Random fields and geometry with applications to brain imaging

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The objective of this research is to develop theoretical and applied statistical methods for the analysis of / signal detection in brain mapping data, specifically those methods related to random fields.The statistical challenge is to develop methods for simultaneous inference to address the problem of detecting significant activation in these images. I have studied this using geometric tools originally developed by Robert Adler twenty years ago, and more recently by Keith Worsley. My work in this area has resulted in (i) a geometric tool known as the Gaussian Kinematic Formula that has unified and simplified much of the existing literature on random fields and (ii) the strongest known results for the accuracy of the approximation in a general setting. I propose to use these tools to study new non-Gaussian models of random fields arising from saddle-point type approximations as well as stable random fields. I also propose to study the behaviour of these approximations in high dimensions and their relations with large random matrices. These results will have applications in the study of non-Gaussian brain imaging data, as well as high-dimensional inference.An alternative approach to this problem is to use empirical Bayes techniques often referred to as False Discovery Rate (FDR) techniques. My work in this area has focused on asymptotic control of the FDR as well as useful summary statistics of FDR: the miss rate, and tail strength of a collection of p-values. In this proposal, I propose to extend the work of Genovese and Wasserman on FDR for random fields by modifying the simultaneous inference problem to one which recognizes the spatial structure inherent in brain imaging data.For any advances in statistical methodology addressing the simultaneous inference problem to be of use to the brain imaging community, they must be integrated into the researcher's software environment. All statistical techniques described in this proposal will be integrated into NiPy, a multi-center neuroimaging software project, of which I am one of the principal developers.