Statistics and Its Interface

Volume 16 (2023)

Number 1

The elliptical Ornstein–Uhlenbeck process

Pages: 133 – 146



Adam Sykulski (Lancaster University, Lancaster, United Kingdom)

Sofia Olhede (École polytechnique fédérale de Lausanne, Switzerland)

Hanna Sykulska-Lawrence (University of Southampton, United Kingdom)


We introduce the elliptical Ornstein–Uhlenbeck (OU) process, which is a generalisation of the well-known univariate OU process to bivariate time series. This process maps out elliptical stochastic oscillations over time in the complex plane, which are observed in many applications of coupled bivariate time series. The appeal of the model is that elliptical oscillations are generated using one simple first order stochastic differential equation (SDE), whereas alternative models require more complicated vectorised or higher order SDE representations. The second useful feature is that parameter estimation can be performed semi-parametrically in the frequency domain using the Whittle Likelihood. We determine properties of the model including the conditions for stationarity, and the geometrical structure of the elliptical oscillations. We demonstrate the utility of the model by measuring periodic and elliptical properties of Earth’s polar motion.


oscillations, complex-valued, widely linear, Whittle likelihood, polar motion

Received 28 December 2020

Accepted 29 November 2021

Published 27 July 2022