Allen, M.R., and Smith, L.A., J. Climate, 9: 3373-3404, 1996.
Abstract
Singular systems (or Singular Spectrum) Analysis, SSA, was originally proposed for noise reduction in the analysis of experimental data and is now becoming widely used to identify intermittent or modulated oscillations in geophysical and climatic time-series. Progress has been hindered by a lack of effective statistics tests to discriminate between potential oscillations and anything but the simplest form of noise, i.e., "white" (independent, identically distributed) noise, in which power is independent of frequency. We show how the basic formalism of SSA provides a natural test for modulated oscillations against an arbitrary "coloured noise" null-hypothesis. This test, Monte Carlo SSA, is illustrated using synthetic data in three situations: (i) where we have prior knowledge of the power-spectral characteristics of the noise, a situation expected in some laboratory and engineering applications, or when the "noise" against which we are testing the data consists of the output of an independently-specified model, such as a climate model; (ii.) where we are testing a simple hypothetical noise model, viz. that the data consists only of white or coloured noise; and (iii) where we are testing a composite hypothetical noise model, assuming some deterministic components have already been found in the data, such as a trend or annual cycle, and we wish to establish whether the remainder may be attributed to noise. We examine two historical temperature records and show that the strength of the evidence provided by SSA for interannual and interdecadal climate oscillations in such data has been considerably over-estimated. In contrast, multiple inter- and sub-annual oscillatory components are identified in an extended Southern Oscillation Index at a high significance level. We explore a number of variations on the Monte Carlo SSA algorithm, and note that it is readily applicable to multivariate series, covering standard Empirical Orthogonal Functions (EOFs) and Multi-channel SSA.