Elastic Functional Principal Component Regression

Abstract

We study regression using functional predictors in situations where these functions contain both phase and amplitude variability. In other words, the functions are misaligned due to errors in time measurements, and these errors can significantly degrade both model estimation and prediction performance. The current techniques either ignore the phase variability, or handle it via pre-processing, i.e., use an off-the-shelf technique for functional alignment and phase removal. We develop a functional principal component regression model which has comprehensive approach in handling phase and amplitude variability. The model utilizes a mathematical representation of the data known as the square-root slope function. These functions preserve the $\mathbf{L}^2$ norm under warping and are ideally suited for simultaneous estimation of regression and warping parameters. Using both simulated and real-world data sets, we demonstrate our approach and evaluate its prediction performance relative to current models. In addition, we propose an extension to functional logistic and multinomial logistic regression