Derivative Estimation of Multivariate Functional Data

TL;DR

This paper introduces two novel methods for estimating derivatives of multivariate functional data, enabling better reconstruction and analysis of complex multivariate signals, especially with densely observed data.

Contribution

The paper proposes derivative MFPCA and DMKL methods for multivariate derivative estimation, extending existing univariate approaches to multivariate functional data.

Findings

DMFPCA outperforms DMKL and direct methods in simulations

Methods effectively recover derivatives from densely observed data

Application to coronary data aids in disease pattern classification

Abstract

Existing approaches for derivative estimation are restricted to univariate functional data. We propose two methods to estimate the principal components and scores for the derivatives of multivariate functional data. As a result, the derivatives can be reconstructed by a multivariate Karhunen-Lo\`eve expansion. The first approach is an extended version of multivariate functional principal component analysis (MFPCA) which incorporates the derivatives, referred to as derivative MFPCA (DMFPCA). The second approach is based on the derivation of multivariate Karhunen-Lo\`eve (DMKL) expansion. We compare the performance of the two proposed methods with a direct approach in simulations. The simulation results indicate that DMFPCA outperforms DMKL and the direct approach, particularly for densely observed data. We apply DMFPCA and DMKL methods to coronary angiogram data to recover derivatives of diameter and quantitative flow ratio. We obtain the multivariate functional principal components and scores of the derivatives, which can be used to classify patterns of coronary artery disease.