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MathGroup Archive 2006

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Re: damped oscilations data fit

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70557] Re: damped oscilations data fit
  • From: "Ray Koopman" <koopman at sfu.ca>
  • Date: Thu, 19 Oct 2006 03:23:20 -0400 (EDT)
  • References: <eh4ntl$7sb$1@smc.vnet.net>

Miroslav Hý?a wrote:
> hello,
> I have one question about data manipulation in mathematica.
> I've got set of experimental data. Data describe damped oscillation. My question is following:
>
> How can I fit these data?
> I would like to get formula of function which will approximately describe my data and plot this function.
>
> my data list:
> {{0, 54}, {120, 56.5}, {230, 56}, {305, 54}, {340, 53}, {360, 52.7}, {378, 52.5}, {405, 52.5}, {443, 53}, {480, 53.5}, {510, 54}, {540, 54.7}, {570, 54.4}, {602, 56}, {643,56.5}, {660, 56.5}, {685, 56.25}, {706, 56}, {727, 55.25}, {743, 55.5}, {756, 55.25}, {775, 55}, {787, 54.75}, {799, 54.5}, {814, 54.25}, {828, 54}, {845, 53.75}, {858, 53.5}, {877, 53.25}, {894, 53}, {923, 52}, {951, 53}, {983, 53.5}, {1014, 54}}
>
> Have anyone an idea?
> I'm mathematica beginner therefore I'll be grateful for any suggestion.
> Thanks
> <<mira

Steps to an answer:

1. ListPlot the data, then guess at the form of the function:
   y = a + b*Sin[t/c]*Exp[-t/d]

2. Repeatedly use pic[c,d] with trial values of c & d,
   until the fit looks good.

pic[c_,d_] := Block[{x,y, mx,my, a,b},
{x,y} = Transpose@data; x = N[Sin[x/c]*Exp[-x/d]];
mx = Mean@x; my = Mean@y; x -= mx; y -= my;
b = y.x/x.x; a = my - b*mx;
Plot[a+b*Sin[t/c]*Exp[-t/d], {t,0,1014}, PlotRange->{51.5,57},
Frame->True, Axes->None, Prolog->{PointSize[.015],Point/@data}];
{a, b, c, d, Sqrt[#.#&[y-b*x]/(Length@data-4)]}]

3. Use NMinimize to polish the fit by minimizing f[c,d].

f[c_?NumericQ, d_?NumericQ] := Block[{x,y,b},
{x,y} = Transpose@data; x = N[Sin[x/c]*Exp[-x/d]];
x -= Mean@x; y -= Mean@y; b = y.x/x.x; #.#&[y-b*x]]

NMinimize[f[c,d],{{c,84,85},{d,2910,2920}}]
{Sqrt[%[[1]]/(Length@data-4)], Block[{x,y, mx,my, b},
{x,y} = Transpose@data; x = N[Sin[x/c/.%[[2]]]*Exp[-x/d/.%[[2]]]];
mx = Mean@x; my = Mean@y; x -= mx; y -= my; b = y.x/x.x;
{my - b*mx, b}]}

{3.20787, {c -> 84.6961, d -> 2917.55}}
{0.327, {54.4977, 2.22006}}


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