Statistical analysis of shape and functional data
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This thesis consists of two topics. First, we present a framework for statistical regression analysis of shape and functional data. Second, we propose a comprehensive Riemannian framework for analyzing white matter fibers. In the first topic, we develop a multivariate regression model when responses or predictors are on nonlinear manifolds, rather than on Euclidean spaces. The nonlinear constraint makes the problem challenging and needs to be studied carefully. By performing principal component analysis (PCA) on tangent space of manifolds at mean, we use principal directions instead in the model. Then, the ordinary regression tools can be utilized. We apply the framework to both shape data (ozone hole contours) and functional data (spectrums of absorbance of meat in Tecator dataset). Specifically, we adopt the square-root velocity representation and parametrization-invariant metric proposed. Experimental results have shown that we can not only perform efficient regression analysis on the non-Euclidean data, but also achieve high prediction accuracy by the constructed model. In the second topic, a quantitative analysis of white matter fibers can be considered as open curves with different physical features (shape, scale, orientation and position). In this topic, we develop a novel comprehensive framework for analyzing, summarizing and clustering whiter matter fibers. We propose a proper Riemannian metric, which is a weighted sum of distances on the product space of shapes and trajectories of functions or symmetric positive definite (SPD) matrices. The metric allows for joint comparison and registration of fibers associated with BOLD signals and DTI data. We apply our framework to detect stimulusrelevant fiber pathways and summarize projection pathways. We also evaluate and compare our method with others on these two real data sets. Experimental results on these two real data sets have shown that we can cluster fiber pathways correctly and summarize projection pathways more effectively. The proposed framework can also be easily generalized to various applications where multi-modality data exist.