Sinusoidal model

A good starting value for C can be obtained by calculating the mean of the data. If the data show a trend, i.e., the assumption of constant location is violated, one can replace C with a linear or quadratic least squares fit. That is, the model becomes

${\displaystyle Y_{i}=(B_{0}+B_{1}T_{i})+\alpha \sin(2\pi \omega T_{i}+\phi )+E_{i}}$

or

${\displaystyle Y_{i}=(B_{0}+B_{1}T_{i}+B_{2}T_{i}^{2})+\alpha \sin(2\pi \omega T_{i}+\phi )+E_{i}}$

The starting value for the frequency can be obtained from the dominant frequency in a periodogram. A complex demodulation phase plot can be used to refine this initial estimate for the frequency.

The root mean square of the detrended data can be scaled by the square root of two to obtain an estimate of the sinusoid amplitude. A complex demodulation amplitude plot can be used to find a good starting value for the amplitude. In addition, this plot can indicate whether or not the amplitude is constant over the entire range of the data or if it varies. If the plot is essentially flat, i.e., zero slope, then it is reasonable to assume a constant amplitude in the non-linear model. However, if the slope varies over the range of the plot, one may need to adjust the model to be:

${\displaystyle Y_{i}=C+(B_{0}+B_{1}T_{i})\sin(2\pi \omega T_{i}+\phi )+E_{i}}$

That is, one may replace α with a function of time. A linear fit is specified in the model above, but this can be replaced with a more elaborate function if needed.