  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.
