problem with NonlinearFit

*To*: mathgroup at smc.vnet.net*Subject*: [mg4705] problem with NonlinearFit*From*: "Brian O'NEILL" <oneill at iiasa.ac.at>*Date*: Sat, 31 Aug 1996 03:57:27 -0400*Sender*: owner-wri-mathgroup at wolfram.com

I am having the following problem with the NonlinearFit function, and would appreciate any tips you might have. (I have appended sample Mathematica code below.) My goal is to fit a non-monotically decreasing time series with a monotonically decreasing function (specifically, a sum of decaying exponenitials plus a constant). I am approaching the problem by using NonlinearFit and specifying a general functional form, then attempting to constrain the search to non-negative coefficients and non-positive exponents. However, in general I have run into the following problems: (1) NonlinearFit will accept parameter specifications of the form {x,xo}, or of the form {x,minx,maxx}, but not of the form {x,xo,xmin,xmax}. This means I can constrain the variables, but I can't start from different points to test robustness. (2)The routine stops when it tries to search outside the constraints instead of continuing the search in a different direction, so that it is important to be able to specify different initial values to look elsewhere. I have also tried using MultiplierMethod and posing the problem as a minimization of least squares errors, but have had even less luck getting that function to accept constraints (but if you have pointers I would be happy to use anything that works). The code is below; any help appreciated. Thanks, Brian O'Neill (* define general functional form - sum of exponentials *) (* plus a constant, such that all coefficients sum to 1 *) fn[t_,a1_,a2_,a3_,tau1_,tau2_,tau3_] := (1.-a1-a2-a3)+a1*Exp[-t/tau1]+a2*Exp[-t/tau2]+ a3*Exp[-t/tau3]; (* generate data with one negative coefficient to *) (* produce a bump in the data *) data = Table[{t,fn[t,0.6,-0.4,0.1,15,30,300]},{t,0,300,10}]; (* take a look at data *) ListPlot[data,PlotRange->All] (* fit data with only positive coefficients allowed, *) (* supplying no starting points. this gives a solution, *) (* but I'd like to look for a better one (depending on the *) (* data, it sometimes stops searching because it exits the *) (* parameter bounds) *) soln = NonlinearFit[data,fn[x,a1,a2,a3,tau1,tau2,tau3], x,{{a1,0,1},{a2,0,1},{a3,0,1}, {tau1,0,Infinity},{tau2,0,Infinity},{tau3,0,Infinity}}, PrecisionGoal->5, AccuracyGoal->5, ShowProgress->True] (* try the same fit again, with same constraints on *) (* parameters but supplying starting points in order *) (* to find a different solution - but the function won't *) (* take both starting points and constraints....I can't *) (* find a syntax that works (this is the syntax give in *) (* the package documentation), nor can I figure out from *) (* looking at the package what the problem is. *) soln = NonlinearFit[data,fn[x,a1,a2,a3,tau1,tau2,tau3], x,{{a1,0.2,0,1},{a2,0.2,0,1},{a3,0.2,0,1}, {tau1,5,0,Infinity},{tau2,50,0,Infinity}, {tau3,100,0,Infinity}}, PrecisionGoal->5, AccuracyGoal->5, ShowProgress->True] (* take a look at the fit *) Show[ ListPlot[data,PlotRange->All,DisplayFunction->Identity], Plot[fn[t,a1,a2,a3,tau1,tau2,tau3]/.soln,{t,0,300}, DisplayFunction->Identity], DisplayFunction->$DisplayFunction]; Brian O'NEILL International Institute for | Email: oneill at iiasa.ac.at Applied Systems Analysis | Phone: +43 2236 807 0 A-2361 Laxenburg, Austria | Fax: +43 2236 71313 ==== [MESSAGE SEPARATOR] ====