Identifiability in penalized function-on-function regression models

Regression models with functional covariates for functional
responses constitute a powerful and increasingly important model
class. However, regression with functional data poses challenging
problems of non-identifiability. We describe these identifiability
issues in realistic applications of penalized linear
function-on-function-regression and delimit the set of
circumstances under which they arise. Specifically, functional
covariates whose empirical covariance has lower effective rank than
the number of marginal basis function used to represent the
coefficient surface can lead to unidentifiability. Extensive
simulation studies validate the theoretical insights, explore the
extent of the problem and allow us to evaluate its practical
consequences under varying assumptions about the data generating
processes. Based on theoretical considerations and our empirical
evaluation, we provide easily verifiable criteria for lack of
identifiability and provide actionable advice for avoiding spurious
estimation artifacts. Applicability of our strategy for mitigating
non-identifiability is demonstrated in a case study on the Canadian
Weather data set.