Derivative Principal Component Analysis for Representing the Time Dynamics of Longitudinal and Functional Data

TL;DR

This paper introduces a nonparametric derivative principal component analysis method for better modeling and representing derivatives of longitudinal data, improving accuracy especially with sparse or dense data, and demonstrating enhanced predictive power.

Contribution

It develops a novel derivative PCA approach that explicitly models derivatives of functional data, with proven consistency and applicability to irregularly spaced data.

Findings

More accurate derivative recovery compared to traditional PCA methods.

Enhanced prediction of wheat protein content using derivative scores.

Effective for both sparse and dense longitudinal data.

Abstract

We propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen--Lo\`eve expansion of the unobserved derivatives and leads to the notion of derivative principal component analysis, which complements functional principal component analysis, one of the most popular tools of functional data analysis. The proposed derivative principal component scores can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studies, as well as for functional data which are densely measured. Novel consistency results and asymptotic convergence rates for the proposed estimates of the derivative principal component scores and other components of the model are derived under a unified scheme for sparse or dense observations and mild conditions. We compare the proposed representations for derivatives with alternative approaches in simulation settings and also in a wallaby growth curve application. It emerges that representations using the proposed derivative principal component analysis recover the underlying derivatives more accurately compared to principal component analysis-based approaches especially in settings where the functional data are represented with only a very small number of components or are densely sampled. In a second wheat spectra classification example, derivative principal component scores were found to be more predictive for the protein content of wheat than the conventional functional principal component scores.