Covariance estimation for derivatives of functional data using an additive penalty in P-splines

TL;DR

This paper introduces a novel method combining the FACE algorithm with additive penalties in P-splines to improve covariance and derivative estimation in functional data, enhancing derivative-based FPCA and extending to multivariate cases.

Contribution

It proposes a new covariance estimation method with additive penalties in P-splines, integrating FACE, and extends derivative-based FPCA to multivariate functional data.

Findings

Improved derivative covariance estimates using the proposed method.

Enhanced differentiation of locomotion tasks in human movement data.

Competitive performance of derivative scores in multivariate FPCA.

Abstract

P-splines provide a flexible and computationally efficient smoothing framework and are commonly used for derivative estimation in functional data. Including an additive penalty term in P-splines has been shown to improve estimates of derivatives. We propose a method which incorporates the fast covariance estimation (FACE) algorithm with an additive penalty in P-splines. The proposed method is used to estimate derivatives of covariance for functional data, which play an important role in derivative-based functional principal component analysis (FPCA). Following this, we provide an algorithm for estimating the eigenfunctions and their corresponding scores in derivative-based FPCA. For comparison, we evaluate our algorithm against an existing function \texttt{FPCAder()} in simulation. In addition, we extend the algorithm to multivariate cases, referred to as derivative multivariate functional principal component analysis (DMFPCA). DMFPCA is applied to joint angles in human movement data, where the derivative-based scores demonstrate strong performance in distinguishing locomotion tasks.