Computation- and Space-Efficient Implementation of SSA

TL;DR

This paper presents efficient algorithms for SSA that exploit the Hankel structure of the trajectory matrix, significantly reducing computational complexity from cubic to near-linear-logarithmic time.

Contribution

It introduces structure-aware algorithms for SSA, utilizing FFT-based methods to improve computational efficiency over traditional blackbox routines.

Findings

Reduced SSA computational complexity from O(N^3) to O(k N log(N))

Demonstrated efficient Hankel matrix-vector multiplication using FFT

Proposed modifications to Lanczos-based SVD for structured matrices

Abstract

The computational complexity of different steps of the basic SSA is discussed. It is shown that the use of the general-purpose "blackbox" routines (e.g. found in packages like LAPACK) leads to huge waste of time resources since the special Hankel structure of the trajectory matrix is not taken into account. We outline several state-of-the-art algorithms (for example, Lanczos-based truncated SVD) which can be modified to exploit the structure of the trajectory matrix. The key components here are hankel matrix-vector multiplication and hankelization operator. We show that both can be computed efficiently by the means of 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 series length and k is the number of eigentriples desired.