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

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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.