Statistical Analysis of Symmetric Positive-definite Matrices

Yuan, Ying

Statistical Analysis of Symmetric Positive-definite Matrices

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March 21, 2019

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Yuan, Ying
- Affiliation: College of Arts and Sciences, Department of Mathematics

Abstract

This dissertation is motivated by addressing the statistical analysis of symmetric positive definite (SPD) matrix valued data, which arise in many applications. Due to the nonlinear structure of such data, it is challenging to apply well-established statistical methods to them. Our goal is to develop statistical models and perform statistical inferences on the Riemannian manifold of the space of SPD matrices. This dissertation has three major parts. In the first part, we develop a local polynomial regression model for the analysis of data with SPD matrix responses on the Riemannian manifold. The independent variable of this model is from Euclidean space. We examine two commonly used metrics including the affine invariant metric and the Log-Euclidean metric on the space of SPD matrices. Under each metric, we develop an associated cross-validation bandwidth selection method, and derive the asymptotic bias, variance, and normality of the intrinsic local constant and local linear estimators and compare their asymptotic mean square errors. Simulation studies are further used to compare the estimators under the two metrics and examine their finite sample performance. In the second part, we develop a functional data analysis framework to model diffusion tensors along fiber bundles as functional responses with a set of covariates of interest, such as age, diagnostic status and gender, in real applications. We propose a statistical model with varying coefficient functions to characterize the dynamic association between functional SPD matrix-valued responses and covariates. We calculate a weighted least squares estimation of the varying coefficient functions under the Log-Euclidean metric in the space of SPD matrices. We also develop a global test statistic to test specific hypotheses about these coefficient functions and construct their simultaneous confidence bands. Simulated data are further used to examine the finite sample performance of the estimated varying coefficient functions. The third part is to develop a varying coefficient model framework under the affine invariant metric. This framework is very similar to that in the second part. However, this metric is more complex than the Log-Euclidean metric, which makes the subsequent estimation of the varying coefficient functions and the theoretical derivations very challenging. Since there is no explicit form formula for the estimators, we developed an optimization method for calculating it. We also derive the asymptotic properties for the estimated coefficient functions, which are important for constructing the simultaneous confidence band and the global test statistic. Moreover, comparisons of the statistical powers of the varying coefficient models under the affine invariant and Log-Euclidean metrics are made by using simulated data.

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