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

Yueyun Zhu, Steven Golovkine, Norma Bargary, Andrew J. Simpkin

TL;DR

A method which incorporates the fast covariance estimation (FACE) algorithm with an additive penalty in P-splines to estimate derivatives of covariance for functional data, which play an important role in derivative-based functional principal component analysis (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.

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

TL;DR

A method which incorporates the fast covariance estimation (FACE) algorithm with an additive penalty in P-splines to estimate derivatives of covariance for functional data, which play an important role in derivative-based functional principal component analysis (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.
Paper Structure (13 sections, 1 theorem, 23 equations, 6 figures, 1 table, 3 algorithms)

This paper contains 13 sections, 1 theorem, 23 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Fast covariance estimation with an additive penalty in P-splines
  3. Covariance estimation
  4. The derivative of covariance estimation
  5. Derivative-based functional principal component analysis
  6. Derivative-based eigenfunctions and eigenvalues
  7. Derivative-based scores
  8. Derivative multivariate functional principal component analysis
  9. Simulation
  10. Univariate functional data
  11. Multivariate functional data
  12. Application
  13. Discussion

Key Result

Lemma 1

Define $\mathbf{A}_0=(\mathbf{B}^{\mathrm{T}} \mathbf{B})^{-1/2}\mathbf{U} \in \mathbb{R}^{c \times c}$, $\mathbf{A}_s=\mathbf{B} \mathbf{A}_0 \in \mathbb{R}^{J \times c}$, $\mathbf{\Sigma}_s = \left\{\mathbf{I}_c + \lambda_{+} \text{diag} (\mathbf{s})\right\}^{-1} \in \mathbb{R}^{c \times c}$. $\ma

Figures (6)

  • Figure 1: A sample of data simulated from the two functions (upper panel) and their corresponding true derivatives (lower panel).
  • Figure 2: Simulation results for $X^{[1]}$. (1) the dense case without noise (upper); (2) the dense case with noise (middle); (3) the sparse case with noise (lower).
  • Figure 3: Simulation results for multivariate functional data under three settings.
  • Figure 4: An example of ankle dorsiflexion, hip flexion and knee flexion for three participants (top panel) and their fitted angular velocities by DMFPCA (bottom panel).
  • Figure 5: The mean functions (black) plus and minus the first two multivariate eigenfunctions of joint angular velocities (i.e., the first two DMFPCs of joint angle trajectories).
  • ...and 1 more figures

Theorems & Definitions (1)

  • Lemma 1