2012/02/27 by Dimitrios Giannakis, Andrew J. Majda, Giannakis, Dimitrios +1
Mathematics · Chemistry · Computer Science · #Statistical and numerical algorithms #Spectroscopy and Chemometric Analyses #Blind Source Separation Techniques
paper · pdf · doi:10.48550/arxiv.1202.6103
We present a technique for spatiotemporal data analysis called nonlinear\nLaplacian spectral analysis (NLSA), which generalizes singular spectrum\nanalysis (SSA) to take into account the nonlinear manifold structure of complex\ndata sets. The key principle underlying NLSA is that the functions used to\nrepresent temporal patterns should exhibit a degree of smoothness on the\nnonlinear data manifold M; a constraint absent from classical SSA. NLSA\nenforces such a notion of smoothness by requiring that temporal patterns belong\nin low-dimensional Hilbert spaces Vl spanned by the leading l Laplace-Beltrami\neigenfunctions on M. These eigenfunctions can be evaluated efficiently in high\nambient-space dimensions using sparse graph-theoretic algorithms. Moreover,\nthey provide orthonormal bases to expand a family of linear maps, whose\nsingular value decomposition leads to sets of spatiotemporal patterns at\nprogressively finer resolution on the data manifold. The Riemannian measure of\nM and an adaptive graph kernel width enhances the capability of NLSA to detect\nimportant nonlinear processes, including intermittency and rare events. The\nminimum dimension of Vl required to capture these features while avoiding\noverfitting is estimated here using spectral entropy criteria.\n