Functional Manifold Data Analysis

Author:

Kang, Hyun Bin

Graduate Program:

Statistics

Degree:

Doctor of Philosophy

Document Type:

Dissertation

Date of Defense:

July 10, 2018

Committee Members:

Matthew Logan Reimherr, Dissertation Advisor/Co-Advisor
Matthew Logan Reimherr, Committee Chair/Co-Chair
Runze Li, Committee Member
Bing Li, Committee Member
Mark Shriver, Outside Member

Keywords:

Functional Data Analysis
Shape Analysis
Object Oriented Data Analysis
Manifold Learning
Reproducing Kernel Hilbert Space

Abstract:

With rapid advances in data collection technologies, many scientific fields are now obtaining more detailed, complex, and structured data. Utilizing such structures to extract more information is increasingly common in fields such as biology, anthropology, forensic science, and meteorology. A great deal of modern statistical work focuses on developing tools for handling such data. Classic statistical tools such as univariate or multivariate methods are often not suitable for such data and in some situations, and in some cases applying them is not even possible because of data structure. In this dissertation, we propose \textit{Functional Manifold Data Analysis (FMDA)}, a subbranch of Functional Data Analysis (FDA) which often extracts additional information contained in the data structure, to deal with such modern complicated data. FDA is a rapidly developing area of statistics for data which can be naturally viewed as a smooth curves or functions. In FDA, the fundamental statistical unit is now function or shape, not the vector of measurements, and the inherent smoothness in the data can be exploited to achieve greater statistical efficiency than typical multivariate methods. In particular, we present an inferential framework when one of the variables being analyzed is a manifold, and thus we assume we have as many manifolds as we have units. To achieve this, we utilize deformation maps from shape analysis and dimension reduction techniques from manifold learning. In doing so, we are able to represent each manifold as a deformation map, which then can be analyzed using functional data methods. Currently, shape analysis methods that go beyond an analysis of landmarks is a very active area of research, but to date, little has been done in terms of shape-on-scalar regression, thus this dissertation will open up exciting avenues for both shape and functional data analysis. To understand how the scalar covariates affect the manifolds, we propose a manifold-on-scalar regression, which is an extension of function-on-scalar regression in FDA. Different algorithms for estimating the parameter functions in the manifold-on-scalar regression are presented and discussed. The optimality of parameter estimates for function-on-scalar regression over complex domains is also established by finding the minimax lower bounds on the estimation rate and proposing a minimax optimal estimator whose upper bounds match the developed lower bounds.