ABSTRACT

Multivariate functional data from a complex system are naturally high-dimensional and have a complex cross-correlation structure. The complexity of data structure can be observed as that (1) some functions are strongly correlated with similar features, while some others may have almost no cross-correlations with quite diverse features; and (2) the cross-correlation structure may also change over time due to the system evolution. With this regard, this article presents a dynamic subspace learning method for multivariate functional data modeling. In particular, we consider that different functions come from different subspaces, and only functions of the same subspace have cross-correlations with each other. The subspaces can be automatically formulated and learned by reformatting the problem as a sparse regression. By allowing but regularizing the regression change over time, we can describe the cross-correlation dynamics. The model can be efficiently estimated by the fast iterative shrinkage-thresholding algorithm, and the features of each subspace can be extracted using the smooth multi-channel functional principal component analysis. Some theoretical properties of the model are presented. Numerical studies, together with case studies, demonstrate the efficiency and applicability of the proposed methodology.

摘要

来自复杂系统的多元函数数据自然是高维的，并且具有复杂的互相关结构。可以观察到数据结构的复杂性： (1) 一些函数与相似的特征有很强的相关性，而另一些则可能几乎没有互相关，但特征非常不同；(2) 互相关结构也可能因系统演进而随时间变化。对此，本文提出了一种用于多元函数数据建模的动态子空间学习方法。特别地，我们认为不同的函数来自不同的子空间，并且只有相同子空间的函数之间才具有互相关性。通过将问题重新格式化为稀疏回归，可以自动制定和学习子空间。通过允许但正则化随时间的回归变化，我们可以描述互相关动态。该模型可以通过快速迭代收缩阈值算法进行有效估计，并且可以使用平滑多通道函数主成分分析提取每个子空间的特征。介绍了该模型的一些理论特性。数值研究与案例研究一起证明了所提出方法的效率和适用性。