Reconciling Functional Data Regression with Excess Bases

Abstract: As the development of measuring instruments and computers has accelerated the collection of massive amounts of data, functional data analysis (FDA) has experienced a surge of attention. The FDA methodology treats longitudinal data as a set of functions on which inference, including regression, is performed. Functionalizing data typically involves fitting the data with basis functions. In general, the number of basis functions smaller than the sample size is selected. This paper casts doubt on this convention. Recent statistical theory has revealed the so-called double-descent phenomenon in which excess parameters overcome overfitting and lead to precise interpolation. Applying this idea to choosing the number of bases to be used for functional data, we show that choosing an excess number of bases can lead to more accurate predictions. Specifically, we explored this phenomenon in a functional regression context and examined its validity through numerical experiments. In addition, we introduce two real-world datasets to demonstrate that the double-descent phenomenon goes beyond theoretical and numerical experiments, confirming its importance in practical applications.