Semiparametric Functional Factor Models with Bayesian Rank Selection

Functional data are frequently accompanied by parametric templates that describe the typical shapes of the functions. Although the templates incorporate critical domain knowledge, parametric functional data models can incur significant bias, which undermines the usefulness and interpretability of these models. To correct for model misspecification, we augment the parametric templates with an infinite-dimensional nonparametric functional basis. Crucially, the nonparametric factors are regularized with an ordered spike-and-slab prior, which implicitly provides consistent rank selection and satisfies several appealing theoretical properties. This prior is accompanied by a parameter-expansion scheme customized to boost MCMC efficiency, and is broadly applicable for Bayesian factor models. The nonparametric basis functions are learned from the data, yet constrained to be orthogonal to the parametric template in order to preserve distinctness between the parametric and nonparametric terms. 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 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.