Eigenvalues of Autocovariance Matrix: A Practical Method to Identify the Koopman Eigenfrequencies

To infer eigenvalues of the infinite-dimensional Koopman operator, we study the leading eigenvalues of the autocovariance matrix associated with a given observable of a dynamical system. For any observable $f$ for which all the time-delayed autocovariance exist, we construct a Hilbert space $\mathcal{H}_f$ and a Koopman-like operator $\mathcal{K}$ that acts on $\mathcal{H}_f$. We prove that the leading eigenvalues of the autocovariance matrix has one-to-one correspondence with the energy of $f$ that is represented by the eigenvectors of $\mathcal{K}$. The proof is associated to several representation theorems of isometric operators on a Hilbert space, and the weak-mixing property of the observables represented by the continuous spectrum. We also provide an alternative proof of the weakly mixing property. When $f$ is an observable of an ergodic dynamical system which has a finite invariant measure $\mu$, $\mathcal{H}_f$ coincides with closure in $L^2(X,d\mu)$ of Krylov subspace generated by $f$, and $\mathcal{K}$ coincides with the classical Koopman operator. The main theorem sheds light to the theoretical foundation of several semi-empirical methods, including singular spectrum analysis (SSA), data-adaptive harmonic analysis (DAHD), Hankel DMD and Hankel alternative view of Koopman analysis (HAVOK). It shows that, when the system is ergodic and has finite invariant measure, the leading temporal empirical orthogonal functions indeed correspond to the Koopman eigenfrequencies. A theorem-based practical methodology is then proposed to identify the eigenfrequencies of $\mathcal{K}$ from a given time series. It builds on the fact that the convergence of the renormalized eigenvalues of the Gram matrix is a necessary and sufficient condition for the existence of $\mathcal{K}-$eigenfrequencies.