Re: Dancing "a la Levenberg-Marquardt" to get the best Logistic Model.

*To*: mathgroup at smc.vnet.net*Subject*: [mg124853] Re: Dancing "a la Levenberg-Marquardt" to get the best Logistic Model.*From*: Ray Koopman <koopman at sfu.ca>*Date*: Thu, 9 Feb 2012 05:41:30 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jgtjaa$6gc$1@smc.vnet.net>

On Feb 8, 2:38 am, Gilmar Rodriguez-pierluissi <peacen... at yahoo.com> wrote: > Dear Math Group: > (**I start with the following population values between 1972 to 2008:**) > In[2]:= popvalues = {217928, 219129, 221577, 227481, 231748, 233514, > 232857, 233664, 235228, 240526, 243310, 249587, 250128, 253383, > 257751, 261999, 258229, 262567, 263272, 267643, 272468, 274035, > 276154, 278323, 282606, 289505, 295243, 293956, 294410, 296399, > 297382, 298289, 299248, 296785, 299359, 300184, 299993}; > In[3]:= L = Length[popvalues]; > (** The highest (projected) value that the population can reach is: **) > In[4]:= pop2020 = 304909; > (** Assemble the data to do build a "scatter plot" **) > In[5]:= popvalues2D = > Join[Table[{i, popvalues[[i]]}, {i, 1, L}], {{49, pop2020}}]; > In[6]:= plt1 = ListPlot[popvalues2D] > Out[6]= (** Plot ommited **) > (** Let: **) > In[7]:= K = pop2020 > Out[7]= 304909 > In[8]:= > Subscript[P, 1] = popvalues[[1]] > Out[8]= 217928 > (** I'm attempting to use a Population Logistic model similar to one \ > found in (where else?) Wikipedia:http://en.wikipedia.org/wiki/Logistic_functionunder the title: "In \ > ecology: modeling population growth". **) > (** Since I need this model to satisfy Logistic[1]= Subscript[P, 1] \ > and Lim t -> Infinity Logistic[t] = K; I came up with the following \ > version of the Logistic Model to handle the above data set \ > appropriately: **) > (** Logistic[t_]=(K Subscript[P, 1]Exp[rt])/(K Exp[r]+ Subscript[P, \ > 1](Exp[er]-Exp[r])); **) > (** If you inspect this model ("by hand") you will see that \ > Logistic[1]= Subscript[P, 1] (the first population data point). Using \ > L'Hopital's Rule; one can show that Lim t -> Infinity (Logistic[t]) = \ > K; by taking the derivative of the numerator and denominator with \ > respect to t and performing the appropriate cancellations. Again; K \ > is the highest value that the population can reach "by design". **) > (** Logistic[t_]=(Subscript[P, 1] E^(r*t))/(E^r+ Subscript[P, 1] \ > (E^(r*t)- E^r)/K); **) > (** The model is equivalent to: **)\[AliasDelimiter] > In[13]:= Logistic[t_] = ( Subscript[P, 1] Exp[r t])/( > Exp[r] + Subscript[P, 1] (Exp[r t] - Exp[r])/K); > (** I' m expecting Logistic[1] = 217928 and indeed : ) > In[14]:= Logistic[1] > Out[14]= 217928 > (** but, unfortunately; **) > In[16]:= Limit[Logistic[t], t -> Infinity] > Out[16]= Limit[(217928 E^(r t))/( > E^r + (217928 (-E^r + E^(r t)))/304909), t -> \[Infinity]] > (** and: **) > In[15]:= Logistic[49] > Out[15]= > = (217928 E^(49 r))/(E^r + (217928 (-E^r + E^(49 r)))/304909) > (** I can see the the function Logistic[t] requires to be "herded" \ > (somehow) so that cancellations of terms can take place. Perhaps \ > using "Hold[]" and "ReleaseHold[]; I just don't know how. **) > (** I need to overcome the above hurdle before evaluating: **) > logisticnlm = NonlinearModelFit[popvalues2D, Logistic[t, r], {r}, t] > (** I want to use the initial point Subscript[P, 1] and end point \ > Subscript[P, 49] as "pivot points" and use NonlinearModelFit to get \ > the Best Fit Non-Linear Regression via a "dance a la Levenberg-Marquardt" \ > similar to the dance shown here:http://www.numerit.com/samples/nlfit/doc.htm**) > (** Thank you for your help! **) Your function has only three parameters. You're using one of them to force the curve to pass through the first data point, and another to set the limit as t -> oo. The remaining parameter controls how fast the curve approaches the limit. The only way to get the curve to pass through the point {49,K} would be to take r so large that 49r = oo, which would make the curve pass far above all the intervening data points. Something's wrong with the theory -- pick one or more of: 304909 is wrong, or 2020 is wrong, or the logistic form is wrong.