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Abstract:
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The statistical analysis of functional magnetic resonance imaging (fMRI) presents many challenges using traditional methodology. The activity of the brain is measured using a blood oxygen level dependent (BOLD) signal, which is broken down as a large number of voxels describing the oxygenated blood flow throughout the entire brain. The individual voxels are not themselves a high dimensional set of variables of interest; rather we wish to examine anatomical regions of the brain using the voxels as proxies. Topological Data Analysis (TDA) allows us to examine the underlying homological features of the data, thus allowing us to analyze these neurological scans in terms of the overall structure instead of treating the data as consisting of a large number of variables. I will discuss the basics of TDA, along with generating persistent homologies using a Morse filtration. This will be followed by examination of homologies generated from fMRI data and how the persistent homology can be used to identify relevant anatomical features of interest.
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