What are some examples that require the use of Sinusoidal Regression? I am interested in knowing where do regression models of the form:
$$
\phi(x,A,B,C)=A\sin(kx+B)+C
$$
for a given $k\in\mathbb{R}$ are applied in domains such as engineering and science. What example are there that requires the use of this model?
I had encountered this model recently on math stack exchange. I would suspect the use of this model would be in a domain that heavily requires the use of signal processing but I can't seem to find any answer on the web of any application of sine regression.
These questions on MATHSE are mainly about how to transform the nonlinear sine model to a linear one via substitution in order to be solved numerically (such as using the normal system)
https://math.stackexchange.com/questions/3926007/least-squares-regression-of-sine-wave
https://math.stackexchange.com/questions/2368821/find-a-sinusoidal-regression-equation-for-some-given-data

I have worked in a few different research areas that dealt with unknown periodic trends. In my experience there are almost always better analytic methods to estimating these trends. It turns out that this model, while seeming quite general, actually has too many stringent assumptions. Add to that, the model can't be uniquely identified (offsetting $B$ by $\pi/2$ is equivalent to changing the sign of $A$). Here are the examples I worked on:

*

*Premise: A colony of frogs are observed to migrate from one end of an aquarium to another at an unknown period. Solution: Frogs are affixed with GPS and Euclidean distance from a reference point is mapped over time, forming a sinusoidal trend. The period is estimated with fast Fourier transform (FFT). This is the scientific question and we are done, the amplitude is not of concern.


*Premise: PM10 pollutant levels are observed to fluctuate seasonally, and researchers wish to have a granular-in-time prediction model. Solution: seasonality is already known. Plus pollutant levels vary according to other important regression features. Universal kriging is used. Land-use variables -- the regression component -- include model terms for season, so amplitude is known.


*Premise: Ionic concentration outside the cardiac membrane is measured as a function of time to assess regular heart beat. The goal is to identify irregular heart beat. Solution: Nothing about the trend is known, the assumptions of sinusoidal regression are too stringent to provide any good alignment of depolarization of heartbeat. A first pass box car filter is used to identify ends of action potential and segment the series. The series are aligned based on depolarization, and smoothing splines are used to regress. Then outliers are identified by MSE for ionic concentration, or by duration.
A good reference for this might be Peter Diggle's Time Series A Biostatistical Introduction
