Wavelet thresholding is an increasingly popular method of nonparametric smoothing. Both real-valued and complex-valued Daubechies wavelets exist. However, to date, complex-valued wavelets have attracted little attention in the statistical literature compared to real-valued wavelets. The broad aim of this thesis is to further the application of complex-valued Daubechies wavelets within the wavelet thresholding framework. Much of the previous work that applied complex-valued wavelets focused upon their application to real-valued data. However, complex-valued data exist and arise in multiple scientific areas. This thesis firstly examines how one method of applying complex-valued wavelets performs on complex-valued data before modifying the methodology to allow for native denoising of complex-valued data. A large number of smoothing regimes exist within the wavelet framework, known as 'thresholding rules'. The majority of these thresholding rules have been designed for use in conjunction with real-valued wavelets. One such thresholding rule is known as 'block thresholding' whereby the data is considered in blocks, rather than as individual data points, to allow for correlation between neighbouring data points. A further thresholding method is known as the 'fiducial thresholding' method which attempts to circumvent perceived problems within the Bayesian approach. Within this thesis these two thresholding rules are developed to allow for the application of complex-valued wavelets; the difference in their performance when using real- and complex-valued wavelets is investigated by simulation.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:634529 |
| Date | January 2012 |
| Creators | Klapper, Jennifer Helen |
| Publisher | University of Leeds |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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