Nonlinear Singular Spectrum Analysis by Neural Networks

Singular spectrum analysis (SSA), a linear univariate and multivariate time series technique , is essentially principal component analysis (PCA) applied to the time series and additional copies of the time series lagged by 1 to K time steps. Neural network theory has meanwhile allowed PCA to be generalized to nonlinear PCA (NLPCA). In this paper, NLPCA is further extended to perform nonlinear SSA (NLSSA): First, SSA is applied to the data, then the leading principal components of the SSA are chosen as inputs to an NLPCA network (with a circular node at the bottleneck), which performs the NLSSA by nonlinearly combining all the input SSA modes into a single NLSSA mode. This nonlinear spectral technique allows the detection of highly anharmonic oscillations, as illustrated by a stretched square wave imbedded in white noise, which shows NLSSA to be superior to SSA and classical Fourier spectral analysis. NLSSA is also applied to the Southern Oscillation Index (a monthly time series of the equatorial Pacific air pressure conditions from 1866 to 2000), which reveals a first mode with period of 52 months, and a second mode of 39 months.