On Parametric Modelling and Inference for Complex-Valued Time Series

This paper introduces new parametric models for the autocovariance functions and spectra of complex-valued time series, together with novel modifications of frequency-domain parameter inference methods appropriate for such time series. In particular, we introduce a version of the frequency-domain 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 and seismic time series, as examples of naturally occurring sampled complex-valued time processes. We demonstrate how to reduce estimation bias of the Whittle likelihood caused by leakage and aliasing, improving parameter estimation of both real-valued and bivariate/complex-valued processes. We also provide techniques for testing the series propriety or isotropy, as well as procedures for model choice and semi-parametric inference.