Regression Models Using Shapes of Functions as Predictors

TL;DR

This paper introduces the elastic functional regression model (EFRM), which integrates phase separation within the regression process to improve prediction robustness when using functional predictors with phase variability.

Contribution

The paper proposes a novel elastic functional regression model that performs phase separation inside the regression, enhancing robustness and prediction accuracy over traditional methods.

Findings

EFRM improves prediction accuracy on gait, NMR, and stock market datasets.

The model is invariant to phase variability in predictors.

EFRM outperforms traditional scalar-on-function regression methods.

Abstract

Functional variables are often used as predictors in regression problems. A commonly-used parametric approach, called {\it scalar-on-function regression}, uses the $\ltwo$ inner product to map functional predictors into scalar responses. This method can perform poorly when predictor functions contain undesired phase variability, causing phases to have disproportionately large influence on the response variable. One past solution has been to perform phase-amplitude separation (as a pre-processing step) and then use only the amplitudes in the regression model. Here we propose a more integrated approach, termed elastic functional regression model (EFRM), where phase-separation is performed inside the regression model, rather than as a pre-processing step. This approach generalizes the notion of phase in functional data, and is based on the norm-preserving time warping of predictors. Due to its invariance properties, this representation provides robustness to predictor phase variability and results in improved predictions of the response variable over traditional models. We demonstrate this framework using a number of datasets involving gait signals, NMR data, and stock market prices.