Covariance estimation is essential yet underdeveloped for analysing multivariate functional data. We propose a fast covariance estimation method for multivariate sparse functional data using bivariate penalized splines. The tensor‐product B‐spline formulation of the proposed method enables a simple spectral decomposition of the associated covariance operator and explicit expressions of the resulting eigenfunctions as linear combinations of B‐spline bases, thereby dramatically facilitating subsequent principal component analysis. We derive a fast algorithm for selecting the smoothing parameters in covariance smoothing using leave‐one‐subject‐out cross‐validation. The method is evaluated with extensive numerical studies and applied to an Alzheimer's disease study with multiple longitudinal outcomes.

中文翻译：

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多元稀疏函数数据的快速协方差估计

协方差估计对于分析多变量函数数据是必不可少的，但尚未得到充分发展。我们提出了一种使用双变量惩罚样条的多元稀疏函数数据的快速协方差估计方法。所提出方法的张量积 B 样条公式能够对相关协方差算子进行简单的谱分解，并将所得特征函数显式表达为 B 样条基的线性组合，从而极大地促进后续的主成分分析。我们推导出了一种快速算法，用于使用留一主题交叉验证在协方差平滑中选择平滑参数。该方法通过广泛的数值研究进行评估，并应用于具有多个纵向结果的阿尔茨海默病研究。