Modeling and estimation of covariance of replicated modulated cyclical time series

This paper introduces the novel class of modulated cyclostationary processes, a class of nonstationary processes exhibiting frequency coupling, and proposes a method of their estimation from repeated trials. Cyclostationary processes also exhibit frequency correlation but have Loève 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 their frequency support in the dual-frequency plane. The spectra and the cross-coherence of the proposed modulated cyclostationary process are first estimated using multitaper methods. A shrinkage procedure is then applied to each trial-specific estimate 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 Loève spectra and cross-coherence across replicates may only be "similar"-but not necessarily identical. 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. The application of the inference methods developed for the modulated cyclostationary model to EEG data also demonstrates that the proposed model captures statistically significant cross-frequency interactions, that ought to be further examined by neuroscientists.