Recursive spectral analysis of natural time series based on eigenvector matrix perturbation for online applications

Singular spectrum analysis (SSA) is a well-studied approach in signal processing. SSA has originally been designed to extract information from short noisy chaotic time series and to enhance the signal-to-noise ratio. SSA is good for offline applications; however, many applications, such as modelling, analysis, and prediction of time-varying and non-stationary time series, demand for online analysis. This study introduces a recursive algorithm called recursive SSA as a modification to regular SSA for dynamic and online applications. The proposed method is based on eigenvector matrix perturbation approach. After recursively calculating the covariance matrix of the trajectory matrix, R-SSA updates the eigenvalues and eigenvectors for new samples by considering the effect of the new sample as perturbation in the covariance matrix and its singular value decomposition. The eigenvalues and eigenvectors adapt simultaneously to track their true values as would be calculated from the current covariance matrix. Analysis of two well-known chaotic time series: Mackey-Glass and Lorenz chaotic time series and two natural time series: Darwin sea-level pressure and Sunspot number as non-stationary processes are considered in this study to examine the performance of the proposed recursive method. The obtained results depict the power of the proposed method in online spectral analysis of non-linear time-varying systems.