On Singular Spectrum Analysis And Stepwise Time Series Reconstruction

Abstract

This article provides a detailed statistical analysis of a new approach to singular spectrum analysis (SSA). It examines SSA constructed using re-scaled trajectories (RT-SSA) and presents a theoretical analysis of RT-SSA under very general conditions concerning the structure of the observed series. The spectral features of population ensemble models implicit in the large sample properties of RT-SSA are investigated, motivating a new time series modelling methodology based on a stepwise application of RT-SSA. The operation of the theoretical results is illustrated via numerical examples involving trend stationary and difference stationary processes, and a random walk with drift. An analysis of the S&P 500 index also serves as a vehicle to demonstrate the practical impact of the stepwise RT-SSA processing methodology.Note that diagonal averaging is a minimum norm linear idempotent operator in the sense that for any matrix A the Hankel matrix (A) minimizes the Frobenius norm of the approximation error, that is, ‖A−(A)‖ ≤ ‖A−H‖ for all conformable Hankel matrices H, and (A ± H) = (A) ± (H) = (A) ± H. After applying diagonal averaging the resulting Hankel matrices S and E = X − S give the signal-noise decomposition X = S + E of the trajectory matrix, and x(t) = s(t) + e(t), t = 1, … , N, yields the associated SSA signal-noise reconstruction of the original time series. The decomposition constructed via (3) and (4) thus defines a signal-plus-noise specification for x(t) with a k(m + 1) element parameter vector ( 1 , … , k , u ′ 1 , … , u ′ k ) ′ (a combined functional-structural model in the terminology of Kendal and Stuart, 1979, chapter 29).Experimental and empirical evidence indicating that SSA performs well relative to more conventional signal extraction techniques has lead to SSA being employed in various discipline areas including, geophysics, the analysis of electro-encephalograms and DNA micro-arrays, applications to economic and financial time series, and in wileyonlinelibrary.com/journal/jtsa 69 image processing, see for example the references in Jolliffe (2002, chapters 12.1 and 12.2), the contributions to the special edition of Statistics and It's Interface (Volume 3, Number 3, 2010), and for a comparative study of SSA and different techniques of trend evaluation in seasonally adjusted series Alexandrov et al. (2012). Despite this broad range of application, however, to this author's knowledge detailed probabilistic and statistical analyses of the theoretical underpinnings of SSA per se, divorced from any substantive practical question or other contextual information, are lacking in the time series analysis literature. The purpose of this article is to address this lacuna.It is well known from Wold's theorem that a stochastic process x(t) can be represented in a unique way as x(t) = y(t) + z(t) where y(t) and z(t) are subordinate to x(t), mutually uncorrelated, and y(t) is singular while z(t) is regular. 1 The spectral distribution of x(t) equals the sum of that of y(t) and z(t), and the Lebesgue deco...

Comparative Assessment of Hierarchical Clustering Methods for Grouping in Singular Spectrum Analysis

“…Owing to its widespread applications in various fields, this non-parametric method has received much attention in recent years. As evidence, examples of the wide variety of SSA applications can be found in [1][2][3][4][5][6][7][8][9]. A whole and precise detailed summary of the theory and applications of SSA can be found in [10,11].…”

Automatic Grouping in Singular Spectrum Analysis

“…The remarkable features of SSA are that neither a parametric model nor stationarity-type conditions have to be assumed for the time series [1]. The SSA method is attracting considerable interest due to its widespread capabilities and it has many applications in a variety of fields such as medicine [2][3][4][5], biology and genetics [6,7], finance and economics [8][9][10][11][12][13][14][15][16], engineering [17][18][19][20][21][22][23], and other fields [24][25][26]. Whole and precise details on the theory and applications of SSA can be found in [1,[27][28][29].…”