Eigenvalues of autocovariance matrix: A practical method to identify the Koopman eigenfrequencies

Zhen, Yicun; Chapron, Bertrand; Mémin, Étienne; Peng, Lin

doi:10.1103/physreve.105.034205

Cited by 9 publications

(5 citation statements)

References 27 publications

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“…The following Proposition [4] demonstrates the correspondence between the eigenfrequencies of the continuous-time time-shift operator and the discrete-time time-shift operator. Please refer to [4] for the notations in the proposition.…”

Section: Theorem 3 (Main Result) Under Assumptions 1 and 2 We Have Fo...mentioning

confidence: 85%

Section: Notes and Commentsmentioning

confidence: 82%

“…The main result as well as the techniques and ideas used for the proof are close in spirit to those developed in [4]. However, the Hilbert-Schmidt operator A τ is defined for continuous time process and the theory developed in [4] does not cover the continuous-time case. The objective of this paper is to confirm that the asymptotic behavior of the Hilbert-Schmidt operator A τ is well related to Koopman theory.…”

Section: Notes and Commentsmentioning

confidence: 82%

“…However, longer time series are sometimes available with long enough data. In this article we generalize the idea and tools developed in [4] and apply them to study of A τ . We shall prove that lim…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Bridging Koopman Operator and Time-Series Auto-Correlation Based Hilbert–Schmidt Operator

Zhen

Chapron

Mémin

2022

Mathematics of Planet Earth

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Given a stationary continuous-time process f(t), the Hilbert–Schmidt operator Aτ can be defined for every finite τ. Let λτ,i be the eigenvalues of Aτ with descending order. In this article, a Hilbert space $$\mathcal {H}_f$$ ℋ f and the (time-shift) continuous one-parameter semigroup of isometries $$\mathcal {K}^s$$ K s are defined. Let $$\{v_i, i\in \mathbb {N}\}$$ { v i , i ∈ ℕ } be the eigenvectors of $$\mathcal {K}^s$$ K s for all s ≥ 0. Let $$f = \displaystyle \sum _{i=1}^{\infty }a_iv_i + f^{\perp }$$ f = ∑ i = 1 ∞ a i v i + f ⊥ be the orthogonal decomposition with descending |ai|. We prove that limτ→∞λτ,i = |ai|2. The continuous one-parameter semigroup $$\{\mathcal {K}^s: s\geq 0\}$$ { K s : s ≥ 0 } is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on L2(X, ν), if the dynamical system is ergodic and has invariant measure ν on the phase space X.

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Section: Theorem 3 (Main Result) Under Assumptions 1 and 2 We Have Fo...mentioning

confidence: 85%

Section: Notes and Commentsmentioning

confidence: 82%

Section: Notes and Commentsmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bridging Koopman Operator and Time-Series Auto-Correlation Based Hilbert–Schmidt Operator

Zhen

Chapron

Mémin

2022

Mathematics of Planet Earth

Self Cite

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“…In what way is the spectral property of A τ related to intrinsic properties of the dynamical system? In this article we generalize the idea and tools developed in [4] and apply them to study of A τ . We shall prove that lim…”

Section: Introductionmentioning

confidence: 99%

Bridging Koopman Operator and time-series auto-correlation based Hilbert-Schmidt operator

Zhen¹,

Chapron²,

Mémin³

2022

Preprint

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Given a stationary continuous-time process f (t), the Hilbert-Schmidt operator Aτ can be defined for every finite τ [3]. Let λτ,i be the eigenvalues of Aτ with descending order. In this article, a Hilbert space H f and the (time-shift) continuous one-parameter semigroup of isometries K s are defined. Let {vi, i ∈ N} be the eigenvectors of K s for all s ≥ 0. Let f = ∞ i=1 aivi + f ⊥ be the orthogonal decomposition with descending |ai|. We prove that lim τ →∞ λτ,i = |ai| 2 . The continuous oneparameter semigroup {K s : s ≥ 0} is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on L 2 (X, ν), if the dynamical system is ergodic and has invariant measure ν on the phase space X.

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Ensemble forecasts in reproducing kernel Hilbert space family

Dufée,

Hug,

Mémin

et al. 2024

Physica D: Nonlinear Phenomena

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Eigenvalues of autocovariance matrix: A practical method to identify the Koopman eigenfrequencies

Cited by 9 publications

References 27 publications

Bridging Koopman Operator and Time-Series Auto-Correlation Based Hilbert–Schmidt Operator

Bridging Koopman Operator and Time-Series Auto-Correlation Based Hilbert–Schmidt Operator

Bridging Koopman Operator and time-series auto-correlation based Hilbert-Schmidt operator

Ensemble forecasts in reproducing kernel Hilbert space family

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