Analysing evolutionary trees

Imagine having information on how "different" a collection of species of plants or animals are. From this information, we might like to draw "family tree" type diagrams to explore when different species diverged. Such trees are called phylogenies and can be constructed in various ways once we have information on how different the various species are. For a more detailed discussion of phylogenies, see the book "Inferring Phylogenies" by Felsenstein (2004).

Unfortunately, the information we have is often imperfect, so our trees are subject to some doubt. Statistically, we might say that our difference information has been corrupted by noise. Wavelet shrinkage (Donoho & Johnstone, 1994, Biometrika pp. 425-455) is a powerful tool for removing noise when we have regularly spaced data in one or two dimensions. The idea of lifting was developed by Sweldens (1997, Siam J. Math. Anal pp. 511-546) as a means of generalising the wavelet shrinkage method to more irregular point sets but has been used to apply the method to more complex data structure; for example Jansen, Nason & Silverman (2009, J. Roy. Stat. Soc. B, pp 91-127) discuss lifting on tree-like structures. Moreover, the lifting algorithm is strikingly similar to the algorithms used to construct phylogenetic trees. This project will explore the use of lifting to construct phylogenies when our information is corrupted by noise.

No prior knowledge of phylogenies or of wavelets / lifting will be assumed as full training will be given.