Singular Spectrum Analysis of Grenander Processes and Sequential Time Series Reconstruction

This paper provides a detailed analysis of the properties of Singular Spectrum Analysis (SSa) under very general conditions concerning the structure of the observed series. It translates the SSA interpretation of the singular value decomposition of the so called trajectory matrix as a discrete Karhunen-Loeve expansion into conventional principle components analysis, and shows how this motivates a consideration of SSA constructed using standardized or re-scaled trajectories (R-SSA). The asymptotic properties of R-SSA are derived assuming that the true data generating process (DGP) satisfies sufficient regularity to ensure that Grenander's conditions are satisfied. The spectral structure of different population ensemble models implicit in the large sample properties so derived is examined and it is shown how the decomposition of the spectrum into discrete and continuous components leads to an application of sequential R-SSA series reconstruction. As part of the latter exercise the paper presents a generalization of Szego's theorem to fractionally integrated processes. The operation of the theoretical results is demonstrated via simulation experiments. The latter serve as a vehicle to illustrate the numerical consequences of the results in the context of different processes, and to assess the practical impact of the sequential R-SSA processing methodology.