Re: damped oscilations data fit

*To*: mathgroup at smc.vnet.net*Subject*: [mg70581] Re: damped oscilations data fit*From*: "Ray Koopman" <koopman at sfu.ca>*Date*: Fri, 20 Oct 2006 05:21:47 -0400 (EDT)*References*: <eh4ntl$7sb$1@smc.vnet.net><eh7aqc$e8r$1@smc.vnet.net>

I forgot, in both what I did and what I posted, something that belongs in step 2: along with pic[c,d], use ContourPlot[-f[c,d],{c,cmin,cmax}, {d,dmin,dmax}] to find approximately optimal values of c & d. (I like working with -f rather than f, but that's a purely personal thing.) Ray Koopman wrote: > 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}}