Elastic functional principal component regression

Abstract

We study regression using functional predictors in situations where these functions contains 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 preprocessing, that is, use an off‐the‐shelf technique for functional alignment and phase removal. We develop a functional principal component regression model which has a 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 double-struckL2 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.

Simultaneously assessing amplitude and temporal effects in biomechanical trajectories using nonlinear registration and statistical nonparametric mapping

“…This study has demonstrated how amplitude-phase separation (Tucker et al, 2013;Lee and Jung, 2017;Tucker et al, 2019) can be used, along with a single hypothesis test, to simultaneously analyze amplitude and timing effects in biomechanical trajectories. This approach requires that (i) amplitude effects are suitably isolated in nonlinearly registered trajectories (Fig.…”

“…There has been extensive recent work on amplitude-phase separation and analysis (Tucker et al, 2013;Marron et al, 2015;Lee and Jung, 2017;Tucker et al, 2019) including hypothesis testing (Henning and Srivastava, 2016). These approaches consider both amplitude, in the form of registered trajectories (Fig.…”

Registration for Exponential Family Functional Data

“…To provide a frame of reference, we compare our approach with two approaches based on the SRSF framework, both of which are implemented in the time_warping() function in the fdasrvf package (Tucker, ). Both implementations use smoothed versions of the binary data but use different optimization methods.…”

“…The Fisher–Rao metric has been the basis for several recent approaches to registration, some which compute parameter values using dynamic programming (Srivastava et al, ; Wu and Srivastava, ), and others which use Riemannian optimization (Huang et al, ). Many of the SRSF‐based approaches are implemented in the fdasrvf package (Tucker, ).…”

Analysis on the Influence of China’s Energy Consumption on Economic Growth

“…In the paper's analysis on the influences of China's energy on economic growth, the independent variables have the multicollinearity, so the paper builds the regression model using the principal component regression method. The principal component regression method can deal with the multicollinearity and build the reasonable regression model practically and feasibly, and many scholars adopt the method to solve various problems [47][48][49]. Studies have proved that the principal component regression not only eliminates the multicollinearity but also improves modeling precision significantly.…”