Approximating the Koopman Operator using Noisy Data: Noise-Resilient Extended Dynamic Mode Decomposition

This paper presents a data-driven method to find a finite-dimensional approximation for the Koopman operator using noisy data. The proposed method is a modification of Extended Dynamic Mode Decomposition which finds an approximation for the projection of the Koopman operator on a subspace spanned by a predefined dictionary of functions. Unlike the Extended Dynamic Mode Decomposition which is based on least squares method, the presented method is based on element-wise weighted total least squares which enables one to find a consistent approximation when the data come from a static linear relationship and the noise at different times are not identically distributed. Even though the aforementioned method is consistent, it leads to a nonconvex optimization problem. To alleviate this problem, we show that under some conditions the nonconvex optimization problem has a common minimizer with a different method based on total least squares for which one can find the solution in closed form.