Novel Lifting Scheme Constructions for Data Collected from Network Edges

As an emerging research area, analysing functions that arise from network (graph) structures attracts researchers in both statistics and signal processing communities. While current literature is rich when data are collected from the graph vertex space, the data collected from graph edges call for new techniques, and in its turn reaches across many application fields, from traffic networks to neuroscience and hydrology. Wavelets are popular tools for understanding the behaviour of the underlying (edge) functions due to their computational efficiency and robust performance in the presence of discontinuities.

In this thesis, we propose three types of new algorithms that provide a multiscale approach developed through lifting scheme wavelet constructions for data collected from the network edges, useful for data compression and signal denoising. We thoroughly investigate the properties of the proposed algorithms through simulation studies. In addition, we analyse the impact on the method performance of different choices of key quantities, such as prediction/update weights and integrals of the scaling functions.

Finally, we propose non-decimated versions for all of the proposed methods, and these are shown to have a significant impact in the context of denoising problems posed in this thesis. We illustrate the advantages of our non-decimated lifting constructions on a simulated dataset previously introduced in the literature as well as on a new hydrological real dataset. The findings and the comparisons with existing results reported in the literature reinforce the superiority of our techniques as well as the wide-reach of our three method types when considering the extent to which data is available, e.g. full or partial information on edge lengths.