  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.
