SSA: theory and methodology

Singular spectrum analysis (SSA) is a technique of time series analysis and forecasting. Let us briefly describe the main idea of the basic version of SSA. Let x₁,x₂, ..., x_N be a time series of length N. Given a window length L (1 < L < N), we construct the L-lagged vectors X_i=(x_i,...,x_i+L-1)^T, 1 <= i <= K=N-L+1, and compose them into one matrix X=[X₁:...:X_K] of size L ×K. This matrix is Hankel and has entries x_ij=x_i+j-2. The columns X_j of X, considered as vectors, lie in the L-dimensional space R^L. The singular value decomposition of the matrix X X^T provides us with a collection of L eigenvalues and eigenvectors. A particular combination of a certain number r < L of these eigenvectors determines an r-dimensional subspace in R^L. The L-dimensional data is then projected onto this r-dimensional subspace and the subsequent averaging over the diagonals gives us an approximation to the original series. The reconstructed series satisfies some linear recurrent formula which we may use to forecast the original series.

SSA can be used for smoothing, filtration, noise reduction, extraction of trends of different resolution, extraction of periodicities in the form of modulated harmonics, change-point detection and other tasks, see Book: Analysis of Time Series Structure: SSA and related techniques, Chapman and Hall/CRC, 2001. It can also be used for the analysis and forecasting of multivariate time series and can help in finding causality between series.

SSA software can be found at the website Caterpillar SSA