I have some astronomical data which I know has a sinusoidal component to it of the form
y = Asin(Ωt+Φ). The period of the sinusoid is equal to a sidereal day. So I know the frequency Ω. So, I just need a way to find out optimal values for A and Φ.
Now, since the period of the sinusoid is so huge and we can track a source only to an extent of about 6 hours, I never get data that would have a full cycle of a sine wave in it. most of the cases I might have only 25 percent of it, at the maximum. In this situation where I don't even have a full cycle, what is the best way to estimate the value of the amplitude and phase of the sinusoid?
Best Answer
Write $y(t) = A\sin(\Omega t)\cos(\phi) + A\cos(\Omega t)\sin(\phi)$. Let $A_1 = A\cos(\phi)$, $A_2 = A\sin(\phi)$. Define the variables $w(t) = \sin(\Omega t)$ and $z(t) = \cos(\Omega t)$. Now you can write $$y(t) = \begin{bmatrix}w(t) & z(t)\end{bmatrix}\begin{bmatrix}A_1 \\ A_2\end{bmatrix}$$
The rest is linear regression. To be more clear you can write your datapoints as $$\begin{bmatrix}y_1 \\ \vdots \\ y_n\end{bmatrix} = \begin{bmatrix}w_1 & z_1\\ \vdots & \vdots\\ w_n & z_n \end{bmatrix}\begin{bmatrix}A_1 \\ A_2\end{bmatrix}$$ Using the matrix notation $$Y = XB$$ You want to solve the following minimization problem $$ \min_{B} \lVert Y - XB \rVert = \min_{B} (Y-XB)^\top (Y-XB) $$
If you differentiate the cost function with respect to $B$ and set it to $0$, you will see that $$B = (X^\top X)^{-1}X^\top Y$$.
Having $B$ you can solve for $A$ and $\phi$.