Trend Filtering for Functional Data

TL;DR

This paper introduces a functional trend filtering method for estimating underlying trends in functional data, extending scalar trend filtering to handle data indexed by time or graphs, with an efficient algorithm and automatic basis selection.

Contribution

It develops a novel functional trend filtering technique with basis adaptation and efficient optimization, filling a gap in flexible functional data trend estimation methods.

Findings

Effective in simulation studies

Successfully applied to real datasets

Automatically adapts the number of basis functions

Abstract

Despite increasing accessibility to function data, effective methods for flexibly estimating underlying functional trend are still scarce. We thereby develop functional version of trend filtering for estimating trend of functional data indexed by time or on general graph by extending the conventional trend filtering, a powerful nonparametric trend estimation technique, for scalar data. We formulate the new trend filtering by introducing penalty terms based on $L_2$-norm of the differences of adjacent trend functions. We develop an efficient iteration algorithm for optimizing the objective function obtained by orthonormal basis expansion. Furthermore, we introduce additional penalty terms to eliminate redundant basis functions, which leads to automatic adaptation of the number of basis functions. The tuning parameter in the proposed method is selected via cross validation. We demonstrate the proposed method through simulation studies and applications to real world datasets.