Complex-valued Gaussian Process Regression for Time Series Analysis

TL;DR

This paper introduces a probabilistic complex-valued Gaussian process regression framework for time series analysis, improving estimation of instantaneous amplitude and frequency in signals like brain oscillations.

Contribution

It develops a novel class of complex-valued covariance functions and models real signals as the real part of a complex Gaussian process, enhancing analysis capabilities.

Findings

Better estimation of amplitude and frequency in simulated signals

Incorporation of prior information improves analysis accuracy

Effective analysis of brain oscillation dynamics

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.