  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.
