Statistical modeling and an adaptive averaging technique for strong convergence of the dynamic mode decomposition

Dynamic mode decomposition (DMD) is known as an efficient analysis method for time series data. We introduce an adaptive averaging technique to the DMD for its strong convergence in the statistical sense under a newly constructed noise model. The new model can represent time series data contaminated with random noise in more general situation than the existing one. However, due to its generality, the model itself is complicated, which makes statistical asymptotic analysis less straightforward. As it stands, under the new statistical model, there is no convergence proof of any existing method as far as the author knows. Although the proposed averaging technique is simple and not so new, it is worth noting that this technique can be generally applied to the above new noise model as the preconditioning, resulting in natural proof of the strong convergence. Moreover, we prove that an estimator of the original DMD is not consistent in the statistical sense, which implies the importance of the averaging technique as the preconditioning.