Identifiability in penalized function-on-function regression models Mathematik, Informatik und Statistik - Open Access LMU - Teil 03/03

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.