Scalar-on-Shape Regression Models for Functional Data Analysis

TL;DR

This paper introduces Scalar-on-Shape regression models that focus on the shape component of functional data, performing registration during analysis to better capture complex predictor-response relationships.

Contribution

It proposes a novel regression framework that isolates shape information, performs registration during modeling, and introduces new concepts like regression phase and mean.

Findings

Effective in predicting COVID outcomes from daily rate curves.

Outperforms traditional scalar-on-function models in shape-based analysis.

Demonstrates applicability to neuro-anatomical object data.

Abstract

Functional data contains two components: shape (or amplitude) and phase. This paper focuses on a branch of functional data analysis (FDA), namely Shape-Based FDA, that isolates and focuses on shapes of functions. Specifically, this paper focuses on Scalar-on-Shape (ScoSh) regression models that incorporate the shapes of predictor functions and discard their phases. This aspect sets ScoSh models apart from the traditional Scalar-on-Function (ScoF) regression models that incorporate full predictor functions. ScoSh is motivated by object data analysis, {\it, e.g.}, for neuro-anatomical objects, where object morphologies are relevant and their parameterizations are arbitrary. ScoSh also differs from methods that arbitrarily pre-register data and uses it in subsequent analysis. In contrast, ScoSh models perform registration during regression, using the (non-parametric) Fisher-Rao inner product and nonlinear index functions to capture complex predictor-response relationships. This formulation results in novel concepts of {\it regression phase} and {\it regression mean} of functions. Regression phases are time-warpings of predictor functions that optimize prediction errors, and regression means are optimal regression coefficients. We demonstrate practical applications of the ScoSh model using extensive simulated and real-data examples, including predicting COVID outcomes when daily rate curves are predictors.