Nonlinear Laplacian spectral analysis：捕获高维数据中的间歇性和低频时空模式
Nonlinear Laplacian spectral analysis: Capturing intermittent and low-frequency spatiotemporal patterns in high-dimensional data
摘要：
我们提出了一种时空数据分析的技术，称为非线性拉普拉斯谱分析（NLSA），广义奇异谱分析（SSA），考虑到复杂的数据集的非线性流形结构。NLSA的关键原则是用于表示时间模式的函数应该在非线性数据流形M上表现出一定程度的平滑性;这是经典SSA中没有的约束。NLSA通过要求时间模式属于由M上的首l个Laplace-Beltrami本征函数所张成的低维Hilbert空间V_l来加强这样的光滑性概念。这些特征函数可以有效地评估在高环境空间维度使用稀疏图论算法。此外，他们提供正交基扩展的线性映射，其奇异值分解导致集的时空模式在逐步精细的分辨率上的数据流形的家庭。M的黎曼测度和自适应的图核宽度增强了NLSA检测重要非线性过程的能力，包括非线性和罕见事件。利用谱熵准则估计了在避免过拟合的同时捕获这些特征所需的V_l的最小维数。
英文摘要：
We present a technique for spatiotemporal data analysis called nonlinear Laplacian spectral analysis (NLSA), which generalizes singular spectrum analysis (SSA) to take into account the nonlinear manifold structure of complex data sets. The key principle underlying NLSA is that the functions used to represent temporal patterns should exhibit a degree of smoothness on the nonlinear data manifold M; a constraint absent from classical SSA. NLSA enforces such a notion of smoothness by requiring that temporal patterns belong in low-dimensional Hilbert spaces V_l spanned by the leading l Laplace-Beltrami eigenfunctions on M. These eigenfunctions can be evaluated efficiently in high ambient-space dimensions using sparse graph-theoretic algorithms. Moreover, they provide orthonormal bases to expand a family of linear maps, whose singular value decomposition leads to sets of spatiotemporal patterns at progressively finer resolution on the data manifold. The Riemannian measure of M and an adaptive graph kernel width enhances the capability of NLSA to detect important nonlinear processes, including intermittency and rare events. The minimum dimension of V_l required to capture these features while avoiding overfitting is estimated here using spectral entropy criteria.
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