Covariance of Replicated Modulated Cyclical Time Series

This paper introduces the class of modulated cyclostationary processes as a new class of non-stationary processes. Cyclostationary processes have dual-frequency spectra whose support lies only on parallel lines in the dual-frequency plane. Such extremely sparse structure does not adequately represent many biological processes. Thus, we propose a model that, in the time domain, modulates the covariance of cyclostationary processes and consequently broadens the support of such processes in the dual-frequency plane. The spectra and the cross-coherence of the proposed modulated cyclostationary process are estimated using multitaper methods. A shrinkage procedure is applied to each trial-specific estimate in order to reduce the estimation risk. Multiple trials of each series are observed. When combining information across trials, we carefully take into account the bias that may be introduced by phase misalignment and the fact that the dual-frequency spectra and cross-coherence across replicates are only “similar” – but not necessarily identical – across replicates. In a simulation study, we illustrate the performance of our estimation method on a model that realistically captures the features observed in the electroencephalogram (EEG) data. An application of the modulated cyclostationary model to the EEG data demonstrates that the proposed model captures statistically significant cross-frequency interactions, that ought to be further examined by neuroscientists. Index Terms Dual frequency coherence; Fourier transform; Harmonizable process; Loève spectrum; Multi-taper estimates; Replicated time series; Spectral Analysis.