FindFit or NonlinearModelfit for sinusoidal function

Question

I am trying to fit this data set into a specific sinusoidal equation.

The equation is shown like this.

modelR2w =  y1 + DL Cos [ \[Phi] + a2 ] +  FL (2 (Cos[\[Phi] + a2]^3) - Cos[\[Phi] + a2]);

And the codes are following like,

DataR2w = Transpose[{Flatten[{dataPhi}], Flatten[{dataR2w}]}];

modelR2w =  y1 + DL Cos [ \[Phi] + a2 ] +  FL (2 (Cos[\[Phi] + a2]^3) - Cos[\[Phi] + a2]);

fit2 = FindFit[DataR2w, modelR2w, {y1, a2, DL, FL}, \[Phi]]

modelf2 = Function[{\[Phi]}, Evaluate[modelR2w /. fit2]]

Plot[modelf2[\[Phi]], {\[Phi], 0, 2 Pi}, Epilog -> Map[Point, DataR2w]]

I use Findfit to find out the parameters, and it works well when the data is shown like this,

However, when the data is out of the figures, that is, $Cos[phi]$ graph,

It does not work well...

Actually, the equation consists of two terms; $cos(phi)$ and $2(cos(phi))^3-cos(phi)$

I think this Findfit is not good with $2(cos(phi))^3-cos(phi)$

Could you give me some advice on how to overcome this problem?

I am trying to find the fitted one with Nonlinearfit.....

Thank you in advance, and if you need any raw data, I shall send you immediately.

---

I added my actual data and Wolfram notebook file.

1) https://docs.google.com/spreadsheets/d/1zYiXrT1l6tgH1d0DNBVCt2iNbVX9tNhq/edit?usp=drive_link&ouid=104846936799445912159&rtpof=true&sd=true,

2) https://drive.google.com/file/d/1dYGqX3Ugs0MaqNVMMHNwV3VMH3Z8AXV9/view?usp=drive_link,

3) https://docs.google.com/spreadsheets/d/1jAdNLq0Kr6ChNMIlwGLHns2sDqwh_Hhx/edit?usp=drive_link&ouid=104846936799445912159&rtpof=true&sd=true,

4) https://docs.google.com/spreadsheets/d/1zDlMJ1pAzO2S2Lhhj1469fUv_nHPc6k_/edit?usp=drive_link&ouid=104846936799445912159&rtpof=true&sd=true

Answer 1

As Sin[w x + phi] are a complete system, you may make a fit with a few such functions.

Given the data:

xs = {0, 0.05236, 0.10472, 0.15708, 0.20944, 0.2618, 0.31416, 0.36652,
    0.41888, 0.47124, 0.5236, 0.57596, 0.62832, 0.68068, 0.73304, 
   0.7854, 0.83776, 0.89012, 0.94248, 0.99484, 1.0472, 1.09956, 
   1.15192, 1.20428, 1.25664, 1.309, 1.36136, 1.41372, 1.46608, 
   1.51844, 1.5708, 1.62316, 1.67552, 1.72788, 1.78024, 1.8326, 
   1.88496, 1.93732, 1.98968, 2.04204, 2.0944, 2.14675, 2.19911, 
   2.25147, 2.30383, 2.35619, 2.40855, 2.46091, 2.51327, 2.56563, 
   2.61799, 2.67035, 2.72271, 2.77507, 2.82743, 2.87979, 2.93215, 
   2.98451, 3.03687, 3.08923, 3.14159, 3.19395, 3.24631, 3.29867, 
   3.35103, 3.40339, 3.45575, 3.50811, 3.56047, 3.61283, 3.66519, 
   3.71755, 3.76991, 3.82227, 3.87463, 3.92699, 3.97935, 4.03171, 
   4.08407, 4.13643, 4.18879, 4.24115, 4.29351, 4.34587, 4.39823, 
   4.45059, 4.50295, 4.55531, 4.60767, 4.66003, 4.71239, 4.76475, 
   4.81711, 4.86947, 4.92183, 4.97419, 5.02655, 5.07891, 5.13127, 
   5.18363, 5.23599, 5.28835, 5.34071, 5.39307, 5.44543, 5.49779, 
   5.55015, 5.60251, 5.65487, 5.70723, 5.75959, 5.81195, 5.86431, 
   5.91667, 5.96903, 6.02139, 6.07375, 6.12611, 6.17847, 6.23083, 
   6.28319};
ys = {4.59 E - 02, 6.05 E - 02, 6.36 E - 02, 6.64 E - 02, 6.91 E - 02,
    7.16 E - 02, 7.39 E - 02, 7.59 E - 02, 7.77 E - 02, 7.92 E - 02, 
   8.06 E - 02, 8.15 E - 02, 8.21 E - 02, 8.23 E - 02, 8.23 E - 02, 
   8.18 E - 02, 8.12 E - 02, 8.02 E - 02, 7.88 E - 02, 7.72 E - 02, 
   7.52 E - 02, 7.30 E - 02, 7.05 E - 02, 6.78 E - 02, 6.50 E - 02, 
   6.18 E - 02, 5.86 E - 02, 5.52 E - 02, 5.17 E - 02, 4.84 E - 02, 
   4.50 E - 02, 4.17 E - 02, 3.83 E - 02, 3.50 E - 02, 3.19 E - 02, 
   2.88 E - 02, 2.62 E - 02, 2.36 E - 02, 2.12 E - 02, 1.91 E - 02, 
   1.73 E - 02, 1.59 E - 02, 1.47 E - 02, 1.39 E - 02, 1.33 E - 02, 
   1.31 E - 02, 1.33 E - 02, 1.38 E - 02, 1.46 E - 02, 1.59 E - 02, 
   1.71 E - 02, 1.84 E - 02, 2.03 E - 02, 2.19 E - 02, 2.41 E - 02, 
   2.67 E - 02, 2.92 E - 02, 3.21 E - 02, 3.50 E - 02, 3.81 E - 02, 
   4.13 E - 02, 4.47 E - 02, 4.79 E - 02, 5.11 E - 02, 5.42 E - 02, 
   5.75 E - 02, 6.04 E - 02, 6.34 E - 02, 6.59 E - 02, 6.84 E - 02, 
   7.06 E - 02, 7.25 E - 02, 7.42 E - 02, 7.57 E - 02, 7.67 E - 02, 
   7.75 E - 02, 7.79 E - 02, 7.81 E - 02, 7.78 E - 02, 7.74 E - 02, 
   7.65 E - 02, 7.53 E - 02, 7.40 E - 02, 7.24 E - 02, 7.04 E - 02, 
   6.84 E - 02, 6.61 E - 02, 6.36 E - 02, 6.10 E - 02, 5.81 E - 02, 
   5.52 E - 02, 5.24 E - 02, 4.96 E - 02, 4.68 E - 02, 4.40 E - 02, 
   4.13 E - 02, 3.88 E - 02, 3.64 E - 02, 3.43 E - 02, 3.23 E - 02, 
   3.06 E - 02, 2.91 E - 02, 2.79 E - 02, 2.70 E - 02, 2.65 E - 02, 
   2.62 E - 02, 2.62 E - 02, 2.66 E - 02, 2.73 E - 02, 2.83 E - 02, 
   2.94 E - 02, 3.10 E - 02, 3.28 E - 02, 3.49 E - 02, 3.72 E - 02, 
   3.97 E - 02, 4.23 E - 02, 4.50 E - 02, 4.80 E - 02, 5.11 E - 02, 
   5.40 E - 02};
d = Rest[Transpose[{xs, ys}]];

We can calculate a fit with the 3 lowest functions:

model = c0 + c1  Sin[w  x + d1 ] + c2  Sin[2 w  x + d2];
f[x_] = model /. FindFit[d, model, {c0, c1, c2, w, d1, d2}, x];

11.4938 - 8.17344  Sin[6.21729 - 1.9709 x] + 
 2.04074  Sin[2.08517 + 0.985452 x]

The result is:

Answer 2

If you just want to "describe" the data with a function (as opposed to obtaining a function with theoretically meaningful and interpretable parameters), then the answer from @DanialHuber is what you want. Here is an example with 4 terms of the form a[i] Sin[w[i] ϕ + θ[i]].

θ = {0, 0.05236, 0.10472, 0.15708, 0.20944, 0.2618, 0.31416, 0.36652, 0.41888, 0.47124, 0.5236, 0.57596, 0.62832, 0.68068, 0.73304, 0.7854, 0.83776, 0.89012, 0.94248, 0.99484, 1.0472, 1.09956, 1.15192, 1.20428, 1.25664, 1.309, 1.36136, 1.41372, 1.46608, 1.51844, 1.5708, 1.62316, 1.67552, 1.72788, 1.78024, 1.8326, 1.88496, 1.93732, 1.98968, 2.04204, 2.0944, 2.14675, 2.19911, 2.25147, 2.30383, 2.35619, 2.40855, 2.46091, 2.51327, 2.56563, 2.61799, 2.67035, 2.72271, 2.77507, 2.82743, 2.87979, 2.93215, 2.98451, 3.03687, 3.08923, 3.14159, 3.19395, 3.24631, 3.29867, 3.35103, 3.40339, 3.45575, 3.50811, 3.56047, 3.61283, 3.66519, 3.71755, 3.76991, 3.82227, 3.87463, 3.92699, 3.97935, 4.03171, 4.08407, 4.13643, 4.18879, 4.24115, 4.29351, 4.34587, 4.39823, 4.45059, 4.50295, 4.55531, 4.60767, 4.66003, 4.71239, 4.76475, 4.81711, 4.86947, 4.92183, 4.97419, 5.02655, 5.07891, 5.13127, 5.18363, 5.23599, 5.28835, 5.34071, 5.39307, 5.44543, 5.49779, 5.55015, 5.60251, 5.65487, 5.70723, 5.75959, 5.81195, 5.86431, 5.91667, 5.96903, 6.02139, 6.07375, 6.12611, 6.17847, 6.23083, 6.28319};

r2w = {0.000376, 0.000526, 0.00052, 0.000503, 0.000527, 0.000541, 0.000502, 0.000551, 0.000541, 0.000542, 0.000539, 0.000547, 0.000545, 0.000544, 0.000559, 0.000545, 0.000544, 0.000555, 0.000558, 0.000539, 0.00054, 0.00056, 0.000529, 0.000532, 0.000524, 0.000509, 0.000499, 0.000442, 0.000474, 0.000467, 0.000465, 0.000409, 0.000355, 0.000341, 0.000297, 0.000269, 0.000266, 0.000245, 0.00022, 0.000229, 0.000263, 0.000246, 0.000247, 0.00018, 0.000249, 0.000254, 0.000253, 0.000277, 0.000297, 0.000302, 0.000295, 0.000294, 0.000336, 0.000378, 0.000415, 0.000431, 0.000445, 0.000443, 0.000471, 0.000448, 0.000454, 0.000449, 0.000426, 0.000433, 0.000454, 0.000447, 0.00043, 0.000437, 0.000446, 0.000456, 0.000429, 0.000438, 0.000433, 0.000422, 0.000407, 0.000422, 0.0004, 0.000404, 0.000439, 0.00043, 0.000415, 0.000432, 0.000427, 0.000426, 0.000472, 0.000436, 0.000418, 0.000406, 0.000453, 0.000389, 0.000369, 0.000364, 0.000324, 0.000274, 0.000262, 0.000251, 0.000235, 0.00022, 0.00025, 0.000221, 0.000229, 0.000266, 0.000264, 0.000255, 0.000252, 0.000268, 0.00028, 0.000306, 0.000377, 0.000357, 0.000348, 0.000411, 0.000421, 0.000407, 0.000427, 0.000474, 0.000446, 0.000484, 0.000502, 0.000537, 0.000519};

DataR2w = Transpose[{θ, r2w}];
k = 4;
model = a0 + Sum[a[i] Sin[w[i]  ϕ + θ[i]], {i, k}]
nlm = NonlinearModelFit[DataR2w, model,
   Join[{{a0, Mean[DataR2w[[All, 2]]]}}, Flatten[Table[{a[i], w[i], θ[i]}, {i, k}]]],
   ϕ, MaxIterations -> 10000];
Show[ListPlot[DataR2w], Plot[nlm[ϕ], {ϕ, 0, 2 π}, WorkingPrecision -> 20]]

If the proposed model is theoretically based and still desired to fit, then the fit is straightforward with NonlinearModelFit:

modelR2w = y1 + DL  Cos[ϕ + a2] + FL  (2  Cos[ϕ + a2]^3 - Cos[ϕ + a2]);
nlm = NonlinearModelFit[DataR2w, modelR2w, {y1, DL, a2, FL}, ϕ]
mpb = nlm["MeanPredictionBands"];
Show[ListPlot[DataR2w], 
 Plot[{nlm[ϕ], mpb}, {ϕ, 0, 2 π}, PlotStyle -> {Black, LightGray, LightGray}]]

This is not a good fit but not the fault of NonlinearModelFit. What appears to be missing is a parameter that @DanielHuber noticed and included. Rewriting the model with that missing parameter provides a much better fit but then doesn't show that the data likely resulted from such a model.

modelR2wx = y1 + DL  Cos[w  ϕ + a2] + FL  (2  Cos[w  ϕ + a2]^3 - Cos[w  ϕ + a2]);
nlmx = NonlinearModelFit[DataR2w, modelR2wx, {y1, DL, a2, FL, {w, 0.7}}, ϕ]
mpbx = nlmx["MeanPredictionBands"];
Show[ListPlot[DataR2w], 
 Plot[{nlmx[ϕ], mpbx}, {ϕ, 0, 2 π}, PlotStyle -> {Black, LightGray, LightGray}]]