Semiparametric Functional Factor Models with Bayesian Rank Selection

TL;DR

This paper introduces a semiparametric functional factor model that combines parametric templates with nonparametric basis functions, learned from data, to reduce bias and improve interpretability in functional data analysis.

Contribution

It proposes a novel Bayesian approach with orthogonal nonparametric basis functions and an ordered spike-and-slab prior for consistent rank selection and bias correction.

Findings

Reduces bias compared to parametric models

Provides reliable inference on the number of nonparametric terms

Achieves these with minimal additional computational cost

Abstract

Functional data are frequently accompanied by a parametric template that describes the typical shapes of the functions. However, these parametric templates can incur significant bias, which undermines both utility and interpretability. To correct for model misspecification, we augment the parametric template with an infinite-dimensional nonparametric functional basis. The nonparametric basis functions are learned from the data and constrained to be orthogonal to the parametric template, which preserves distinctness between the parametric and nonparametric terms. This distinctness is essential to prevent functional confounding, which otherwise induces severe bias for the parametric terms. The nonparametric factors are regularized with an ordered spike-and-slab prior that provides consistent rank selection and satisfies several appealing theoretical properties. The versatility of the proposed approach is illustrated through applications to synthetic data, human motor control data, and dynamic yield curve data. Relative to parametric and semiparametric alternatives, the proposed semiparametric functional factor model eliminates bias, reduces excessive posterior and predictive uncertainty, and provides reliable inference on the effective number of nonparametric terms--all with minimal additional computational costs.