Domain selection and family-wise error rate for functional data: a unified framework
Tuesday 26th March 2019
Abramowicz, K.; Pini, A.; Schelin, L.; Sjostedt de Luna, S.; Stamm, A.; Vantini, S.
Functional data are smooth, often continuous, random curves, which can be seen as an extreme case of multivariate data with infinite dimensionality. Just as component-wise inference for multivariate data naturally performs feature selection, subset-wise inference for functional data performs domain selection. In this paper, we present a unified null-hypothesis testing framework for domain selection on populations of functional data. In detail, $p$-values of hypothesis tests performed on point-wise evaluations of functional data are suitably adjusted for providing a control of the family-wise error rate (FWER) over a family of subsets of the domain. We show that several state-of-the-art domain selection methods fit within this framework and differ from each other by the choice of the family over which the control of the FWER is provided. In the existing literature, these families are always defined a priori. In this work, we also propose a novel approach, coined threshold-wise testing, in which the family of subsets is instead built in a data-driven fashion. The method seamlessly generalizes to multidimensional domains in contrast to methods based on a-priori defined families. We provide theoretical results with respect to exactness, consistency, and strong and weak control of FWER for the methods within the unified framework.