Re: Unstable solutions to NonlinearFit

*To*: mathgroup at smc.vnet.net*Subject*: [mg32443] Re: Unstable solutions to NonlinearFit*From*: adam.smith at hillsdale.edu (Adam Smith)*Date*: Mon, 21 Jan 2002 02:54:55 -0500 (EST)*References*: <a2b4lp$ljd$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Without having your data sets to work with it is difficult to give specific details. But what you describe is a common topic when working with nonlinear fits. My first hint would be to look at your initial "guesses" for the parameters. In your case these are: {{b, -1}, {c, -1}, {d, -1}, {f, -1}}. By examining a data set can you give mathematica starting parameters that are nearer to what gives you a reasonable fit. If your initial guesses are far from the "correct" values the code may be getting stuck in a local minimum that is far from the best solution. ANother thing to try. In your function Exp[b r1^2] + Exp[c + d r1] + 10^f, I would avoid the 10^f. Such a term means that small changes in f will result in very large changes in the "constant" term. This can cause difficulties in nonlinear fits. You might try replacing it by a simple constant say g, let Mathematica do the fit then convert g into your paramter f at the end via f = Log[g]. In a similar manner. Exp[c + d r1] can be rewritten as Exp[c]Exp[d r1]. Have a new parameter say "amp" = Exp[c] then fit to: Exp[b r1^2] + amp*Exp[d r1] + g. I won't promise this will work. But it is worth a try. My experience is that you sometimes have to play around with nonlinear fits and see what happens. Adam ashcroft at remss.com (Peter Ashcroft) wrote in message news:<a2b4lp$ljd$1 at smc.vnet.net>... > 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.