  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.
