Elastic Handling of Predictor Phase in Functional Regression Models

Functional variables serve important roles as predictors in a variety of pattern recognition and vision applications. Focusing on a specific subproblem, termed scalar-on-function regression, most current approaches adopt the standard L2 inner product to form a link between functional predictors and scalar responses. These methods may perform poorly when predictor functions contain nuisance phase variability, i.e., predictors are temporally misaligned due to noise. While a simple solution could be to prealign predictors as a pre-processing step, before applying a regression model, this alignment is seldom optimal from the perspective of regression. We propose a new approach, termed elastic functional regression, where alignment is included in the regression model itself, and is performed in conjunction with the estimation of other model parameters. This model is based on a norm-preserving warping of predictors, not the standard time warping of functions, and provides better prediction in situations where the shape or the amplitude of the predictor is more useful than its phase. We demonstrate the effectiveness of this framework using simulated and stock market data.