Least squares regression of sine wave

Question

Let's say that a sine-like function of a fixed frequency and zero-mean can only vary in amplitude and offset.

That is, for variables $a \in R^+$ and $z \in [0..2\pi]$

$$ f(t) = a(\sin(kt + z))$$

(where $k$ is a constant proportional to the fixed frequency)

For some set of $n$ samples $(t_1,y_1),(t_2,y_2)...(t_n, y_n)$ we want to fit $a$ and $z$ such that they minimize:

$$ L(a,z) = \sum(f(t_i)-y_i)^2 $$

If f was a linear function we could just use ordinary least squares regression.

Is there some similar technique for the above function? What approach would you use to minimize $L$ ?

Answer 1

In general because of the presence of periodic functions, this is not the simplest problem in the domain of curve fitting. However, this case is simple because $k$ is not a tunable parameter but a fixed constant.

You have $n$ data points $(t_i,y_i)$ and you want to perform a least square fit based on the model $$y=a \sin(kt+z)$$ Rewrite is as $$y=a \cos (z) \sin (k t)+a \sin (z) \cos (k t)$$ and define $$A=a \cos (z)\qquad B=a \sin (z)\qquad S_i=\sin (k t_i)\qquad C_i=\cos (k t_i)$$ which makes the model to be $$y=A \,S+B \,C$$ which is just a bilinear regression without intercept.

When solved for $(A,B)$, you have $$a^2=A^2+B^2\qquad\qquad\text{and}\qquad\qquad z=\tan ^{-1}\left(\frac{B}{A}\right)$$

I tried an example : using $n=21$ and $k=2$, I generated the following table $$\left( \begin{array}{cc} t & y \\ 0.0 & 42 \\ 0.5 & 121 \\ 1.0 & 89 \\ 1.5 & -25 \\ 2.0 & -115 \\ 2.5 & -99 \\ 3.0 & 8 \\ 3.5 & 108 \\ 4.0 & 109 \\ 4.5 & 10 \\ 5.0 & -98 \\ 5.5 & -115 \\ 6.0 & -27 \\ 6.5 & 87 \\ 7.0 & 121 \\ 7.5 & 44 \\ 8.0 & -73 \\ 8.5 & -123 \\ 9.0 & -59 \\ 9.5 & 59 \\ 10.0 & 124 \end{array} \right)$$

This gives $$A=116.1789094\qquad\qquad\text{and}\qquad\qquad B=41.88715050$$ from which $$a=123.4992808\qquad\qquad\text{and}\qquad\qquad z=0.3460335933$$

In fact the data were generated using $$y_i=\lceil 123.456 \sin (2 t+0.345678)\rceil$$