Data-driven spectral decomposition and forecasting of ergodic dynamical systems

Abstract

We develop a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic dynamical systems. This framework is based on a representation of the Koopman and Perron-Frobenius groups of unitary operators in a smooth orthonormal basis of the L 2 space of the dynamical system, acquired from time-ordered data through the diffusion maps algorithm. Using this representation, we compute Koopman eigenfunctions through a regularized advection-diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. In systems with pure point spectra, we construct a decomposition of the generator of the Koopman group into mutually commuting vector fields that transform naturally under changes of observation modality, which we reconstruct in data space through a representation of the pushforward map in the Koopman eigenfunction basis. We also establish a correspondence between Koopman operators and Laplace-Beltrami operators constructed from data in Takens delay-coordinate space, and use this correspondence to provide an interpretation of diffusion-mapped delay coordinates for this class of systems. Moreover, we take advantage of a special property of the Koopman eigenfunction basis, namely that the basis elements evolve as simple harmonic oscillators, to build nonparametric forecast models for probability densities and observables. In systems with more complex spectral behavior, including mixing systems, we develop a method inspired from time change in dynamical systems to transform the generator to a new operator with potentially improved spectral properties, and use that operator for vector field decomposition and nonparametric forecasting.

Extended-range statistical ENSO prediction through operator-theoretic techniques for nonlinear dynamics

“…This behavior indicates that including previous states by delay-coordinate mapping plays a significant role in the performance of the method at long leads, which challenge many traditional approaches. This is consistent with previous work, which has shown that incorporating delays increases the capability of kernel algorithms to extract dynamically intrinsic coherent patterns in the spectrum of the Koopman operator 15,16 , including ENSO and its linkages with seasonal and decadal variability of the climate system [24][25][26] . Given our main focus here on the extended-range regime, Figure 1 Another important consideration in ENSO prediction is the seasonal dependence of skill, exhibiting the so-called "spring barrier" 6 .…”

“…The KAF approach employed in this work utilizes nonlinear-kernel algorithms to enhance the information content of the predictor vectors to beyond linear functions of the input data, and leverages that information through the use of statistical learning theory 10 and operator-theoretic ergodic theory 11,12 to optimally capture the evolution of a response variable (here, an ENSO index) under partially observed nonlinear dynamics. Here, we build KAF models using the class of kernels introduced in the context of nonlinear Laplacian spectral analysis (NLSA) 13,14 algorithms, which provably capture modes of high temporal coherence 15,16 , evolving at intrinsic timescales in the spectrum of the Koopman evolution operator 17 of the underlying dynamical system. This capability is realized without ad hoc prefiltering of the input data through the use of delay-coordinate maps 18 and a nonlinear (normalized Gaussian) kernel function 19 , measuring similarity between delay-embedded sequences.…”

The Antarctic circumpolar wave and its seasonality: Intrinsic travelling modes and El Niño–Southern Oscillation teleconnections

“…Moreover, in the SIC field, these modes exhibit appreciable wavenumber-2 variability in the meridional direction (Figure 3a), consistent with the seasonal growth and melting of Antarctic sea ice in response to the seasonal cycle. Analogous relationships between annual-cycle modulated patterns and sea-ice reemergence phenomena (Blanchard-Wrigglesworth et al, 2011;Holland et al, 2013) have also been identified in the Arctic (Bushuk et al, 2014;2015;2017), but have not received similar attention in the Southern Ocean.…”

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

“…Issues pertaining to non-compactness or continuous spectra of Koopman operators associated with systems of high complexity are beyond the scope of this paper. Although such cases can theoretically be handled, the numerical analysis is often challenging and typically requires regularization, which is, for instance, implicitly given by Galerkin projections [25]. This is discussed in detail in the aforecited work by Giannakis.…”