Complex-valued Gaussian Process Regression for Time Series Analysis

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.