Computation- and space-efficient implementation of SSA

The computational complexity of different steps of the Basic SSA is discussed. It is shown that the use of the general-purpose “blackbox” routines which can be found in packages like LAPACK leads to a huge waste of time since the Hankel structure of the trajectory matrix is not taken into account.

We outline several state-of-the-art algorithms including the Lanczos-based truncated Singular Value Decomposition (SVD) which can be modified to exploit the structure of the trajectory matrix. The key components here are Hankel matrix-vector multiplication and the hankelization operator. We show that both operations can be computed efficiently by means of the Fast Fourier Transform.

The use of these methods yields the reduction of the worst-case computational complexity from $O(N^{3})$ to $O(k N \log{N})$, where $N$ is the series length and $k$ is the number of desired eigentriples.

Keywords

basic SSA, computational complexity, truncated SVD, Hankelization, Fast Fourier transform

2010 Mathematics Subject Classification

65C60, 68Q17

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Published 1 January 2010