = a.*(x./b).^(-c) + d = a/b^(-c) * x^(-c) + d

Now define as independent variables to be fitted

a_tilde = a/b^(-c), c_tilde = -c and d_tilde = d.

Then your model takes the form

curvem(x) = a_tilde * x^(c_tilde) + d_tilde

with 3 parameters instead of 4 and the same explanatory power.

[Peter Gorham]

I agree that the parameters are not independent, and can be reduced as you have done. However, the physics of the problem gives an equation of the form tau = taumax*(f/fmax)^-p + tau0 where taumax is the delay at the low end of the range, and fmax is the frequency corresponding to taumax.

Fits in general do not require independent parameters, since the covariance matrix can capture any interdependence. My concern is more with the fact that the fit results depend on the units chosen, and also that the confidence intervals are not sensible.

When I try your decomposition, the fit does not converge at all, whereas it does converge nicely for the case where the units are chosen so that the numbers are in a moderate numerical range. For your decomposition, the expected best value of the lead coefficient is around 1.82e22 in the SI units and the lower and upper bounds become [1, 1e30]. So I think something about the fit algorithm is not happy with such large numbers.

[Torsten]

It's no problem:

Fit the parameters a, b, c and d of your 4-parameter model.

Define as starting guesses

a_tilde0 = a/b^(-c), c_tilde09 = -c and d_tilde0 = d.

and start the fitting anew for the three-parameter model.

The fitting process should converge immediately (together with confidence bounds for a_tilde, c_tilde and d_tilde).

That the physics is of the form

tau = taumax*(f/fmax)^-p + tau0

does not mean that you should choose 4 (dependent) parameters for the mathematical fitting.

[Peter Gorham]

I did try this, but in fact your approach does not converge to a sensible result at all. The fit does not work.

Here is what I used:

fo = fitoptions('Method','NonlinearLeastSquares',...

'StartPoint',[50e-9/(100e6)^(-1), -1, 5e-9]);

ft = fittype('a.*(x).^c + d','option',fo);

The result is a poor fit:

However the confidence intervals are at least not completely crazy:

General model:

curve1(x) = a.*(x).^(c) + d

Coefficients (with 95% confidence bounds):

a = 5.625 (1.782, 9.468)

c = -1.006 (-1.043, -0.9696)

d = 7.758e-09 (7.332e-09, 8.184e-09)

They are within the right range, and did not saturate, but somehow the curve still did not line up.

[Peter Gorham]

I misspoke about the large values of the combined parameters however; in fact the values are numerically moderate after the combination you prescribe. I will try a similar approach with the MHz, nanosecond units and see what happens.

[Torsten]

The only thing I can say is that a 4-parameter fit mathematically does not make sense.

If a, b, c and d are the parameters determined by the fitting, then the new parameters a',b',c' and d' with

a' = a*constant , b' = b*constant^(-1/c), c' = c and d' = d

for any value "constant" > 0 give the same approximation to the group delay.

Thus the "partition" of the quantity a/b^(-c) on the parameters a and b is arbitrary.

Of course, fixing "fmax" or "taumax" in your model equation before fitting will change things.

[Peter Gorham]

I believe that using a completely independent set of parameters does solve the issue of the crazy confidence intervals.

The remaining problem, which is that the fits do not converge unless I change to units that do not involve very large or small numbers, is still a mystery. I was able to get sensible confidence intervals by using independent parameters, but only the fits with MHz and nanoseconds will actually converge to a correct curve.

[Peter Gorham]

Anyway, Torsten thanks for giving this some attention, your responses are very helpful. I will try to create a simpler example to demonstrate if there is still an issue with the magnitude of the numbers, and circle back to you if I can't sort it out on my own.