Multivariate Functional Regression and Selection

Abstract

With the pervasiveness of sensor data, real-time physiological signals and behavioral data are often collected in many biomedical studies. This thesis is motivated by data collected from a tri-axis accelerometer ActiGraph GT3X, a device that measures acceleration in the 3-D directions with a sampling frequency of 30-100 Hz. The central task is to relate this multivariate functional quantity with various scalar health outcomes of interest in the presence of other scalar covariates. In the first project, we propose a new methodological framework of semi-parametric regression models that allow the study of a non-linear relationship between a scalar response and multiple functional predictors in the presence of scalar covariates. The proposed methodology is termed as MFRS (Multivariate Functional Regression and Selection). Utilizing functional principal components analysis (FPCA) and least squares kernel machine methods (LSKM), we substantially extend the classical semiparametric regression model of scalar responses on scalar predictors, in which multiple functional predictors are included in the non-linear model. Regularization is established for feature selection in the setting of reproducing kernel Hilbert spaces. The proposed method enables us to perform simultaneous model fitting and variable selection on functional features. For implementation, we propose an effective algorithm to solve related optimization problems, in that iterations take place between both linear mixed models and a variable selection procedure (e.g. sparse group lasso). We show algorithmic convergence results and theoretical guarantees for the proposed methodology. We illustrate its performance through extensive simulation experiments. In the second project we apply our MFRS framework developed in project I to perform a comprehensive mobile health application. This is a study conducted in Mexico City where participants wore an ActiGraph (a tri-axis accelerometer) for seven days with no interruption. We investigated various ways of preprocessing the raw accelerometer data and focused on an important comparative analysis. This comparison concerns methods that treat either the full accelerometer data of seven days as one functional or average the seven days of data into a one day functional. We extend the LSKM framework developed in project I to handle an additive model for multiple functional covariates and compare the extension with our MFRS method given in project I. In the third project we adopt structural principal component analysis (SFPCA) for an alternative analysis of the accelerometer data to that done in project II. SFPCA allows us to treat the functional data of seven days into seven repeats of one day functional. Utilizing the MFRS framework, we demonstrate the benefits of allowing a non-linear and non-additive relationship between health outcomes and repeated functional predictors. Taken together, the second and third projects collectively provide some useful approaches to preprocessing functional data from a mobile device and performing non-linear and non-additive regression with functional covariates. In the fourth project we briefly describe how to extend the MFRS framework to the case where the outcome of interest is binary. In addition, we present a method on how to select import points in the context of kernel logistic regression (KLR) by extending the elastic net via Tikhonov regularization. This project should demonstrate the general approach on how to extend the MFRS framework to other outcomes in the GLM family as well.