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Re: FindFit issue
*To*: mathgroup at smc.vnet.net
*Subject*: [mg86422] Re: [mg86395] FindFit issue
*From*: Darren Glosemeyer <darreng at wolfram.com>
*Date*: Tue, 11 Mar 2008 02:56:56 -0500 (EST)
*References*: <200803100704.CAA24763@smc.vnet.net>
t.dubaj at gmail.com wrote:
> Hi,
> I am trying to fit experimental data
>
> d = {{1000, 44.576*10^(-3)},
> {600, 49.546*10^(-3)},
> {800, 46.757*10^(-3)},
> {400, 55.023*10^(-3)},
> {200, 61.209*10^(-3)},
> {10, 71.584*10^(-3)},
> {100, 65.367*10^(-3)}}
>
> and model is:
>
> y = 0.072214 - a*Log[10, 1+b*x]
>
> but Mathematica after typing
>
> FindFit[d, 0.072214 - a*Log[10, 1 + b*x], {a, b}, x]
>
> return
>
> FindFit::nrlnum: The function value {-0.0226973-0.0208168 \
> \[ImaginaryI],<<5>>,-0.0282009-<<21>> \[ImaginaryI]} is not a list of
> \
> real numbers with dimensions {7} at {a,b} = {0.0152574,-1.99204}.
>
> {a -> 0.0152574, b -> -1.99204}
>
> (especialy value of "b" is really awful - negative number in Log)
>
> But I know, there is Fit (calculated in Scientist) with coefficients
> a = 0.033706
> b = 0.0057934
>
> Can someone explain what I'm doing wrong?
>
> Best regards,
>
> T.D.
>
>
>
To obtain a good fit, non-default starting values are needed for the
parameters. By default 1 is used as the starting value for each
parameter if no starting value is given. In this case, the default
starting values are pretty far off from the optimal values, as can be
seen by looking at
Show[ListPlot[d, PlotMarkers -> Automatic],
Plot[0.072214 - Log[10, 1 + x], {x, 10, 1000}], PlotRange -> All]
With better starting values, a good fit is obtained.
In[1]:= d = {{1000, 44.576*10^(-3)}, {600, 49.546*10^(-3)}, {800,
46.757*10^(-3)}, {400, 55.023*10^(-3)}, {200,
61.209*10^(-3)}, {10,
71.584*10^(-3)}, {100, 65.367*10^(-3)}}
Out[1]= {{1000, 0.044576}, {600, 0.049546}, {800, 0.046757}, {400,
0.055023},
> {200, 0.061209}, {10, 0.071584}, {100, 0.065367}}
In[2]:= ff = FindFit[d, 0.072214 - a*Log[10, 1 + b*x], {{a, .1}, {b,
.01}}, x]
Out[2]= {a -> 0.0337413, b -> 0.00578092}
In[3]:= scientist = {a -> 0.033706, b -> 0.0057934}
Out[3]= {a -> 0.033706, b -> 0.0057934}
FindFit has a slightly better fit (smaller sum of squared errors) than
the quoted result. This may just be an indication of a difference in the
tolerances used by the two applications.
In[4]:= Total[((0.072214 - a*Log[10, 1 + b*x] /. ff /. x -> d[[All, 1]]) -
d[[All, 2]])^2]
-7
Out[4]= 9.81295 10
In[5]:= Total[((0.072214 - a*Log[10, 1 + b*x] /. scientist /.
x -> d[[All, 1]]) - d[[All, 2]])^2]
-7
Out[5]= 9.81367 10
Darren Glosemeyer
Wolfram Research
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