Identifiability in Penalized Function-on-Function Regression Models

• Activity Number: 474
• Type: Topic Contributed
• Date/Time: Wednesday, August 7, 2013 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Sonja Greven*+ and Fabian Scheipl
• Companies: Ludwig-Maximilians-Universität München and Ludwig-Maximilians-Universität München
• Keywords: functional regression model ; identifiability ; penalized regression ; penalizes splines ; covariance operator

Abstract

Regression models for functional responses with functional covariates constitute a powerful and increasingly important model class. However, they also pose well known and challenging problems of non-identifiability. 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 arise in practice. Specifically, functional covariates whose empirical covariance has a kernel that overlaps that of the roughness penalty of the spline estimator can lead to non-identifiability. 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 immediately applicable diagnostics for lack of identifiability and offer practicable advice for avoiding spurious estimation artifacts. The practical relevance of these issues and the applicability of our strategy for mitigating them is demonstrated in a case study on the well-known Canadian Weather data set.