Functional data analysis in an operator-based mixed-model framework

TL;DR

This paper introduces an operator calculus-based mixed-effects model framework for functional data analysis, offering a flexible alternative to penalized likelihood methods and simplifying computational procedures.

Contribution

It develops an operator calculus approach for functional data analysis in mixed models, eliminating the need for functional basis and improving computational stability.

Findings

Operator approximations enable efficient matrix computations.

Semi-explicit, numerically stable formulas with linear complexity are derived.

The approach clarifies the role of boundary conditions in functional analysis.

Abstract

Functional data analysis in a mixed-effects model framework is done using operator calculus. In this approach the functional parameters are treated as serially correlated effects giving an alternative to the penalized likelihood approach, where the functional parameters are treated as fixed effects. Operator approximations for the necessary matrix computations are proposed, and semi-explicit and numerically stable formulae of linear computational complexity are derived for likelihood analysis. The operator approach renders the usage of a functional basis unnecessary and clarifies the role of the boundary conditions.