Multivariate and 2D Extensions of Singular Spectrum Analysis with the Rssa Package

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 basis for implementation of MSSA and other generalizations. Efficient implementation of operations with Hankel and Hankel-block-Hankel matrices through the fast Fourier transform is suggested. Examples with code fragments in R, which explain the methodology and demonstrate the proper use of Rssa, are presented.

[1]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[2]  Konstantin Usevich,et al.  An Algebraic View on Finite Rank in 2D-SSA , 2009 .

[3]  Thomas Kailath,et al.  ESPRIT-estimation of signal parameters via rotational invariance techniques , 1989, IEEE Trans. Acoust. Speech Signal Process..

[4]  Anatoly Zhigljavsky,et al.  Forecasting UK Industrial Production with Multivariate Singular Spectrum Analysis , 2013 .

[5]  Anatoly A. Zhigljavsky,et al.  Singular Spectrum Analysis for Time Series , 2013, International Encyclopedia of Statistical Science.

[6]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[7]  Yang Wang,et al.  Comments on "Estimation of frequencies and damping factors by two-dimensional ESPRIT type methods" , 2005, IEEE Trans. Signal Process..

[8]  Nina Golyandina,et al.  Basic Singular Spectrum Analysis and forecasting with R , 2012, Comput. Stat. Data Anal..

[9]  B. Weare,et al.  Examples of Extended Empirical Orthogonal Function Analyses , 1982 .

[10]  Nina Golyandina,et al.  Variations of singular spectrum analysis for separability improvement: non-orthogonal decompositions of time series , 2013, 1308.4022.

[11]  Kesheng Wu,et al.  Adaptive Projection Subspace Dimension for the Thick-Restart Lanczos Method , 2010, ACM Trans. Math. Softw..

[12]  N. Golyandina,et al.  SSA-based approaches to analysis and forecast of multidimensional time series , 2012 .

[13]  F. Ade,et al.  Characterization of textures by ‘Eigenfilters’ , 1983 .

[14]  Gabor Grothendieck,et al.  Lattice: Multivariate Data Visualization with R , 2008 .

[15]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[16]  Luciano da Fontoura Costa,et al.  Gene Expression Noise in Spatial Patterning: hunchback Promoter Structure Affects Noise Amplitude and Distribution in Drosophila Segmentation , 2011, PLoS Comput. Biol..

[17]  A. Swindlehurst,et al.  Subspace-based signal analysis using singular value decomposition , 1993, Proc. IEEE.

[18]  Konstantin Usevich,et al.  2D-extension of Singular Spectrum Analysis: algorithm and elements of theory , 2010 .

[19]  Victor Y. Pan,et al.  Multivariate Polynomials, Duality, and Structured Matrices , 2000, J. Complex..

[20]  Nina Golyandina,et al.  On the choice of parameters in Singular Spectrum Analysis and related subspace-based methods , 2010, 1005.4374.

[21]  Georg Heinig,et al.  Generalized inverses of Hankel and Toeplitz mosaic matrices , 1995 .

[22]  R. Vautard,et al.  Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series , 1989 .

[23]  Michael Ghil,et al.  ADVANCED SPECTRAL METHODS FOR CLIMATIC TIME SERIES , 2002 .

[24]  Gene H. Golub,et al.  Matrix computations , 1983 .

[25]  Licesio J. Rodríguez-Aragón,et al.  Singular spectrum analysis for image processing , 2010 .

[26]  David M. Holloway,et al.  Measuring Gene Expression Noise in Early Drosophila Embryos: Nucleus-to-nucleus Variability , 2012, ICCS.

[27]  Igor V. Florinsky,et al.  Filtering of digital terrain models by 2D Singular Spectrum Analysis , 2007 .

[28]  Theodore Alexandrov,et al.  A METHOD OF TREND EXTRACTION USING SINGULAR SPECTRUM ANALYSIS , 2008, 0804.3367.

[29]  A. Zeileis,et al.  zoo: S3 Infrastructure for Regular and Irregular Time Series , 2005, math/0505527.

[30]  Anatoly A. Zhigljavsky,et al.  Analysis of Time Series Structure - SSA and Related Techniques , 2001, Monographs on statistics and applied probability.

[31]  Amirhassan Monadjemi,et al.  Towards efficient texture classification and abnormality detection , 2004 .

[32]  Stewart Trickett,et al.  F-xy Cadzow Noise Suppression , 2008 .

[33]  Stéphanie Rouquette-Léveil,et al.  Estimation of frequencies and damping factors by two-dimensional ESPRIT type methods , 2001, IEEE Trans. Signal Process..

[34]  Michael Elad,et al.  ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: Where SENSE meets GRAPPA , 2014, Magnetic resonance in medicine.

[35]  N. Golyandina,et al.  The "Caterpillar"-SSA method for analysis of time series with missing values , 2007 .

[36]  Ian T. Jolliffe,et al.  Empirical orthogonal functions and related techniques in atmospheric science: A review , 2007 .

[37]  Ivan Markovsky,et al.  Software for weighted structured low-rank approximation , 2014, J. Comput. Appl. Math..