Long memory estimation for complex-valued time series

Abstract

Long memory has been observed for time series across a multitude of fields, and the accurate estimation of such dependence, for example via the Hurst exponent, is crucial for the modelling and prediction of many dynamic systems of interest. Many physical processes (such as wind data) are more naturally expressed as a complex-valued time series to represent magnitude and phase information (wind speed and direction). With data collection ubiquitously unreliable, irregular sampling or missingness is also commonplace and can cause bias in a range of analysis tasks, including Hurst estimation. This article proposes a new Hurst exponent estimation technique for complex-valued persistent data sampled with potential irregularity. Our approach is justified through establishing attractive theoretical properties of a new complex-valued wavelet lifting transform, also introduced in this paper. We demonstrate the accuracy of the proposed estimation method through simulations across a range of sampling scenarios and complex-and real-valued persistent processes. For wind data, our method highlights that inclusion of the intrinsic correlations between the real and imaginary data, inherent in our complex-valued approach, can produce different persistence estimates than when using real-valued analysis. Such analysis could then support alternative modelling or policy decisions compared with conclusions based on real-valued estimation. Keywords Complex-valued time series • Hurst exponent • Irregular sampling • Long-range dependence • Wavelets 1 Introduction Complex-valued time series arise in many scientific fields of interest, for example digital communication and signal processing (Curtis 1985; Martin 2004), environmental series (Gonella 1972; Lilly and Gascard 2006; Adali et al. 2011) and physiology (Rowe 2005). Modelling and analysis of such series in the complex domain is not only natural, but also convenient. In addition, complex-valued time series models are often able to represent more realistic behaviour in observed physical processes; see, for example, Mandic and Goh (2009) and Sykulski et al. (2017). A particular modelling Electronic supplementary material The online version of this article (

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