Geometric and topological methods in statistics

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Statistics is challenged by massive multi-dimensional data. The challenges are in representation, visualization, interpretation and analysis, requiring the development of new statistical methodologies, and these advances are dependent on ever more increasing technical sophistication. The main objective of this proposed research program is to examine two such emerging areas: geometric statistics; and, statistical topology. Geometric statistics is concerned with parameter estimation over Riemannian manifolds, where the recovery of the underlying geometric parameter that generates the data is of quantitative statistical interest. Statistical topology, is concerned with recovering topological parameters which are qualitative global features, such as connectedness, or the number of holes, or the existence of obstructions to certain constructions. With the quantitative approach one can obtain qualitative topological information where the pathway between them can be established by using elements of Morse theory. This framework thus provides an abundant enough statistical structure for examining the statistical properties of topological parameters through geometric parameters. One of the areas where we have seen explosive growth of massive multi-dimensional object data is in medical imaging. Due to technological advances in medical scanners, the resolution of voxel data is now extremely detailed. Techniques using geometric statistics have witnessed tremendous proliferation with the need for greater technical sophistication being evident. This research program intends to build upon this and provide contributions both through quantitative approach, as well the qualitative approach. The latter is really frontier, and is only now starting to be appreciated, particularly so for it's ability to clinically discriminate between sub-populations. Other application areas where attention will be given are in bioinformatics, biomechanics, microwave engineering, orthodontal data, and quantum information processing.