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Abstract
In this dissertation, I discuss and propose several geostatistical methods for functionalMagnetic Resonance Imaging (fMRI) data. Geostatistics is a branch of applied statistics thatfocuses on providing quantitative descriptions of natural variables distributed in space or intime and space. Nowadays geostatistics is popular in many elds of science such as mining,environmental sciences, remote sensing and ecology. Functional Magnetic Resonance Imaging(fMRI) is a relatively new non-invasive technique for studying the workings of the activehuman brain. To date there has not been much work using geostatistical methods to analyzethe brain in spite of the similarities of data types and questions of interest. Some recentexceptions are Spence et al. (2007), who used the variogram function to nd neighbors ofvoxels of interest and Bowman (2007), who used the empirical variogram to dene the spatialdistance structure. My dissertation topic is applying geostatistical methods more broadly infMRI data analysis.There are three interrelated parts in geostatistics: Classication, Structural analysis, andKriging. My research explores these three parts in detail as they apply to fMRI.In clustering, I use geostatistical methods and sparse principal component analysis toanalyze the fMRI data and establish a special clustering method for fMRI data time series;my results show that both techniques can eectively identify regions of similar activations.A byproduct of my analysis is the nding that masking prior to clustering, as is commonlydone in fMRI, may degrade the quality of the discovered clusters, and I oer an explanationfor this phenomenon.In structural analysis, I rst introduce an alternative point of view of an axial imageof the brain based on the empirical variograms during dierent time points, which gives agood understanding of how the brain reacts to the experimental task. I then deal with thevariogram modeling of the same axial image, and use parametric and nonparametric holeeect models to look at the spatial character of the data. The models I use consider bothphysical and functional relations among the dierent parts of the brain, which distinguishesthem from previous attempts to use variograms in fMRI. I show the eectiveness of the holeeect model compared with the regular monotonical model in describing the structure of thefMRI data.In kriging, I choose ltered kriging as an alternative to spline smoothing to remove themeasurement errors at the observed sites of the data, and maintain temporal consistency bycontrolling the noise to signal ratio of the smoothness { an idea borrowed from the smoothingfunction approach. This proposed new method incorporates combining both spatial andtemporal information of the data into the smoothing procedure and can reduce the noise ofthe data in an intelligent way.