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MathGroup Archive 2000

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Re: Q:CurveFitting

  • To: mathgroup at smc.vnet.net
  • Subject: [mg21945] Re: Q:CurveFitting
  • From: "Kevin J. McCann" <kevin.mccann at jhuapl.edu>
  • Date: Sat, 5 Feb 2000 00:49:19 -0500 (EST)
  • Organization: Johns Hopkins University Applied Physics Lab, Laurel, MD, USA
  • References: <87e1tl$s2g@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Another job for a least squares fit.  I have attached a NB at the bottom.

Hope this helps you.

Kevin

--

Kevin J. McCann
Johns Hopkins University APL

FRESSENGEAS Nicolas <nicolas.fressengeas at supelec.fr> wrote in message
news:87e1tl$s2g at smc.vnet.net...
> Hello out there
>
> I am looking for any ideas to fit data to :
> Exp[1-Exp[-t/tau]] : the fit variable is tau
>
> Thanks in advance
>

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Cell[CellGroupData[{
Cell["LSQ Fit", "Title"],

Cell[CellGroupData[{

Cell["Original Message", "SectionFirst"],

Cell[BoxData[{
    \(Hello\ out\ there\), "\n",
    \(I\ am\ looking\ for\ any\ ideas\ to\ fit\ data\ \(to : \(\(\
\[ExponentialE]\^\(1 - \[ExponentialE]\^\(-\(t\/tau\)\)\)\) :
          the\ fit\ variable\ is\ tau\)\)\)}], "Text"]
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Cell["\<\
There are lots of routines out there for doing this, but it is so simple to
\
do that this is how I almost always do it. The same technique will work with
\
a multivariate fit. The only real problem is that the FindMinimum below \
requires an initial guess.

Your function. We wish to determine the \[Tau] that best fits the data\
\>", "Text"],

Cell[BoxData[
    \(f[t_, \[Tau]_] := \[ExponentialE]\^\(1 - \
\[ExponentialE]\^\(-\(t\/\[Tau]\)\)\)\)], "Input",
  CellLabel->"In[7]:="],

Cell[CellGroupData[{

Cell["Some \"data\"", "Subsection"],

Cell["\<\
Generate a random list of x-values -  maybe your data is not equispaced\
\>", "Text"],

Cell[BoxData[
    \(\(xlist = Table[Random[Real, {0, 5}], {100}];\)\)], "Input",
  CellLabel->"In[3]:="],

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and the corresponding data list with \[Tau]=3.14 and some noise added\
\>", "Text"],

Cell[BoxData[
    \(\(dataList = \(({#,
                f[#, 3.14] + Random[Real, {\(- .5\),  .5}]} &)\)\  /@ \
          xlist;\)\)], "Input",
  CellLabel->"In[8]:="],

Cell["Plot the data", "Text"],

Cell[BoxData[
    \(\(pdata =
        ListPlot[dataList,
          PlotStyle \[Rule] {RGBColor[0, 0, 1],
              AbsolutePointSize[5]}];\)\)], "Input",
  CellLabel->"In[10]:="]
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Cell[CellGroupData[{

Cell["LSQ fit Module", "Subsection"],

Cell["\<\
Here is a Module to do a least-squares sum for a given \[Tau]\
\>", "Text"],

Cell[BoxData[
    \(lsFit[\[Tau]_] :=
      Module[{}, \[IndentingNewLine]Plus\  @@ \
          Apply[\((f[#1, \[Tau]]\  - \ #2)\)\^2 &,
            dataList, {1}]\[IndentingNewLine]]\)], "Input",
  CellLabel->"In[11]:="],

Cell["\<\
What the module does is to take the first argument in each {t,data} pair, \
compute f(t,\[Tau]), subtract the data value from it and square.  Then sum \
over all the data pairs. A least-squares fit looks for the \[Tau] that \
minimizes this.\
\>", "Text"],

Cell["\<\
Let's plot this for a range of \[Tau]:\
\>", "Text"],

Cell[BoxData[
    \(Plot[lsFit[\[Tau]], {\[Tau],  .001, 10},
      PlotStyle \[Rule] RGBColor[1, 0, 0], Frame \[Rule] True,
      FrameLabel \[Rule] {"\<\[Tau]\>", "\<LSQ(\[Tau])\>"},
      GridLines \[Rule] Automatic]; \)], "Input",
  CellLabel->"In[16]:="],

Cell["\<\
The is a clear minimum. Just use FindMinimum to get it. Initial guess is 3 \
based on the plot\
\>", "Text"],

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    \(ans = FindMinimum[lsFit[\[Tau]], {\[Tau], 3}]\)], "Input",
  CellLabel->"In[18]:="],

Cell[BoxData[
    \(\[Tau]0 = \[Tau] /. ans[\([2]\)]\)], "Input",
  CellLabel->"In[19]:="],

Cell["\<\
Not too bad, since the real \[Tau] = 3.14.\
\>", "Text"],

Cell["\<\
Now, plot the function with this \[Tau] and compare with the data. Also note
\
that ans[[1]]=8.38 is the measure of how close the fit is. Here it is not so
\
close because of the large noise.\
\>", "Text"],

Cell[BoxData[
    \(\(Plot[f[t, \[Tau]0], {t, 0, 5},
        PlotStyle \[Rule] {AbsoluteThickness[3], RGBColor[1, 0, 0]},
        Epilog \[Rule] pdata[\([1]\)],
        PlotRange \[Rule] {0.5, 2.5}];\)\)], "Input",
  CellLabel->"In[23]:="]
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