Random fields and geometry with applications to brain imaging

本研究的目的是开发理论和应用的统计方法，用于分析/信号检测的脑映射数据，特别是那些方法相关的随机field.The统计的挑战是开发方法，同时推理，以解决检测显着激活这些图像中的问题。我研究这一问题时使用的几何工具最初是由罗伯特·阿德勒（Robert Adler）在20年前开发的，后来又由基思·沃斯利（Keith Worsley）开发。我在这方面的工作已经导致（i）一个几何工具，称为高斯运动公式，统一和简化了现有的文献中的随机场和（ii）最强的已知结果的精度近似在一般设置。我建议使用这些工具来研究新的非高斯模型的随机场所产生的鞍点型近似以及稳定的随机场。我还建议研究这些近似的行为在高维和它们的关系与大型随机矩阵。这些结果将在非高斯脑成像数据的研究中有应用，以及高维inference.An替代方法，这个问题是使用经验贝叶斯技术，通常被称为错误发现率（FDR）技术。我在这一领域的工作主要集中在FDR的渐近控制以及FDR的有用的汇总统计：未命中率和p值集合的尾部强度。在这个建议中，我建议扩展Genovese和Wasserman的FDR的工作，通过修改的同时推理问题，认识到在脑成像数据中固有的空间结构。对于任何进步的统计方法解决的同时推理问题是有用的脑成像社区，他们必须集成到研究人员的软件环境。本提案中描述的所有统计技术将被集成到NiPy中，这是一个多中心神经成像软件项目，我是其中的主要开发人员之一。

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.