Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

Dimitrios Giannakis, Abbas Ourmazd, Joanna Slawinska, Zhizhen Zhao

Research output: Contribution to journalArticlepeer-review


We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called vector-valued spectral analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for multitask machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto–Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

Original languageEnglish (US)
Pages (from-to)2385-2445
Number of pages61
JournalJournal of Nonlinear Science
Issue number5
StatePublished - Oct 1 2019
Externally publishedYes


  • Dynamical symmetries
  • Dynamical systems
  • Kernel methods
  • Koopman operators
  • Spatiotemporal patterns
  • Spectral decomposition

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Engineering
  • Applied Mathematics


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