Title: Topological Data Analysis for Two Applications in Computational Biology
As the pace and scale and of data availability becomes more significant across all areas of biology, there is a growing need for effective and principled data scientific methods for the analysis of the resulting data. In this talk, I will overview some results and subsequently inspired ongoing projects applied to two important applications in computational biology.
The first is a problem of biomedical imaging informatics for glioblastoma multiforme (brain cancer). In quantitative cancer radiogenomics, the goal is to combine molecular data (such as gene expression) and imaging data for better informed phenotypic analysis. Using topological data analysis, I will show how an appropriately constructed summary statistic of GBM MRIs integrates into functional data analytic methodology for improved prediction of clinical outcome over existing imaging quantifications. I will then overview how such ideas can then be extended to histopathology imaging.
The second extends topological summary statistics to sufficient statistics. Topological data analysis has shown to be an innovation in studying genetic reassortment events in RNA viruses --- a key factor in studying viral mutations, such as those in HIV. I will show how a sufficient statistic for an important invariant in topological data analysis constructed using tropical algebraic geometry allows for the first time parametric statistical analysis of inter- and intra-subtype reassortment events in both HIV and avian influenza. An important question that extends from such work is whether it is possible to define the distribution, and in particular, the mean and variance of such mutations? In such a rapid and dynamically evolving virus such as HIV, the answer to such a question has huge implications on the development of therapies; for other viruses, this may allow for predictions of outbreaks. I will discuss how theory from tropical geometry may lend towards a probabilistic model for such quantities.