Giannakis, Dimitrios; Ourmazd, Abbas; Slawinska, Joanna; Zhao, Zhizhen Spatiotemporal pattern extraction by spectral analysis of vector-valued observables. (English) Zbl 1428.37084 J. Nonlinear Sci. 29, No. 5, 2385-2445 (2019); correction ibid. 30, No. 2, 711 (2020). This paper is a self-contained analysis on a data-driven framework for extracting complex spatio-temporal patterns generated by ergodic dynamical systems. More precisely, the authors present a Vector-valued Spectral Analysis (VSA) for spatio-temporal pattern extraction using operator-valued kernels. The main purpose of VSA is to construct a decomposition of the observation map \(\vec{F}\) via an expansion of the form \(\vec{F}\approx \sum_{j=0}^{l-1}\vec{F}_{j}\), \(\vec{F}_j=c_{j}\vec{\phi}_{j}\), where \(c_{j}\) and \(\vec{\phi}_{j}\) are real-valued coefficients and vector-valued observables. The authors describe operator-valued kernel, vector-valued eigenfunctions, operator-valued kernels with delay-coordinate maps and Markov normalization. The authors also give some general observations on the topological structure of spatio-temporal data in a delay-coordinate space. The application of VSA to the Kuramoto-Sivashinsky model in periodic and chaotic regimes demonstrates a significant performance. Reviewer: Mohammad Sajid (Buraidah) Cited in 1 ReviewCited in 5 Documents MSC: 37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) 28B05 Vector-valued set functions, measures and integrals 37M10 Time series analysis of dynamical systems 47B34 Kernel operators Keywords:spatio-temporal patterns; spectral decomposition; kernel methods; Koopman operators; dynamical symmetries Software:ARPACK PDFBibTeX XMLCite \textit{D. Giannakis} et al., J. 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