Re: Finding a sine wave

*To*: mathgroup at smc.vnet.net*Subject*: [mg95390] Re: [mg95341] Finding a sine wave*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Sat, 17 Jan 2009 05:30:11 -0500 (EST)*References*: <200901161107.GAA14041@smc.vnet.net>*Reply-to*: drmajorbob at longhorns.com

Your constraint on the wavelength will NOT make solutions unique, I think, and not all data will yield a solution at all. Here are three solvers you might try; the first is yours, except that I've put constraints on the wavelength. Your version set a starting point, only, for the wavelength. Clear[findSin, fitSin, minSin] findSin[xx_List, yy_List] := findSin[xx, yy] = Block[{a, ph, wl, x}, a Sin[ph + 2 Pi x/wl] /. FindRoot[ yy - a Sin[ph + 2 Pi xx/wl] // Evaluate, {{a, Max[Abs[yy]]}, {wl, 2 Max@xx, Max@xx, 10 Max@xx}, {ph, 0}}]] fitSin[xx_List, yy_List] := fitSin[xx, yy] = Block[{a, ph, wl, x}, a Sin[ph + 2 Pi x/wl] /. FindFit[Rest@Transpose@{xx, yy}, {a Sin[ph + 2 Pi x/wl], wl > Max@Abs@xx, 0 < 2 Pi ph < Max@Abs@xx}, {a, ph, wl}, x] ] minSin[xx_List, yy_List] := minSin[xx, yy] = Block[{a, ph, wl, x}, a Sin[ph + 2 Pi x/wl] /. Last at NMinimize[{#.# &[Thread[yy - a Sin[ph + 2 Pi xx/wl]]], wl > Max@xx, 0 < ph < (1/2 Pi) Max@xx}, {a, ph, wl}] ] xx=Sort[Join[{0},RandomReal[{0,10},2]]]; yy=RandomReal[{-25,25},3]; f[x_]=findSin[xx,yy] g[x_]=fitSin[xx,yy] h[x_]=minSin[xx,yy] -27.4879 Sin[0.456353-1.55521 x] FindFit::eit: The algorithm does not converge to the tolerance of 4.806217383937354`*^-6 in 500 iterations. The best estimated solution, with feasibility residual, KKT residual or complementary residual of {1.0772*10^-10,0.0000226396,3.80526*10^-11}, is returned. >> -37.2328 Sin[0.527053+0.0356086 x] -27.4879 Sin[2.68524+1.55521 x] Plot[{f@x, g@x, h@x}, {x, 0, 10}, Epilog -> {PointSize[0.02], Point /@ Transpose@{xx, yy}}, PlotRange -> {{0, 10}, {-25, 25}}] minSin will (almost always?) return a solution without errors, but it won't always fit the data points. The other two will fail in ALL SORTS of ways. In only three random draws, I found an example for which none of the solvers actually fit the data: xx={0, 1.5565, 6.32643} yy={0.305144, 3.21477, 17.9622} f[x_] = findSin[xx, yy] g[x_] = fitSin[xx, yy] h[x_] = minSin[xx, yy] FindRoot::reged: The point {21.1716,63.2643,0.000205665} is at the edge of the search region {6.32643,63.2643} in coordinate 2 and the computed search direction points outside the region. >> 21.1716 Sin[0.000205665+0.0993164 x] 16.5886 Sin[1.00688+0.993164 x] -9.61001 Sin[4.39622+0.993164 x] findSin fit the first two points, fitSin fit NONE of the points, and minSin fit only the second point. I'm pretty sure NO Sin function fits those data points, with a wavelength longer than the interval. Bobby On Fri, 16 Jan 2009 05:07:49 -0600, Hugh Goyder <h.g.d.goyder at cranfield.ac.uk> wrote: > An experiment will give me three coordinates which lie on a sine wave. > I have to find the sine wave efficiently. There are three unknowns the > sine wave amplitude, A, the wavelength of the sine wave, L, and the > phase, ph. I also know that the wavelength is larger than the interval > containing my measurement points. I think this condition removes > possible multiple solutions. > > Below I give two methods which need improving. The first method uses > FindRoot. This methods works about 60% of the time. I give six > examples where it fails. The failure may be due to poor initial > guesses. In the second method I use FindFit. This is not quite the > correct method because I have an equal number of equations and > unknowns. Some of the failures here, I think, are due to there being > no error for the algorithm to work with. I give examples of failures. > I have also tried FindInstance, Reduce and NSolve but I don't think > these are appropriate. > > Here are some questions > > 1. I would really like a symbolic solution rather than an iterative > one. Is such a solution possible? > 2. Can anyone improve on the methods below to make them more robust? > 3. I have some control over my x-locations. How can I work out best x > locations given an estimate of the wavelength L? > > Many thanks for all answers. > > > xx = Sort[Join[{0}, RandomReal[{0, 10}, 2]]] > > yy = RandomReal[{-25, 25}, 3] > > sol = FindRoot[{yy[[1]] - A*Sin[ph], > yy[[2]] - A*Sin[2*Pi*(xx[[2]]/L) + ph], > yy[[3]] - A*Sin[2*Pi*(xx[[3]]/L) + ph]}, {{A, > Max[Abs[yy]]}, {L, Max[xx]}, {ph, 0}}] > > Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10}, > Epilog -> {PointSize[0.02], (Point[#1] & ) /@ > Transpose[{xx, yy}]}, PlotRange -> {{0, 10}, {-25, 25}}] > > FindRootFailures = {{{0, 3.781264462608982, > 3.797055100562352}, {-22.948348087068737, 1.4581744078038472, > -14.242676740704574}}, {{0, 4.424069570670131, > 8.861716396743098}, {-6.444538773775843, -10.787309362608688, > 15.579334094330942}}, {{0, 4.424069570670131, > 8.861716396743098}, {-5.944536471563289, 10.627536107497393, > -21.294232497202316}}, {{0, 1.3680556047878967, > 8.546115267250002}, {4.128528863849845, 2.9017293848933923, > -22.51610539815371}}, {{0, 1.3738869718371616, > 5.689309423462079}, {20.553993773523437, -16.972841620064592, > 16.61185061676568}}, {{0, 1.0408828831133632, > 8.484645515699821}, {-3.7267589861478045, 4.1016610850387, > -24.600635061804443}}}; > > (({xx, yy} = #1; > sol = FindRoot[{yy[[1]] - A*Sin[ph], > yy[[2]] - A*Sin[2*Pi*(xx[[2]]/L) + ph], > > yy[[3]] - A*Sin[2*Pi*(xx[[3]]/L) + ph]}, {{A, > Max[Abs[yy]]}, {L, Max[xx]}, {ph, 0}}]; > Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10}, > > Epilog -> {PointSize[0.02], (Point[#1] & ) /@ > Transpose[{xx, yy}]}, > PlotRange -> {{0, 10}, {-25, 25}}]) & ) /@ > FindRootFailures > > xx = Sort[Join[{0}, RandomReal[{0, 10}, 2]]] > > yy = RandomReal[{-25, 25}, 3] > > sol = FindFit[ > Transpose[{xx, yy}], {A*Sin[2*Pi*(x/L) + ph], > Max[Abs[xx]] < L}, {A, L, ph}, x] > > Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10}, > Epilog -> {PointSize[0.02], (Point[#1] & ) /@ > Transpose[{xx, yy}]}, PlotRange -> {{0, 10}, {-25, 25}}] > > FindFitFailures = {{{0, 2.463263668380835, > 4.3190892163093615}, {-1.8407827676541144, -8.736574079785198, > 12.661520984622932}}, {{0, 3.894521446091823, > 9.937403619870642}, {12.381369822165155, 15.840399165432128, > -6.914634137727327}}, {{0, 8.087369725271945, > 8.343899312282815}, {24.73795103895976, -13.396248713970305, > 7.150470311065216}}, {{0, 1.5480031866178834, > 9.575255260205617}, {-8.163720246784278, 13.373468882892958, > -18.018462091502098}}, {{0, 0.6152784485896601, > 0.818296772602134}, {-1.858698140836046, 3.695113783904491, > -1.3186989232026658}}, {{0, 4.743093154057316, > 6.028314583406327}, {-16.60893277597446, -17.413392198343093, > -18.54081837965986}}}; > > (({xx, yy} = #1; > sol = FindFit[ > Transpose[{xx, yy}], {A*Sin[2*Pi*(x/L) + ph], > Max[Abs[xx]] < L}, {A, L, ph}, x]; > Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10}, > > Epilog -> {PointSize[0.02], (Point[#1] & ) /@ > Transpose[{xx, yy}]}, > PlotRange -> {{0, 10}, {-25, 25}}]) & ) /@ > FindFitFailures > > > > -- DrMajorBob at longhorns.com

**References**:**Finding a sine wave***From:*Hugh Goyder <h.g.d.goyder@cranfield.ac.uk>