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.
Highlights
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)
We analyse the degree of persistence exhibited by complexvalued wind measurements, i.e. series which have both wind speed and direction, using new complex-valued Hurst estimation methodology we propose here
Even when only considering real-valued data, Knight et al (2017) show that methods designed for regularly spaced data often fail to deliver a robust estimate if the time series is subject to missing observations or has been sampled irregularly, and in this context they propose a lifting-based approach for Hurst estimation
Summary
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). 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). Complex-valued processes, both proper (circular) and improper (noncircular), are relevant across fields such as oceanography and geophysics (Adali et al 2011; Sykulski et al 2017), where data are typically difficult to acquire and will frequently suffer from omissions/ missingness or be irregularly sampled (see, e.g. Fig. 1). We note here that data from other scientific areas may benefit from analysis with our proposed methodology; see Sect. 6 for further discussion
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