Re: Rationalized Fitting

*To*: mathgroup at smc.vnet.net*Subject*: [mg125562] Re: Rationalized Fitting*From*: Antonio Alvaro Ranha Neves <aneves at gmail.com>*Date*: Mon, 19 Mar 2012 04:55:10 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jhienb$b3o$1@smc.vnet.net> <jjmuv3$cen$1@smc.vnet.net>

On Thursday, March 15, 2012 6:36:56 AM UTC+1, Ray Koopman wrote: > On Mar 13, 10:41 pm, Ray Koopman <koop... at sfu.ca> wrote: > > First, the data you generate has multiplicative error, > > so you should specify Weights->(1/#&) in NonlinearModelFit. > > With multiplicative error, the error variance is proportional > to the square of the expected value, so the weight function > should be (#^-2&), because the weights should be inversely > proportional to the error variance. Dear Darren and Ray, I've tried, del = 10^-3; NoisyParabola = Table[{x, (Prime[7]/Prime[8] + x Prime[9]/Prime[10] + Prime[11]/Prime[12] x^2)*RandomReal[{98/100, 102/100}]}, {x, 1, 10, 1/4}]; NLMFit = NonlinearModelFit[NoisyParabola, Rationalize[a, del] + x Rationalize[b, del] + Rationalize[c, del] x^2, {a, b, c}, x, Weights -> (Rationalize[#^-2, del] &), VarianceEstimatorFunction -> (Mean[Abs[Rationalize[#, del]]] &)]; NLMFit["ParameterTable"] Rationalize[NLMFit["ParameterTable"], del] As suggested with no success. Here I'm trying to find the best fit rational ized coefficients. The function is linear, but just an example since I would latter need a NonlinerModelFit. The brute-force solution would be something like, PossibleList = Prepend[Table[Prime[i], {i, 1, 26}], 1]; CheatedList = {Prime[7], Prime[8], Prime[9], Prime[10], Prime[11], Prime[12]}; PossibleList = CheatedList; (* Here I cheated, so it'll be fast *) PossibleCoef = Tuples[PossibleList, 2]; {xData, yData} = Transpose[NoisyParabola]; Error = 99999999; For[Coef1 = 1, Coef1 <= Length[PossibleCoef], Coef1++, For[Coef2 = 1, Coef2 <= Length[PossibleCoef], Coef2++, For[Coef3 = 1, Coef3 <= Length[PossibleCoef], Coef3++, temp = Total[Abs[(yData - ParallelMap[(PossibleCoef[[Coef1, 1]]/ PossibleCoef[[Coef1, 2]] + # PossibleCoef[[Coef2, 1]]/ PossibleCoef[[Coef2, 2]] + PossibleCoef[[Coef3, 1]]/PossibleCoef[[Coef3, 2]] #^2) &, xData])]^2]; If[temp < Error, Error = temp; BestCoef = {PossibleCoef[[Coef1, 1]]/PossibleCoef[[Coef1, 2]], PossibleCoef[[Coef2, 1]]/PossibleCoef[[Coef2, 2]], PossibleCoef[[Coef3, 1]]/PossibleCoef[[Coef3, 2]]}] ] ] ] Now BestCoef holds the desired coefficients that satisfies least square fitting. I was hopping to obtain such a result via NonlinearModelFit, but had no success so far. Also, the brute force method takes too long, when considering other ensemble of rationals.