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Re: FindFit, Weibull

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92900] Re: [mg92868] FindFit, Weibull
  • From: Darren Glosemeyer <darreng at wolfram.com>
  • Date: Fri, 17 Oct 2008 05:25:03 -0400 (EDT)
  • References: <200810160904.FAA18690@smc.vnet.net>

P_ter wrote:
> I found some data which seemed to fit a WeibullDistribution. The data are:
> pr= {16, 34, 53, 75, 93, 120};
> After adding the median rank to the data:
> praxis={{16, 0.109101}, {34, 0.26445}, {53, 0.421407}, {75, 0.578593}, {93, 0.73555}, {120, 0.890899}};
> I checked with LeastSquares the parameters: 
> \[Alpha]=1.4301;\[Beta] = 76.318. 
> So far ok.
> But the following did not work:
>  fit2 = {a, b} /. FindFit[praxis, CDF[WeibullDistribution[a, b], x], {a ,b}, {x}]
>
> So, I used the parameters in the Weibull CDF and the times from pr to get:
> praxis1= {{16, 0.106157}, {34, 0.27559}, {53, 0.451308}, {75, 0.623149}, {93, 0.73256}, {120, 0.848082}}
> (something went wrong with editing, so maybe I sent a by accident an unfinished message)
> I checked with FindFit and it worked. But then I gave extra some variations:
> praxis2 = {#, RandomReal[{0.95, 1.05}] CDF[WeibullDistribution[1.4, 76.318], #]} & /@ pr;
> fit4 = {a,b} /. FindFit[praxis2, CDF[WeibullDistribution[a,b], x], {a,b}, {x}]
> After a few times trying the two above lines FindFit did not work.
> It could be ok. But it seems to me that FindFit with a such a small relative variation in the data from 0.95 to 1.05 should still work.
> How can I use FindFit in the case of praxis in a safe way?
> Thanks in advance.
> P_ter
>   

There are two problems I can see. First, a and b need to be constrained 
to be positive. The FindFit::nrlnum message and failure happen because 
the algorithm reaches a negative value of b, which results in complex 
values.

Second, the default starting values of 1 for each parameter are far from 
the optimum curve. With better non-default starting values a good result 
can be obtained.

In[1]:= praxis = {{16, 0.109101}, {34, 0.26445}, {53, 0.421407}, {75,
            0.578593}, {93, 0.73555}, {120, 0.890899}};

In[2]:= fit2 = FindFit[
          praxis, {CDF[WeibullDistribution[a, b], x],
           a > 0 && b > 0}, {{a, 1}, {b, 80}}, {x}]

Out[2]= {a -> 1.52051, b -> 77.4766}

Note that the result given is different from the least squares result 
quoted in your message.

In[3]:= ls = {a -> 1.4301, b -> 76.318}

Out[3]= {a -> 1.4301, b -> 76.318}

If we check the sum of squared errors, we see that the FindFit result 
gives a smaller value and so is a better fit in the sense of distance 
from the cdf curve.

In[4]:= Map[((CDF[WeibullDistribution[a, b], #[[1]]] /. fit2) - 
#[[2]])^2 &,
          praxis] // Total

Out[4]= 0.0032187

In[5]:= Map[((CDF[WeibullDistribution[a, b], #[[1]]] /. ls) - #[[2]])^2 &,
          praxis] // Total

Out[5]= 0.00426419

I suspect the least squares solution given solves a slightly different 
problem, that is to say the condition for optimality for that solution 
is probably something other than minimizing the sum of squared errors 
from the cdf.


Darren Glosemeyer
Wolfram Research


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