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I need you to help me understand the Singular Spectrum Analysis algorithm. I already read a lot of articles about the subject but they never answered my questions like what is the mathematical reason for embedding the time series into a trajectory matrix and why applying the SVD gives us access to such trend and periodic and noise functions.

In fact, a lot of people compare SSA to PCA for time series but one could easily explain PCA by saying that we want to find the most relevant direction for explaining the variance of the dataset and thus we are aiming to maximize the variance of the projection of the data on this particular direction we are looking for leading to this well-known optimization and eigenvector and diagonalization problem. But I am absolutely unable to find such an explanation for the SSA algorithm even by trying to explain it myself.

So if someone could help me with that I would really appreciate, in fact, it is really important for me to understand deeply how the things work in order to, for instance, understand what window length to choose or how the eigenvalues are related to the importance of the principal components.

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