Simultaneous registration and modelling for multi-dimensional functional data

Abstract

Functional data analysis (FDA) has many applications in almost every branch of science, such as engineering, medicine and biology. It aims to cope with the analysis of data in the form of images, curves and shapes. In this thesis, we study the 2D trajectories of hyoid bone movement from X-ray image. Those curves are seen as the observations of multi-dimensional functional data. We rstly develop an all-in-one platform for the data acquisition and preprocessing. However, analyzing the data arises a lot of challenges. In this thesis, we provide solutions to solve some of those challenging problems. We propose one new registration method for handling those raw 2D curves. It basically integrates Generalized Procrusts analysis and self-modelling registration method (GPSM). However, the application reveals that the classi cation followed by registration does not work well. Therefore, we propose two-stage functional models for joint curve registration and classi cation (JCRC). In the rst stage, we use a functional logistic regression model where the aligned curves are estimated from the second stage. The latter uses a nonlinear warping function while modelling the 2D curves, i.e. resolving the misaligned problem and modelling problem simultaneously. This two-stage model takes into account both the scalar variables and the multi-dimensional functional data. For the functional data clustering, we propose mixtures of Gaussian process functional regression with time warping and logistic allocation model, allowing the use of both types of variables and also allowing simultaneous registration and clustering (SRC). A two-level model is introduced. For the data collected from subjects in di erent groups, a Gaussian process functional regression model is used as the rst level model; an allocation model depending on scalar variables is used as the second level model providing further information over the groups. Those three methods, i.e., GPSM, JCRC and SRC are all examined on both simulated data and real data.