EOFs of One-Dimensional Cyclostationary Time Series: Computations, Examples, and Stochastic Modeling

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Abstract
Many climatic time series seem to be a mixture of unpredictable fluctuations and changes that occur at a known frequency, as in the case of the annual cycle. Such a time series is called a cyclostationary process. The lagged covariance statistics of a cyclostationary process are periodic in time with the frequency of the nested undulations, and the eigenfunctions are no longer Fourier functions. In this study, examination is made of the properties of cyclostationary empirical orthogonal functions (CSEOFs) and a computational algorithm is developed based on Bloch's theorem for the one-dimensional case. Simple examples are discussed to test the algorithm and clarify the nature and interpretation of CSEOFs. Finally, a stochastic model has been constructed, which reasonably reproduces the cyclostationary statistics of a 100-yr series of the globally averaged, observed surface air temperature field. The simulated CSEOFs and the associated eigenvalues compare fairly with those of the observational data. Abstract Many climatic time series seem to be a mixture of unpredictable fluctuations and changes that occur at a known frequency, as in the case of the annual cycle. Such a time series is called a cyclostationary process. The lagged covariance statistics of a cyclostationary process are periodic in time with the frequency of the nested undulations, and the eigenfunctions are no longer Fourier functions. In this study, examination is made of the properties of cyclostationary empirical orthogonal functions (CSEOFs) and a computational algorithm is developed based on Bloch's theorem for the one-dimensional case. Simple examples are discussed to test the algorithm and clarify the nature and interpretation of CSEOFs. Finally, a stochastic model has been constructed, which reasonably reproduces the cyclostationary statistics of a 100-yr series of the globally averaged, observed surface air temperature field. The simulated CSEOFs and the associated eigenvalues compare fairly with those of the observational data.

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