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.

翻译：P样条提供了一种灵活且计算高效的平滑框架，常用于函数型数据的导数估计。在P样条中加入加法惩罚项已被证明能改进导数的估计。我们提出一种方法，将快速协方差估计（FACE）算法与P样条中的加法惩罚相结合。所提方法用于估计函数型数据的协方差导数，这在基于导数的函数型主成分分析（FPCA）中起着重要作用。随后，我们提供了一种算法，用于估计基于导数的FPCA中的特征函数及其对应的得分。为进行比较，我们在模拟中评估了我们的算法与现有函数 \texttt{FPCAder()} 的性能。此外，我们将该算法扩展至多元情形，称为导数多元函数型主成分分析（DMFPCA）。DMFPCA应用于人体运动数据中的关节角度，其中基于导数的得分在区分运动任务方面表现出色。