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I have a set of data (included below) that I want to fit to a sinusoidal envelope to see how closely it follows this pattern. The data is:

Data1={0.309071, 0.33515, 0.509679, 0.635752, 0.623349, 0.376292, 0.00702311, -0.247732, -0.288283, 0.0392362, 0.424462, 0.185524, 0.157014, -0.193954, 0.123598, 0.495541, 0.624529, 0.44186, 0.0727319, -0.20929, -0.179794, 0.149309, 0.184027, 0.180771, -0.162702, -0.0318286, 0.280962, 0.561242, 0.518519, 0.177428, -0.13503, -0.17717, 0.0472761, 0.187547, 0.197198, -0.101749, 0.0631292, 0.382762, 0.52321, 0.422893, 0.133342, -0.068269, -0.0465858, 0.211891, 0.205344, 0.18894, 0.190942, -0.057206, -0.018655, 0.161711, 0.407714, 0.44038, 0.314649, 0.0467926, -0.0846743, 0.197518, 0.202479, 0.147582, 0.0171371, 0.0711502, 0.206846, 0.366559, 0.356282, 0.256973, 0.0971488, 0.00206637, 0.190533, 0.21806, 0.226478, 0.298721, 0.284167, 0.192472, 0.110618, 0.0577778, 0.121448, 0.249658, 0.341592, 0.199183, 0.19315, 0.195232, 0.237282, 0.285631, 0.257908, 0.163031, 0.104114, 0.119962, 0.17615, 0.227583, 0.162638, 0.17692, 0.188076, 0.238142, 0.215275, 0.241336, 0.206312, 0.195018, 0.167541, 0.188019, 0.196394, 0.233276, 0.211437, 0.196104, 0.196805, 0.235119, 0.211475, 0.207491, 0.212745, 0.181352, 0.230285, 0.185054, 0.200672, 0.209156, 0.234059, 0.175968, 0.178012, 0.128228, 0.178259, 0.211292, 0.18542, 0.185805, 0.155408, 0.221904, 0.222765, 0.20993, 0.203248, 0.218133, 0.209671, 0.207186, 0.237372, 0.271545, 0.223714, 0.203778, 0.194717, 0.209947, 0.1795, 0.158245, 0.231815, 0.270194, 0.229028, 0.214345, 0.178967, 0.147468, 0.203715, 0.213661, 0.213279, 0.172309, 0.143847, 0.147826, 0.230358, 0.210431, 0.261791, 0.207865, 0.156438, 0.131477, 0.222499, 0.205672, 0.233718, 0.23413, 0.204448, 0.157382, 0.164233, 0.216498, 0.24449, 0.258621, 0.230947, 0.208378, 0.211446, 0.220309, 0.226997, 0.165953, 0.160112, 0.213768, 0.239196, 0.271349, 0.252204, 0.165489, 0.20023, 0.178357, 0.203997, 0.230865, 0.276264, 0.248909, 0.228895, 0.143152, 0.195306, 0.210185, 0.229889, 0.218724, 0.235904, 0.246969, 0.173281, 0.177513, 0.208786, 0.221224, 0.261342, 0.262841, 0.212207, 0.160277, 0.229143, 0.202113, 0.203961, 0.249503, 0.238434, 0.213757, 0.25397, 0.229209, 0.182187, 0.216175, 0.23034, 0.229909, 0.202815, 0.242284, 0.218549, 0.256111, 0.221454, 0.205487, 0.207981, 0.237192, 0.204759, 0.217205, 0.221754, 0.200389, 0.238324, 0.243082, 0.217888, 0.201374, 0.198798, 0.189414, 0.228472, 0.227368, 0.2442, 0.200464, 0.193633, 0.159162, 0.209135, 0.262962, 0.278533, 0.253368, 0.203009, 0.135398, 0.123181, 0.204614, 0.20029, 0.222763, 0.224768, 0.299789, 0.261344, 0.210455, 0.13148, 0.146101, 0.17487, 0.249837, 0.30369, 0.202594, 0.248258, 0.294337, 0.304376, 0.2777, 0.220565, 0.111778, 0.106131, 0.144807, 0.235485, 0.283122, 0.19426, 0.21609, 0.0931201, 0.14347, 0.211519, 0.311878, 0.312163, 0.272518, 0.17686, 0.0848083, 0.120368, 0.200508, 0.197732, 0.25629, 0.339277, 0.309214, 0.256057, 0.174898, 0.0983865, 0.125015, 0.185359, 0.285428, 0.244608, 0.242815, 0.146848, 0.122946, 0.122114, 0.179348, 0.285598, 0.329529, 0.281544, 0.219595, 0.115432, 0.233389, 0.211577, 0.15348, 0.109439, 0.0885411, 0.185185, 0.300772, 0.335077, 0.292058, 0.207472, 0.101263, 0.223388, 0.216962, 0.188757, 0.0858888, 0.173627, 0.248867, 0.329886, 0.334462, 0.267857, 0.149218, 0.0939199, 0.205009, 0.204213, 0.180405, 0.220802, 0.299368, 0.319144, 0.311618, 0.192534, 0.149804, 0.0977141, 0.170934, 0.211812, 0.230457, 0.146043, 0.147742, 0.214149, 0.313017, 0.310056, 0.295263, 0.232353, 0.192809, 0.131725, 0.205076, 0.181839, 0.254358, 0.215729, 0.189337, 0.118086, 0.133628, 0.191204, 0.245922, 0.293993, 0.261041, 0.189409, 0.240371, 0.269324, 0.296122, 0.161646, 0.133229, 0.141756, 0.226917, 0.231886, 0.246886, 0.227295, 0.223463, 0.190347, 0.155447, 0.16225, 0.220588, 0.221309, 0.260858, 0.245372, 0.224228, 0.19766, 0.211557, 0.211861, 0.207691, 0.232002, 0.177278, 0.195422, 0.214852, 0.207025, 0.206766, 0.235009, 0.221069, 0.17925, 0.235326, 0.204984, 0.248338, 0.269855, 0.263508, 0.236014, 0.23071, 0.24772, 0.223968, 0.228462, 0.168735, 0.213937, 0.220366, 0.263239, 0.225068, 0.219187, 0.24838, 0.22138, 0.199448, 0.224209, 0.233003, 0.231807, 0.199574, 0.197298, 0.215652, 0.196667, 0.182118, 0.214451, 0.261312, 0.219363, 0.20681, 0.174271, 0.179097, 0.201217, 0.214468, 0.221119, 0.217908, 0.155635, 0.169446, 0.200914, 0.230277, 0.224512, 0.211738, 0.188873, 0.210123, 0.187306, 0.217841, 0.275159, 0.246055, 0.23013, 0.168858, 0.201035, 0.220072, 0.230789, 0.244695, 0.219828, 0.198347, 0.268312, 0.274949, 0.210661, 0.200858, 0.200994, 0.192099, 0.212076, 0.229709, 0.267034, 0.216645, 0.199789, 0.248571, 0.21012, 0.187611, 0.213273, 0.20734, 0.245058, 0.231637, 0.20592, 0.18638, 0.242921, 0.233676, 0.209621, 0.19435, 0.192956, 0.194512, 0.204144, 0.224806, 0.235779, 0.251603, 0.2053, 0.22598, 0.224877, 0.232533, 0.211332, 0.224546, 0.197367, 0.187678, 0.212382, 0.249761, 0.233594, 0.19201, 0.201084, 0.213968, 0.249826, 0.234896, 0.217118, 0.176485, 0.189911, 0.208698, 0.213005, 0.221254, 0.238774, 0.193031, 0.214946, 0.194444, 0.21477, 0.22283, 0.2485, 0.247803, 0.227941, 0.158324, 0.161366, 0.169061, 0.208237, 0.205385, 0.17892, 0.19634, 0.187137, 0.219079, 0.234997, 0.270446, 0.227186, 0.19585, 0.189501, 0.192457, 0.226828, 0.205672, 0.242684, 0.250321, 0.264016, 0.217203, 0.181052, 0.164646, 0.150746, 0.192029, 0.212988, 0.220697, 0.213644, 0.148598, 0.125564, 0.189824, 0.230994, 0.263338, 0.250261, 0.271088, 0.210566}

and the plot is given by

l = Table[l, {l, 0, 538}];
data = MapThread[List, {l, Data1}];
ListLinePlot[data, PlotRange -> All]

This gives the plot shown in the figure.

enter image description here

I want to see how well a damped sinusoidal envelope can be fitted to this. How can I do this?

|improve this question
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  • $\begingroup$ What have you tried? $\endgroup$ – David G. Stork May 16 '18 at 15:56
  • $\begingroup$ You could use some sort of Fourier Transformation to get the main frequencies. $\endgroup$ – kiara May 16 '18 at 17:02
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Here's one way to approach this: begin by finding the upper envelope of the curve, then fit that envelope to a nonlinear model of the desired form:

ListLinePlot[{Abs[Data1], 
  env = GaussianFilter[MaxFilter[Abs[Data1], 7], 7]}, PlotRange -> All]

enter image description here

NonlinearModelFit[env, d + c Exp[-a x] Abs[Cos[b x]], 
   {{a, .005}, {b, .01}, {c, 0.6}, {d, 0.2}}, x]
eqn[x_] = Normal[%];

fit = Table[eqn[x], {x, 1, Length[Data1]}];
ListLinePlot[{Abs[Data1], fit}, PlotRange -> All]

enter image description here

Not a very good fit... so you will need to figure out what nonlinearity you really want to use.

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You can do this using Quantile Regression.

In order to get a good fit I used Cos instead of Sin for the upper envelope.

In order to get a different/better fit, you can experiment with different number of Sin/Cos functions and the scales of their arguments.

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/QuantileRegression.m"]

qs = {0.01, 0.99};
qFuncs1 = 
 QuantileRegressionFit[data, 
  Table[Sin[2 \[Pi]/3000*i*x], {i, 0, 100, 1}], x, {0.01}]; 
qFuncs2 = 
 QuantileRegressionFit[data, 
  Table[Cos[2 \[Pi]/3000*i*x], {i, 0, 100, 1}], x, {0.99}];
qFuncs = Join[qFuncs1, qFuncs2];

qFuncs3 = qFuncs /. {(x_?NumberQ /; Abs[x] < 10^-4) -> 0};

Show[
 Plot[qFuncs3, {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}, PlotStyle -> Red, PlotRange -> All], 
 ListLinePlot[data, PlotRange -> All]]

enter image description here

Here is a screenshot of the envelope functions:

enter image description here

|improve this answer
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  • $\begingroup$ I challenge the downvoter to give a reasonable explanation for his downvote. :) $\endgroup$ – Anton Antonov May 19 '18 at 15:32
  • $\begingroup$ When I use Data1 rather than data I get an error about "The first argument is expected to be a matrix of numbers with two columns". What are the missing steps used to create data from Data1? Thank you in advance. $\endgroup$ – Jack LaVigne May 19 '18 at 23:32
  • 1
    $\begingroup$ I created data from Data using data = Transpose[Join[{Range[Length@Data1]}, {Data1}]]; and it seemed to be the missing step. $\endgroup$ – Jack LaVigne May 19 '18 at 23:39
  • $\begingroup$ @JackLaVigne Yes, you are correct. QuantileRegression requires the first argument to have the form {{_?NumericQ, _?NumericQ}..}. I was considering to have Transpose[{Range[Length[arg1]],arg1}] in case the first argument is a vector, but in a fair amount of cases Rescale[Range[Length[arg1]]] was a better choice than Range[Length[arg1]]. So, I left the t-axis addition to be done by the user. $\endgroup$ – Anton Antonov May 20 '18 at 0:41

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