# Complex-valued Gaussian Process Regression for Time Series Analysis

arXiv: https://arxiv.org/abs/1611.10073 · HTML: https://mlsanity.com/paper/1611.10073/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2016.

## Built on by (most cited first, 3 of 3)

- [Gaussian process deconvolution](https://mlsanity.com/paper/2305.04871/) (2023)
- [Stationarity without mean reversion in improper Gaussian processes](https://mlsanity.com/paper/2310.02877/) (2023)
- [Operator-Informed Gaussian Processes for Complex Helmholtz Wavefields: From Synthetic Benchmarks to In Vivo Brain Elastography](https://mlsanity.com/paper/2607.14193/) (2026)

## Abstract (arXiv)

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.