PhD Dissertation Abstracts: 2009PhD Dissertation Abstracts: 2009
Statistics PhD Alumni 2009:
Ci-Ren Jiang (2009)
ADVISER: Jane-Ling Wang
TITLE: Covariate Adjusted Functional Principal Component Analysis
ABSTRACT: Classical multivariate principal component analysis has been extended to functional data and termed Functional principal component analysis (FPCA), but most existing FPCA approaches do not accommodate covariate information. The goal of this thesis is to develop alternative approaches to incorporate covariate information in FPCA and to develop specific approaches for dynamic positron emission tomography (PET) data. The thesis consists of two projects.
Two approaches are studied in the first project. The first focuses on the conditional distribution of the functional data given the value of a covariate $Z$, thereby leading to a modelling approach where both the mean and covariance functions depend on the covariate $Z$ and time scale. The second approach can be motivated by a marginal approach that pools all the centered functional data together into one single population and thereby average out the influence of the covariate. Both new approaches can accommodate additional measurement errors and functional data sampled at regular time grids as well as sparse longitudinal data sampled at irregular time grids. We develop general asymptotic theory for both approaches and provide numerical support through simulations. The two approaches are also compared numerically through simulations and a data set consisting of the egg-laying trajectories of 567 Mexican Flies.
The covariate adjusted FPCA in the first project is adapted in the second project for Dynamic PET data, which are collected in four dimensions, three spatial and one temporal, and it will be the time dimension that is of particular interest here. We take the viewpoint that the observed PET time-course data at each voxel are generated by a smooth random function measured with additional noise on a time grid. By borrowing information across space and accounting for this pooling through the use of a non-parametric covariate adjustment, it is possible to smooth the PET time course data thus reducing the noise. We found that a multiplicative nonparametric random-effects model more accurately account for the variation in the data. The use of this model to smooth the data then allows subsequent analysis by methods such as Spectral Analysis to be dramatically improved in terms of their mean squared error.
Qian Weng (2009)
ADVISER: Laurel Beckett
TITLE: Modeling Progression of Neurodegenerative Disease with Longitudinal Neuroimaging Data
ABSTRACT: Conventional methods in the study of neurodegenerative disease, such as cognitive function tests, are indirect measures of brain function. The development of neuroimaging techniques enables researchers to visualize brain structural change through magnetic resonance imaging (MRI), or quantify brain activity through positron emission tomography (PET). Studies are beginning to collect serial images on subjects and there is a need for the development of a strategy for modeling change in the brain over time using high-dimensional, spatially structured neuroimaging data. In this dissertation, two methods have been proposed.
The first method is a novel conditional approach that defines a transition model addressing the high dimensionality, spatial temporal correlations and potential measurement error in the data. We first gave a theoretical argument for the asymptotic properties of the parameter estimates based on the exact likelihood, then proposed a near MLE estimation strategy which approximates the likelihood when the calculation becomes infeasible in high dimensional cases. The second approach is a marginal model that adapts the widely used Bayesian hierarchical model in disease mapping to the longitudinal setting and to the brain anatomy, so it can be applied to neuroimaging data. We evaluated the small sample properties of parameter estimates under both models and each model’s robustness under model misspecification through simulations.
We then modeled the progression of metabolism reduction that has been proposed as a tool in diagnosing AD by applying our methods to data from sequential PET scans obtained in a large scale longitudinal imaging study.
Yanhua Zhang (2009)
ADVISER: Jiming Jiang
TITLE: Fence Methods in Model and Moment Condition Selection in Generalized Method of Moments
ABSTRACT: This thesis proposes to use the Fence methods as new procedures for model and moment condition selection for the Generalized Method of Moments (GMM) estimation. Simulation results show that the Fence methods have superior performance than the existing model and moment selection criteria in terms of higher probabilities in selecting a good model and more accurate parameter estimation. Empirical studies on the US stock return also demonstrate that models and moment conditions selected by the Fence method have better performance not only in sample but also out of sample.