Distinguishing modulated oscillations from coloured noise in multivariate datasets

* Permanent address: Space Science Department, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK and Department of Physics, University of Oxford, UK (e-mail: m.allen16physics.oxford.ac.uk) Abstract. Extended empirical orthogonal functions (EEOFs), alternatively known as multi-channel singular systems (or singular spectrum) analysis (MSSA), provide a natural method of extracting oscillatory modes of variability from multivariate data. The eigenfunctions of some simple non-oscillatory noise processes are, however, also solutions to the wave equation, so the occurrence of stable, wave-like patterns in EEOF/MSSA is not sufficient grounds for concluding that data exhibits oscillations. We present a generalisation of the “Monte Carlo SSA” algorithm which allows an objective test for the presence of oscillations at low signal-to-noise ratios in multivariate data. The test is similar to those used in standard regression, examining directions in state-space to determine whether they contain more variance than would be expected if the noise null-hypothesis were valid. We demonstrate the application of the test to the analysis of interannual variability in tropical Pacific sea-surface temperatures.