Sensitivity of NonlinearModelFit to model

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.

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.

In short the two models presented are identical for the particular dataset.

"Intuition is sometimes tricky on fitting procedures." in the answer by @mikuszefsky is right on the money. So is the comment by @MarcoB about increasing the number of iterations.

The two fits are found just fine by NonlinearModelFit:

data = With[{a = 43.45582489316203`, 
    f = 94.92003941300389`, ϕ = 431.155471523826`}, 
   SeedRandom[1234];
   Table[{t, a Cos[2 π f t + ϕ] + RandomReal[{-0.1, 0.1}]},
   {t, 13.439999656460714`, 13.479799655455281`, 0.0002}]];
m1 = a Cos[2 π f t] + b Sin[2 π f t];
m2 = a Cos[2 π f t + ϕ];
nlm1 = NonlinearModelFit[data, m1, {{a, -30}, {b, 30}, {f, 95}}, t, MaxIterations -> 1000]
nlm2 = NonlinearModelFit[data, m2, {{a, 40}, {f, 100}, {ϕ, 3.9}}, t]

Both models are essentially identical but one cannot tell that from looking at the ParameterConfidenceIntervalTable.

The estimated variances are the same:

nlm1["EstimatedVariance"]
(* 0.0037219 *)
nlm2["EstimatedVariance"]
(* 0.0037219 *)

The $AIC_c$ values are identical:

nlm1["AICc"]
(* -543.167 *)
nlm2["AICc"]
(* -543.167 *)

The predictions are identical:

ListPlot[{Transpose[{nlm1["PredictedResponse"], 
nlm2["PredictedResponse"]}], {{-45, -45}, {45, 45}}}, Joined -> {False, True}]

For m1 the numerical instabilities can be also seen by examining the parameter correlation matrix:

nlm1["CorrelationMatrix"]//MatrixForm

When the values in the off-diagonal are this close to 1 or -1, that suggests that there will either be numerical instability in the estimates and/or over-parameterization of the model for the particular data set.

The correlation matrix for m2 is also not without issues:

nlm2["CorrelationMatrix"]//MatrixForm

Addition:

Given the values of the parameters for one model, one can find the parameters for the other model that give identical predictions. Consider the following slight deviation from the original definitions:

m1 = a Cos[2 π f t] + b Sin[2 π f t];
m2 = c Cos[2 π f t + ϕ];

(Only m2 is changed: a is replaced with c.) Then

{a, b} = c {Cos[ϕ], -Sin[ϕ]}

and