2D-Extension of Singular Spectrum Analysis: Algorithm and Elements of Theory

• Published in: Matrix Methods: Theory, Algorithms And Applications pp. 449 - 473
• Main Authors: Golyandina, N. E., Usevich, K. D.
• Format: Book Chapter
• Language: English; Japanese
• Published: WORLD SCIENTIFIC 01.04.2010
• Subjects: Matrices and Applications; Singular Value Decomposition; Kronecker-product SVD; Singular Spectrum Analysis; separability; finite rank; image analysis; Hankel-block-Hankel matrix
• ISBN: 9812836020; 9789812836014; 9812836012; 9814469556; 9789812836021; 9789814469555
• DOI: 10.1142/9789812836021_0029

Singular Spectrum Analysis is a nonparametric method, which allows one to solve problems like decomposition of a time series into a sum of interpretable components, extraction of periodic components, noise removal and others. In this paper, the algorithm and theory of the SSA method are extended to analyse two-dimensional arrays (e.g. images). The 2D-SSA algorithm based on the SVD of a Hankel-block-Hankel matrix is introduced. Another formulation of the algorithm by means of Kronecker-product SVD is presented. Basic SSA notions such as separability are considered. Results on ranks of Hankel-block-Hankel matrices generated by exponential, sine-wave and polynomial 2D-arrays are obtained. An example of 2D-SSA application is presented.