Statistical Analysis of Non-Linear Spatio-Temporal Signals with particular application to Functional Neuroimaging

高分辨率时空数据正变得越来越普遍，为统计学家提供了大量数据集的乐趣和挑战。然而，在这些数据集中所研究的信号往往是复杂和非线性的，既有平稳变化的成分，也有突然变化的成分。在功能磁共振成像和正电子发射断层扫描等应用中尤其如此，在实验时间框架内反复进行三维空间测量。分析这些数据集的常用方法是基于使用大量单变量线性模型。最近，研究表明，在估计信号方面，通过考虑非参数函数平滑方法，特别是如果使用数据中的空间信息，可以取得很大的改进。然而，这种方法目前仅限于简单的空间模型和仅平滑变化的信号。该项目旨在提供一个统计框架，用于分析同一信号内的平滑变化和突变的时空数据，无论这些变化是跨空间还是时间发生的。功能主成分方法学将扩展到包含隐马尔可夫随机场成分。这将允许将数据聚类到空间受限的类似功能区域，或者将结果过程的完整4-D时空模型。特别注意将这种方法应用于功能性神经成像数据。大脑解剖结果需要一个平滑变化的空间模型来应对突变，而神经化学反应和实验挑战可以导致信号在时间上的平滑变化和突变。此外，需要考虑的大量数据要求所确定的所有方法都必须伴随着计算效率高的算法。通过关注常见的实验范例，本项目的目标是提供创新的一般统计方法，这对神经影像学数据的分析具有实际和直接的好处。

High resolution spatio-temporal data is becoming increasingly common, providing statisticians with both the joys and challenges of massive data sets. However the signals under investigation in these data sets are often complex and non-linear with both smoothly and abruptly changing components. This is especially true in applications such as functional magnetic resonance imaging and positron emission tomography, where three dimensional spatial measurements are taken repeatedly during the experimental time frame. Common approaches to the analysis of these data sets are based on the use of mass univariate linear models. Recently, work has shown that great improvements can be made, in terms of estimating the signal, by considering a non-parametric functional smoothing approach, particularly if use is made of the spatial information in the data. However, this methodology is currently limited to simple spatial models and to signals that are only smoothly varying.This projects aims to provide a statistical framework for analysis of spatio-temporal data which is subject to both smooth variations and abrupt changes within the same signal, whether these changes are occurring across space or time. Functional principal component methodology will be extended to incorporate a hidden Markov random field component. This will allow either a clustering of the data into regions of similar function constrained in space, or a fully 4-D spatio-temporal model of the resulting process.Particular attention will be paid to the application of this methodology to functional neuroimaging data. Brain anatomy results in the need for a smoothly changing spatial model subject to abrupt changes while neurochemical reactions and experimental challenges can result in both smoothly varying and abruptly changing signals in time. In addition, the massive amounts of data that need to be considered require that all the methodologies determined must be accompanied by computationally efficient algorithms. By focusing on common experimental paradigms, the goal of this project is to deliver innovative general statistical methodology that is of real and immediate benefit to the analysis of neuroimaging data.