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edited Sep 2, 2015 at 21:51

1.6k
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EDIT

The code to obtain the 3d plot was:

m1 = A Sin[2 \[Pi] f t] + B Cos[2 \[Pi] f t];
m2 = A Cos[2 \[Pi] f t + B];

data = Block[{A = 43, B = 94, f = 1/8},
    Table[{t, m1 + 5 RandomReal[II]}, {t, 0, 30, 0.1}]
];
gdata = ListLinePlot[data]

fit1a = NonlinearModelFit[data, m1, {A, B, f}, t, Method -> Automatic];
Show[gdata, Plot[fit1a[t], {t, 0, 30}, PlotStyle -> Red]]

fit1b = NonlinearModelFit[data, m1, {A, B, f}, t, 
    Method -> "NMinimize"
];
Show[gdata, Plot[fit1b[t], {t, 0, 30}, PlotStyle -> Red]]

{x,y} = Transpose[data];
My = Mean[y];
ClearAll[error];
error[model_, {AA_, BB_, ff_}] :=
    Block[{A = AA, B = BB, f = ff},
        Norm[y - model /. t -> x]/My
    ];

The error plot above comes from:

Plot3D[error[m2, {103.3, B, f}], {B, -\[Pi], \[Pi]}, {f, 0, 1}, 
    PlotRange -> All, AxesLabel -> {"B", "f"}, 
    ColorFunction -> "Rainbow", MaxRecursion -> 3
]

With respect to Fourier:

afy = Abs@Fourier[y];
Pick[Range[0, Length[afy] - 1], afy, Max[afy]] // N

{4,297}

Note that 30 is the sampling time of the signal:

m1f = A Sin[2 \[Pi] (4/30 + f) t] + B Cos[2 \[Pi] 4/30 + f) t]

fit1f = NonlinearModelFit[data, m1f, {A, B, {f, 0}}, t,
    Method -> Automatic
];
Show[gdata, Plot[fit1f[t], {t, 0, MT}, PlotStyle -> Red]]

Now, even with the Automatic method, since the frequency is centered around a close-to-optimal value, the fit comes out correctly.

EDIT

The code to obtain the 3d plot was:

m1 = A Sin[2 \[Pi] f t] + B Cos[2 \[Pi] f t];
m2 = A Cos[2 \[Pi] f t + B];

data = Block[{A = 43, B = 94, f = 1/8},
    Table[{t, m1 + 5 RandomReal[II]}, {t, 0, 30, 0.1}]
];
gdata = ListLinePlot[data]

fit1a = NonlinearModelFit[data, m1, {A, B, f}, t, Method -> Automatic];
Show[gdata, Plot[fit1a[t], {t, 0, 30}, PlotStyle -> Red]]

fit1b = NonlinearModelFit[data, m1, {A, B, f}, t, 
    Method -> "NMinimize"
];
Show[gdata, Plot[fit1b[t], {t, 0, 30}, PlotStyle -> Red]]

{x,y} = Transpose[data];
My = Mean[y];
ClearAll[error];
error[model_, {AA_, BB_, ff_}] :=
    Block[{A = AA, B = BB, f = ff},
        Norm[y - model /. t -> x]/My
    ];

The error plot above comes from:

Plot3D[error[m2, {103.3, B, f}], {B, -\[Pi], \[Pi]}, {f, 0, 1}, 
    PlotRange -> All, AxesLabel -> {"B", "f"}, 
    ColorFunction -> "Rainbow", MaxRecursion -> 3
]

With respect to Fourier:

afy = Abs@Fourier[y];
Pick[Range[0, Length[afy] - 1], afy, Max[afy]] // N

{4,297}

Note that 30 is the sampling time of the signal:

m1f = A Sin[2 \[Pi] (4/30 + f) t] + B Cos[2 \[Pi] 4/30 + f) t]

fit1f = NonlinearModelFit[data, m1f, {A, B, {f, 0}}, t,
    Method -> Automatic
];
Show[gdata, Plot[fit1f[t], {t, 0, MT}, PlotStyle -> Red]]

Now, even with the Automatic method, since the frequency is centered around a close-to-optimal value, the fit comes out correctly.

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answered Sep 2, 2015 at 21:12

user8074

1.6k
9
7

It a small help, but I also did some analyses on this interesting problem and I found that the methods "NMinimize" and "ConjugateGradient" of NonlinearModelFit can find the real solution when the other methods fail, at least on MMA 10.2 (while on MMA 9 this did not seem to me to be effective).

With respect to the difficulty of getting the frequency right, the chart below shows the model error as a function of amplitude and f (the axis in front, between 0 and 1).

The fitting value should be 1/8 and it can be seen that the error decreases only for f values in a very thin strip around the optimal value. A little away, the error is almost constant. Consequently, only a thorough examination of the error behavior can point to the correct frequency, otherwise it will not be found, and amplitude and f will be practically arbitrary.

My suggestion is to fix the correct frequency with a Fourier analysis before any attempt to fitting the other parameters.