Abstract We introduce a nonlinear additive functional principal component analysis (NAFPCA) for vector-valued functional data. This is a generalization of functional principal component analysis and allows the relations among the random functions involved to be nonlinear. The method is constructed via two additively nested Hilbert spaces of functions, in which the first space characterizes the functional nature of the data, and the second space captures the nonlinear dependence. In the meantime, additivity is imposed so that we can avoid high-dimensional kernels in the functional space, which causes the curse of dimensionality. Along with the NAFPCA, we also develop a method of selection of the number of principal components and the tuning parameters that determines the degree of nonlinearity, as well as the asymptotic results for both the fully observed and the incompletely observed functional data. Simulation results show that the new method performs better than functional principal component analysis when the relations among random functions are nonlinear. We apply the new method to online handwritten digits and electroencephalogram (EEG) data sets.

中文翻译：

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函数数据的非线性和可加主成分分析

摘要 我们介绍了向量值函数数据的非线性加性函数主成分分析（NAFPCA）。这是函数主成分分析的推广，并允许所涉及的随机函数之间的关系是非线性的。该方法是通过两个相加嵌套的 Hilbert 函数空间构建的，其中第一个空间表征数据的函数性质，第二个空间捕获非线性相关性。同时，强加了可加性，以便我们可以避免功能空间中的高维内核，这会导致维数灾难。与 NAFPCA 一起，我们还开发了一种选择主成分数量和确定非线性程度的调谐参数的方法，以及完全观察和不完全观察的功能数据的渐近结果。仿真结果表明，当随机函数之间的关系为非线性时，新方法的性能优于泛函主成分分析。我们将新方法应用于在线手写数字和脑电图 (EEG) 数据集。