Strong consistency of the projected total least squares dynamic mode decomposition for datasets with random noise

Dynamic mode decomposition (DMD) has attracted much attention in recent years as an analysis method for time series data. In this paper, we perform asymptotic analysis on the DMD to prove strong consistency in the statistical sense. More specifically, we first give a statistical model of random noise for data with observation noise. Among many variants of the DMD, the total least squares DMD (TLS-DMD) is known as a robust method for the random noise in the observation. We focus on consistency analysis of the TLS-DMD in the statistical sense and aim to generalize the analysis for projected methods. This paper gives a general framework for designing projection methods based on efficient dimensionality reduction analogously to the proper orthogonal decomposition under the statistical model and proves its strong consistency of the estimation.