基于流形上时间自适应的高斯过程回归新方法
Time-adaptive functional Gaussian Process regression
- 利用流形等距变换下的协方差核不变性，构建时间可变谱的函数回归模型
- 通过截断低频特征实现降维，在小样本下仍具良好预测性能
- 适用于时空随机场中的函数型数据建模，适合高维动态系统分析
本文提出一种在流形上基于经验贝叶斯框架的函数型高斯过程回归新方法，适用于时空随机场场景。通过紧致高斯测度在可分希尔伯特空间中的理论，结合流形等距群下协方差核的不变性，将这些测度与无穷乘积高斯测度关联，依赖于流形上拉普拉斯-贝尔特拉米算子的特征函数。所涉及的时间可变角谱成为实现该回归方法降维的关键工具，采用根据函数样本量设计的截断方案。模拟研究与合成数据应用验证了所提函数回归预测器在有限样本及渐近条件下的性质。
原文摘要 · Abstract (English)
This paper proposes a new formulation of functional Gaussian Process regression in manifolds, based on an Empirical Bayes approach, in the spatiotemporal random field context. We apply the machinery of tight Gaussian measures in separable Hilbert spaces, exploiting the invariance property of covariance kernels under the group of isometries of the manifold. The identification of these measures with infinite-product Gaussian measures is then obtained via the eigenfunctions of the Laplace-Beltrami operator on the manifold. The involved time-varying angular spectra constitute the key tool for dimension reduction in the implementation of this regression approach, adopting a suitable truncation scheme depending on the functional sample size. The simulation study and synthetic data application undertaken illustrate the finite sample and asymptotic properties of the proposed functional regression predictor.
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