The Whittle Likelihood for Complex-Valued Time Series

This paper introduces a version of the Whittle likelihood for complex-valued processes. This represents a nontrivial extension of the Whittle likelihood for bivariate real-valued processes, as complex-valued models can capture structure that is only evident by separating negative and positive frequency behaviour. Flexible inference methods for such parametric models are proposed, and the properties of such methods are derived. The methods are applied to oceanographic time series, as a naturally occurring sampled complex-valued time process. We demonstrate how to reduce estimation bias and ameliorate sampling effects semi-parametrically, thus advancing the state-of-the-art in frequency domain modelling and estimation of time series in general. We test the hypothesis that the negative and positive frequency behaviour of the process are independent, or equivalently that the process is isotropic or proper, and show the performance of the methodology on simulated and real-world data.