Generalized Singular Spectrum Time Series Analysis

This paper is a study of continuous time Singular Spectrum Analysis (SSA). We show that the principal eigenfunctions are solutions to a set of linear ODEs with constant coefficients. We also introduce a natural generalization of SSA, constructed using local (Lie-) transformation groups. The time translations used in standard SSA is a special case. The eigenfunctions then satisfy a simple type of linear ODE with time dependent coefficient, determined by the infinitesimal generator of the transformation group. Finally, more general one parameter mappings are considered. Singular Spectrum Analysis (SSA) is a relatively recent method for nonlinear time series analysis. The original idea behind SSA was first presented by Broomhead and King [1], in the context of time series embedding. During the last decade this technique has been very successful and has become a standard tool in many different scientific fields, such as climatic [2], meteorological [3], and astronomical [4] time series analysis. For introductions to the SSA technique, see e.g., [5, 6]. In practical applications, a time series is a result of a sampled measurement, and is therefore discrete. This paper is a theoretical study of SSA, and it is therefore more natural to consider the general case of continuous time. We start the paper by a (semi) formal expansion of the SSA procedure to continuous time. Let f(t) be a function representing a continuous time signal on an interval ΩT = [0, T ]. We assume that f(t) ∈ L2(ΩT ), where L2(ΩT ) is a Hilbert space with an inner product defined as (f, g)ΩT = ∫ ΩT dtf(t)g(t) and a norm ‖f‖Ω = √ (f, f)Ω. We define a trajectory function X(ξ, t) X(ξ, t) = f(ξ + t) (1) where t ∈ ΩW = [0,W ] and ξ ∈ ΩT−W = [0, T − W ]. The parameter W is fixed and referred to as the window length. By construction X(ξ, t) ∈ L2(ΩT−W ) × L2(ΩW ) and the norm is defined as ‖X‖2 = ∫ ΩT−W dξ ∫ ΩW dtX(ξ, t)X(ξ, t). A Schmidt decomposition (the continuous equivalent to a Singular Value Decomposition of a matrix) of the trajectory function is defied as