Re: Problems with NonlinearRegress

*To*: mathgroup at smc.vnet.net*Subject*: [mg32248] Re: Problems with NonlinearRegress*From*: Adam Smith<adam.smith at hillsdale.edu>*Date*: Wed, 9 Jan 2002 03:18:02 -0500 (EST)*References*: <a1633l$pp0$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

This was a very interesting study for me and the use of non-linear fits in Mathematica. I apologize for the length of this, but I wanted to point out all the pitfalls one can encounter in doing fits of this type. Since I am not sure what your data "test4.txt" looked like I created a data set based on your function as follows. func = base + amplitude/(((freq - wo)/gamma)^2 + 1) Note that I avoided the capital letters on "base" and "gamma". Capital letters are often reserved for functions and special values. Then to create the data with an error of 1% in each point. mysub = {base -> 0.65, gamma -> 0.008, amplitude -> 1.31, wo -> 8.82} temprange = 0.008 data = Table[{freq, func*(1 + Random[Real, {-.01, 0.01}]) /. mysub}, {freq, 8.82 - 15*temprange, 8.82 + 15*temprange, temprange/2.5}]; Clear[temprange]; Now check that the fake data looks OK dataplot = ListPlot[data, PlotStyle -> PointSize[0.02], PlotRange -> All, DisplayFunction -> Identity]; exactplot = Plot[func /. mysub, {freq, 8.65, 8.95}, PlotRange -> All, DisplayFunction -> Identity]; Show[exactplot, dataplot, DisplayFunction -> $DisplayFunction]; The first problem that I see is that your function as defined is very sensitive to the starting value of w0. I was able to get reasonable results with your other starting parameters only for starting values of w0 = 8.81 to 8.83 fit2 = NonlinearRegress[data, func, {freq}, {{base, 0.8}, {amplitude, 1}, {wo, 8.81}, {gamma, 0.009}}, MaxIterations -> 100, RegressionReport -> ParameterCITable] Estimate Asymptotic SE base 0.649568 0.000711214 amplitude 1.30892 0.00353695 wo 8.81999 0.0000216297 gamma -0.00801306 0.0000342016 Anything less that 8.81 and greater than 8.83 produces unrealistic values of all parameters with very large Asymptotic SE. This makes sense if you look at the distribution. You need a starting value of wo that is at lest somewhere in the "central peak". Also note that the gamma parameter is negative but of the correct magnitude. This is because of the square on gamma in the function. Now I found that a modification of the function definition produces a more robust routine. Based upon the definition I found of the Lorentzian: 1/Pi (g/2)/( (x-m)^2 + (g/2)^2) I took you definition and massaged it a bit with your gamma = g/2 to get: myfunc = base + amplitude*(gamma/((freq - wo)^2 + gamma^2)) Here I absorbed the 1/Pi into the "amplitude" so instead of 1.31 my exact amplitude is 1.31*.008 = .01048. Anyhow, I checked that this gave the same data as your function and it did. Here the fit was much less sensitive to the starting parameters and worked for starting values of w0 from 8.78 to 8.86 (vs. 8.81 to 8.83). You still need to be careful in the starting value, but now it seems to me this is a much easier range to "eyeball" to pick the value. myfit = NonlinearRegress[mydata, myfunc, {freq}, {{base, 0.8}, {amplitude, 1.*.009}, {wo, 8.8}, {gamma, 0.009}}, MaxIterations -> 100] Estimate Asymptotic SE base 0.65063 0.00069442 amplitude 0.0105122 0.0000343314 wo 8.82 0.0000210092 gamma 0.00799454 0.000033212 Hope this helps. In article <a1633l$pp0$1 at smc.vnet.net>, Martin Duemling says... > >Hi, >I have the following problem: >I try to do a Lorentzian-fit with my data. It works very good if the datas are >simulated and very close to the ideal curve. But as soon as I use my real data >(they have still Lorentzian shape) I don't get any result which makes sense >(See example). > >Maybe somebody can help, Thank you. >Martin > > >Input: > >Clear["Global`*"] >Needs["Statistics`NonlinearFit`"] >data = ReadList["test4.txt", {Number, Number}]; >fit2 = NonlinearRegress[data, \ > Base + Amplitude/((1 + ((((freq - wo))/gamma))^2)), > freq, {{Base, 0.8}, {Amplitude, 1}, {wo, 9}, {gamma, 0.009}}, > Weights -> Equal, MaxIterations -> 100, WorkingPrecision -> 16, > RegressionReport > -> {ParameterTable, ANOVATable, EstimatedVariance}, > AccuracyGoal -> 16, PrecisionGoal -> 16, ShowProgress -> True] > > >Output >Iteration:1 ChiSquared:37.193 Parameters:{0.8, 1., 9., 0.009} > >Iteration:2 ChiSquared:37.5594 Parameters:{0.863029, 17.9184, 11.1608, >0.0648881} > >Iteration:3 ChiSquared:38.099 Parameters:{0.878984, 46.7423, 16.506, >0.117541} > >Iteration:4 ChiSquared:38.7406 Parameters:{0.900703, 29.929, 22.6742, >0.096431} > >Iteration:5 ChiSquared:38.7053 Parameters:{0.901547, -25.3515, 44.1125, >0.0073759} >... >And than the values are staying approx constant > > > >For comparison the correct values are (other fit program): >Base= 0.65 >Amplitude= 1.31 >wo= 8.82 >gamma= 0.008 > > > Adam Smith Dept. of Physics Hillsdale College adam.smith at hillsdale.edu