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Statistics, Geometry and Brain Mapping

Degree(s) New students per year/session 0
Director Keith Worsley
Duration (yrs)
University of Science and Technology of China. Shanghai, China
The geometry in the title is not the geometry of lines and angles but the geometry of topology, shape and knots. For example, galaxies are not distributed randomly in the universe, but they tend to form clusters, or sometimes strings, or even sheets of high galaxy density. How can this be handled statistically? The Euler characteristic (EC) of the set of high density regions has been used to measure the topology of such shapes; it counts the number of connected components of the set, minus the number of `holes’, plus the number of `hollows'. Despite its complex definition, the exact expectation of the EC can be found for some simple models, so that observed EC can be compared with expected EC to check the model. A similar problem arises in functional magnetic resonance imaging (fMRI), where the EC is used to detect local increases in brain activity due to an external stimulus.

We are also interested the analysis of brain shape: does brain shape change with disease, age or gender? Three types of data are now available: 3D binary masks, 2D triangulated surfaces, and trivariate 3D vector displacement data from the non-linear deformations required to align the structure with an atlas standard. Again the Euler characteristic of the excursion set of a random field is used to test for localised shape changes. We extend these ideas to scale space, where the scale of the smoothing kernel is added as an extra dimension to the random field. Extending this further still, we look at fields of correlations between all pairs of voxels, which can be used to assess brain connectivity. Shape data is highly non-isotropic, that is, the effective smoothness is not constant across the image, so the usual random field theory does not apply. We propose a solution that warps the data to isotropy using local multidimensional scaling. We then show that the subsequent corrections to the random field theory can be done without actually doing the warping – a result guaranteed in part by the famous Nash Embedding Theorem.

Finally we shall look in some detail at the statistical analysis of fMRI data. Our proposed method seeks a compromise between validity, generality, simplicity and execution speed. The method is based on linear models with local AR(p) errors fitted via the Yule-Walker equations with a simple bias correction that is similar to the first step in the Fisher scoring algorithm for finding ReML estimates. The resulting effects are then combined across runs in the same session, across sessions in the same subject, and across subjects within a population by a simple mixed effects model. The model is fitted by ReML using the EM algorithm after re-parameterization to reduce bias, at the expense of negative variance components. The residual degrees of freedom are boosted using a form of pooling by spatial smoothing. Activation is detected using Bonferroni, False Discovery Rate, and non-isotropic random field methods for local maxima and spatial extent. We conclude with some suggestions for the optimal design of fMRI experiments.
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