Complex-valued Gaussian Process Regression for Time Series Analysis
This work addresses the need for more accurate and flexible complex-valued signal estimation in time series analysis, particularly for applications like brain oscillation studies, but it is incremental as it builds on existing Hilbert transform methods.
The authors tackled the problem of constructing synthetic complex-valued signals from real-valued time series by proposing a probabilistic generalization of the Hilbert transform, modeling the signal as the real part of a latent complex-valued Gaussian process with new covariance functions. They showed that this method provides better estimates of instantaneous amplitude and frequency than established approaches, with improvements demonstrated on simulated chirplets and stochastic oscillations, and applied it to analyze non-stationary brain oscillations in magneto-encephalography data.
The construction of synthetic complex-valued signals from real-valued observations is an important step in many time series analysis techniques. The most widely used approach is based on the Hilbert transform, which maps the real-valued signal into its quadrature component. In this paper, we define a probabilistic generalization of this approach. We model the observable real-valued signal as the real part of a latent complex-valued Gaussian process. In order to obtain the appropriate statistical relationship between its real and imaginary parts, we define two new classes of complex-valued covariance functions. Through an analysis of simulated chirplets and stochastic oscillations, we show that the resulting Gaussian process complex-valued signal provides a better estimate of the instantaneous amplitude and frequency than the established approaches. Furthermore, the complex-valued Gaussian process regression allows to incorporate prior information about the structure in signal and noise and thereby to tailor the analysis to the features of the signal. As a example, we analyze the non-stationary dynamics of brain oscillations in the alpha band, as measured using magneto-encephalography.