Complex-valued gaussian process regression for time series analysis

Abstract

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.

Energy consumption forecasting using a stacked nonparametric Bayesian approach

“…We performed the time series GP prediction by applying a single GP on each data sample, as given in Eq. (4)(5)(6)(7)(8)(9). In order to capture the periodic random functions that vary over time, we used an exponential kernel with a periodic kernel as given in Eq.…”

“…GPs are nonparametric Bayesian models which have been employed in a large number of fields for a diverse range of applications [4,32]. The work reported by Swastanto [40] highlights the inflexibility of using parametric models in time series forecasting, thus encouraging the use of nonparametric models.…”

The Gaussian Process Hilbert Transform (GP-HT): Testing the Consistency of Electrochemical Impedance Spectroscopy Data

“…One important point to note is that the GP-HT can be further expanded following the methodology developed herein. This can be done, for example, by constructing new kernels 46 or adding depth to the GP. 47 For this reason, we foresee that this article will serve as a starting point for the development of other probabilistic and GP-inspired HT and KK-based frameworks.…”