[Paper Review] Multivariate and 2D Extensions of Singular Spectrum Analysis with the Rssa Package

Nina Golyandina, Anton Korobeynikov+2 more|arXiv (Cornell University)|Sep 19, 2013

Statistical and numerical algorithms39 references68 citations

TL;DR

This paper presents the Rssa package's implementation of multivariate and 2D extensions of Singular Spectrum Analysis (SSA), introducing Shaped 2D-SSA for non-rectangular image analysis. It leverages efficient Hankel matrix operations via FFT and provides a unified framework for MSSA, 2D-SSA, and ShSSA, with R code examples demonstrating decomposition, forecasting, and signal reconstruction across time series and images.

ABSTRACT

Implementation of multivariate and 2D extensions of Singular Spectrum Analysis (SSA) by means of the R-package Rssa is considered. The extensions include MSSA for simultaneous analysis and forecasting of several time series and 2D-SSA for analysis of digital images. A new extension of 2D-SSA analysis called Shaped 2D-SSA is introduced for analysis of images of arbitrary shape, not necessary rectangular. It is shown that implementation of Shaped 2D-SSA can serve as a base for implementation of MSSA and other generalizations. Efficient implementation of operations with Hankel and Hankel-block-Hankel matrices through FFT is suggested. Examples with code fragments in R, which explain the methodology and demonstrate the proper use of Rssa, are presented.

Motivation & Objective

• To provide a comprehensive guide to the Rssa package for multivariate and multidimensional SSA applications.
• To introduce Shaped 2D-SSA (ShSSA) as a generalization enabling analysis of non-rectangular images, such as circular or irregularly shaped arrays.
• To demonstrate efficient implementation of Hankel and Hankel-block-Hankel matrix operations using the fast Fourier transform (FFT) for improved computational performance.
• To unify the theoretical and algorithmic framework of MSSA, 2D-SSA, and ShSSA under a common SSA scheme with specialized embedding operators.
• To support practical use through detailed R code examples for decomposition, forecasting, and visualization in time series and image processing.

Proposed method

• Adapt the general SSA framework—trajectory matrix construction, SVD, grouping, and reconstruction—for multivariate time series (MSSA), 2D arrays (2D-SSA), and arbitrary-shaped arrays (ShSSA).
• Use window-based embedding to generate trajectory matrices with Hankel, stacked Hankel, or quasi-Hankel structures from input data.
• Apply singular value decomposition (SVD) to the trajectory matrix to extract principal components and signal subspaces.
• Implement efficient SVD and matrix operations using the fast Fourier transform (FFT) to accelerate computations on large Hankel and block-Hankel matrices.
• Develop a fast rank-one hankelization algorithm to reconstruct time series or image components from grouped matrices.
• Formulate vector and row forecasting via subspace-based prediction using shift matrices derived from pseudo-inverses (P†P), enabling recursive forecasting with low-rank updates.

Experimental results

Research questions

• RQ1 How can singular spectrum analysis be extended to handle multivariate time series and 2D image data within a unified computational framework?
• RQ2 What is the role of the embedding operator in adapting SSA to different data types, such as multivariate series and non-rectangular images?
• RQ3 How does the Shaped 2D-SSA method enable the analysis of images with arbitrary shapes beyond rectangular grids?
• RQ4 What computational optimizations, particularly using FFT, can be applied to accelerate the SVD and reconstruction steps in multidimensional SSA?
• RQ5 To what extent can the ShSSA framework serve as a foundational implementation for other SSA extensions, such as MSSA and 2D-SSA?

Key findings

• Shaped 2D-SSA enables effective decomposition and analysis of images with non-rectangular shapes, such as circular or irregularly shaped arrays, by generalizing the trajectory matrix construction.
• The implementation of ShSSA provides a unifying base for MSSA and 2D-SSA, demonstrating that all multidimensional SSA extensions can be viewed as special cases of a single, generalized algorithm.
• Efficient computation of Hankel and Hankel-block-Hankel matrices via FFT reduces the computational complexity of SVD and reconstruction, significantly improving performance for large datasets.
• The fast vector and row forecasting algorithms using pseudo-inverse-based shift matrices (P†P) achieve high-speed recursive forecasting with O(r²) operations per step, where r is the signal subspace dimension.
• The Rssa package offers a consistent, high-level interface across all SSA extensions, with R code examples that illustrate the full workflow from data input to decomposition, forecasting, and visualization.
• Theoretical correctness of the fast forecasting algorithm is proven by showing equivalence to standard projection-based forecasting, ensuring reliable and accurate predictions.