Identifiability in penalized function-on-function regression models

Regression models with functional responses and covariates
constitute a powerful and increasingly important model class.
However, regression with functional data poses well known and
challenging problems of non-identifiability. This
non-identifiability can manifest itself in arbitrarily large errors
for coefficient surface estimates despite accurate predictions of
the responses, thus invalidating substantial interpretations of the
fitted models. We offer an accessible rephrasing of these
identifiability issues in realistic applications of penalized
linear function-on-function-regression and delimit the set of
circumstances under which they are likely to occur in practice.
Specifically, non-identifiability that persists under smoothness
assumptions on the coefficient surface can occur if the functional
covariate's empirical covariance has a kernel which overlaps that
of the roughness penalty of the spline estimator. Extensive
simulation studies validate the theoretical insights, explore the
extent of the problem and allow us to evaluate their practical
consequences under varying assumptions about the data generating
processes. A case study illustrates the practical significance of
the problem. Based on theoretical considerations and our empirical
evaluation, we provide immediately applicable diagnostics for lack
of identifiability and give recommendations for avoiding estimation
artifacts in practice.