Generalizations, extensions and applications for principal component analysis

Principal component analysis (PCA) is one of the most important dimension reduction technique. It is widely used in many applications including economics, finance and medical research. In this research, several novel generalizations of PCA are proposed to adapt the technique to more complicated scenarios. In the first project, we propose a principal surface model for manifold-like datasets in 3D space. In the second part, a new concept of graphical intra-class correlation coefficient (GICC) is defined and a Markov Chain Monte Carlo Expectation-Maximization (mcmcEM) algorithm is used for likelihood optimization. In the third part, we propose multilevel binary principal component analysis (MBPCA) models for finding the principal components of multilevel binary dataset. A variational expectation maximization algorithm is used for likelihood optimization

Functional Regression Methods for Densely-Sampled Biomarkers in the ICU

This thesis develops methods for modeling longitudinal predictors by treating them as functional covariates in regression models. First, I introduce Variable-Domain Functional Regression, which extends the generalized functional linear model by allowing for functional covariates that have subject-specific domain widths. I then propose a blueprint for the inclusion of baseline functional predictors in Cox proportional hazards models. Finally, I propose the Historical Cox Model, which introduces a new way of modeling time-varying covariates in survival models by including them as historical functional terms. Methods were motivated by and applied to a study of association between daily measures of the Intensive Care Unit (ICU) Sequential Organ Failure Assessment (SOFA) score and mortality, and are generally applicable to a large number of new studies that record a continuous variables over time

Regional variation models of white matter microstructure

Diffusion-weighted MRI (DW-MRI) is a powerful in vivo imaging technique that is particularly sensitive to the underlying microstructure of white matter tissue in the brain. Many models of the DW-MRI signal exist that allow us to relate the signals we measure to various aspects of the tissue structure, including measures of diffusivity, cellularity and even axon size. From histology, we know that many of these microstructure measures display distinct patterns of variation on length scales greater than the average voxel size. However very few methods exist that use this spatial coherence to inform and guide parameter estimation. Instead, most techniques treat each voxel of data independently. This is particularly problematic when estimating parameters such as axon radius which only weakly influence the signal, as the resulting estimates are noisy. Several methods have been proposed that spatially smooth parameter estimates after fitting the model in each voxel. However if the parameter estimates are very noisy, the underlying trend is likely to be obscured. These methods are also unable to account for spatial coupling that may exist between the various parameters. This thesis introduces a novel framework, the Regional Variation Model (RVM), which exploits the underlying spatial coherence within white matter tracts to estimate trends of microstructure variation across large regions of interest. We fit curves describing parameter variation directly to the diffusion-weighted signals which should capture spatial changes in a more natural way as well as reducing the effects of noise. This allows for more precise estimates of a range of microstructure indices, including axon radius. The resulting curves, which show how microstructure parameters vary spatially through white matter regions, can also be used to detect groupwise differences with potentially greater power than traditional methods