Metric for distance of sinewaves

Question

I have time-series data consisting of the sum of 2 sinewaves and my goal is to predict their frequencies and their amplitudes.

I would like to know what are the best distance metrics/loss functions I can use.

You can see from the following plots that the mean-square-error (MSE) is not an appropriate metric. In blue the predicted reconstructed sinewave, in orange the target one. As the title of the subplots, you can find the value of the MSE. In the first figure, the reconstruction is kindly worse, however, the MSE tells a different story.

First:

Second:

Is there a metric that can better emphasize this?

Answer 1

When talking sine waves, the first thing that comes to my mind would be the observe them in the frequency domain. Each of your curves can be described as:

$$y(t) = a_0 + a_1 \cos(2 \pi f_1 t) + b_1 \sin(2 \pi f_1 t) + a_2 \cos(2 \pi f_2 t) + b_2 \sin(2 \pi f_2 t)$$

with $f_k$ denoting the frequencies of the two harmonics, and $a_k$ and $b_k$ the Fourier coefficients (cosine and sine amplitudes at the corresponding frequency). I'd construct the target and the reconstructed vector, $t$ and $r$:

$$t = \left[ a_{0t}, f_{1t}, a_{1t}, b_{1t}, f_{2t}, a_{2t}, b_{2t} \right]$$ $$r = \left[ a_{0r}, f_{1r}, a_{1r}, b_{1r}, f_{2r}, a_{2r}, b_{2r} \right]$$

with frequencies and the Fourier coefficients as the components. I would then simply calculate the Euclidean distance between the two vectors, perhaps with some scaling of the frequency components, because frequency and amplitude are measured in different units.