Geostatistical methods for spatio-temporal analysis of fMRI data

Abstract

In this dissertation, I discuss and propose several geostatistical methods for functional Magnetic Resonance Imaging (fMRI) data. Geostatistics is a branch of applied statistics that focuses on providing quantitative descriptions of natural variables distributed in space or in time 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 active human brain. To date there has not been much work using geostatistical methods to analyze the brain in spite of the similarities of data types and questions of interest. Some recent exceptions are Spence et al. (2007), who used the variogram function to ¯nd neighbors of voxels of interest and Bowman (2007), who used the empirical variogram to de¯ne the spatial distance structure. My dissertation topic is applying geostatistical methods more broadly in fMRI data analysis. There are three interrelated parts in geostatistics: Classi¯cation, Structural analysis, and Kriging. My research explores these three parts in detail as they apply to fMRI. In clustering, I use geostatistical methods and sparse principal component analysis to analyze the fMRI data and establish a special clustering method for fMRI data time series; my results show that both techniques can e®ectively identify regions of similar activations. A byproduct of my analysis is the ¯nding that masking prior to clustering, as is commonly done in fMRI, may degrade the quality of the discovered clusters, and I o®er an explanation for this phenomenon. In structural analysis, I ¯rst introduce an alternative point of view of an axial image of the brain based on the empirical variograms during di®erent time points, which gives a good understanding of how the brain reacts to the experimental task. I then deal with the variogram modeling of the same axial image, and use parametric and nonparametric hole e®ect models to look at the spatial character of the data. The models I use consider both physical and functional relations among the di®erent parts of the brain, which distinguishes them from previous attempts to use variograms in fMRI. I show the e®ectiveness of the hole e®ect model compared with the regular monotonical model in describing the structure of the fMRI data. In kriging, I choose ¯ltered kriging as an alternative to spline smoothing to remove the measurement errors at the observed sites of the data, and maintain temporal consistency by controlling the noise to signal ratio of the smoothness { an idea borrowed from the smoothing function approach. This proposed new method incorporates combining both spatial and temporal information of the data into the smoothing procedure and can reduce the noise of the data in an intelligent way.