Nonlinear Laplacian spectral analysis: Capturing intermittent and low-frequency spatiotemporal patterns in high-dimensional data

TL;DR

Nonlinear Laplacian spectral analysis (NLSA) extends classical SSA by incorporating the nonlinear manifold structure of data, enabling detection of complex spatiotemporal patterns like intermittency and rare events in high-dimensional datasets.

Contribution

NLSA introduces a manifold-aware spectral analysis method that efficiently captures nonlinear dynamics in high-dimensional data sets, improving pattern detection over traditional linear approaches.

Findings

NLSA effectively detects intermittent and rare events.

The method estimates the minimal embedding dimension using spectral entropy.

NLSA outperforms classical SSA in nonlinear data scenarios.

Abstract

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.