External Seminar: Marina Knight (York)

[Statistics & Probability Seminar]

Complex-valued wavelet lifting and some applications

Signals with irregular sampling structures naturally arise in many fields. In applications such as nonparametric regression, spectral decomposition and long-memory estimation, classical methods often assume a regular sampling pattern, thus cannot be applied without prior data processing. Such sampling structures and subsequent data processing have been shown to cause bias in a range of analysis tasks, including Hurst estimation. On the other hand, many physical processes, such as wind series, are more naturally expressed as complex-valued data to represent e.g. speed and directional (or magnitude and phase) information. Based on the wavelet lifting scheme, this work proposes two new complex-valued analysis techniques able to operate on real- and complex-valued signals and naturally accommodating irregular data sampling. As our new lifting scheme produces complex-valued wavelet coefficients, it provides an alternative to the Fourier transform for irregular designs, allowing for coherence and phase quantification. We also establish attractive theoretical properties that justify a new approach for long-memory intensity estimation for complex-valued processes and demonstrate the potential of this flexible methodology.

Several R packages accompany this work – CNLTreg, CNLTtsa (both on CRAN) and CLiftLRD (release on CRAN pending upon manuscript publication).