Random fields and geometry with applications to brain imaging

本研究的目的是发展脑成像数据中/信号检测分析的理论和应用统计方法，特别是与随机场相关的方法。统计上的挑战是开发同时推理的方法，以解决检测这些图像中显著激活的问题。我使用几何工具来研究这个问题，这些工具最初是由罗伯特·阿德勒在20年前开发的，最近是由基斯·沃斯利开发的。我在这一领域的工作产生了(i)一个被称为高斯运动公式的几何工具，它统一和简化了许多关于随机场的现有文献；（ii）在一般情况下近似精度的最强已知结果。我建议使用这些工具来研究由鞍点近似和稳定随机场产生的新的非高斯随机场模型。我还建议研究这些近似在高维中的行为及其与大随机矩阵的关系。这些结果将应用于非高斯脑成像数据的研究，以及高维推理。解决这个问题的另一种方法是使用经验贝叶斯技术，通常称为错误发现率（FDR）技术。我在这个领域的工作主要集中在FDR的渐近控制以及FDR的有用汇总统计：缺失率和p值集合的尾部强度。在本提案中，我建议扩展Genovese和Wasserman关于随机场的FDR的工作，将同时推理问题修改为识别脑成像数据固有的空间结构的问题。对于任何解决同步推理问题的统计方法的进步，都必须将其集成到研究人员的软件环境中，以供脑成像社区使用。本提案中描述的所有统计技术都将集成到NiPy中，NiPy是一个多中心神经成像软件项目，我是该项目的主要开发者之一。

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.