Dimension reduction is crucial in functional data analysis (FDA). The key tool to reduce the dimension of the data is functional principal component analysis. Existing approaches for functional principal component analysis usually involve the diagonalization of the covariance operator. With the increasing size and complexity of functional datasets, estimating the covariance operator has become more challenging. Therefore, there is a growing need for efficient methodologies to estimate the eigencomponents. Using the duality of the space of observations and the space of functional features, we propose to use the inner-product between the curves to estimate the eigenelements of multivariate and multidimensional functional datasets. The relationship between the eigenelements of the covariance operator and those of the inner-product matrix is established. We explore the application of these methodologies in several FDA settings and provide general guidance on their usability.

翻译：在函数型数据分析中，降维至关重要。降低数据维度的核心工具是函数型主成分分析。现有的函数型主成分分析方法通常涉及协方差算子的对角化。随着函数型数据集规模和复杂性的不断增加，估计协方差算子变得更具挑战性。因此，对估计特征分量的高效方法的需求日益增长。利用观测空间与函数型特征空间的对偶性，我们提出使用曲线间的内积来估计多元及多维函数型数据集的特征元素。本文建立了协方差算子的特征元素与内积矩阵的特征元素之间的关系。我们探讨了这些方法在多种函数型数据分析场景中的应用，并就其适用性提供了通用指导原则。