De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets

Abstract

The Dynamic Mode Decomposition (DMD)-a popular method for performing data-driven Koopman spectral analysis-has gained increased adoption as a technique for extracting dynamically meaningful spatiotemporal descriptions of fluid flows from snapshot measurements. Often times, DMD descriptions can be used for predictive purposes as well, which enables informed decision-making based on DMD modelforecasts. Despite its widespread use and utility, DMD regularly fails to yield accurate dynamical descriptions when the measured snapshot data are imprecise due to, e.g., sensor noise. Here, we express DMD as a two-stage algorithm in order to isolate a source of systematic error. We show that DMD's first stage, a subspace projection step, systematically introduces bias errors by processing snapshots asymmetrically. To remove this systematic error, we propose utilizing an augmented snapshot matrix in a subspace projection step, as in problems of total least-squares, in order to account for the error present in all snapshots. The resulting unbiased and noise-aware total DMD (TDMD) formulation reduces to standard DMD in the absence of snapshot errors, while the two-stage perspective generalizes the de-biasing framework to other related methods as well. TDMD's performance is demonstrated in numerical and experimental fluids examples.

Subspace dynamic mode decomposition for stochastic Koopman analysis

“…An output of totalleast-squares DMD [22,23] is unbiased even for noisy observations as long as the dynamics are deterministic, but it is biased as a realization of K Ω for the RDS. These inconsistencies in the existing methods are partly revealed in the numerical examples in Section IV.…”

“…With subspace DMD, we can naturally conduct a low-rank approximation of dynamics by replacing the compact SVD in Step 3 with a truncated SVD. In contrast, in Algorithm 1 and total-least-squares DMD [22,23], the low-rank approximation is achieved via the truncated proper orthogonal decomposition (POD). Note that there is also a line of research on the low-rank approximation of DMD, such as [24][25][26][27].…”

“…We estimated the continuous-time eigenvalues using subspace DMD (Algorithm 2), standard DMD (Algorithm 1), total-least-squares DMD (tls-DMD) [22,23], and noise-corrected DMD (nc-DMD) [22]. Figure 1b shows the eigenvalues without any process noise (i.e., σ p = 0), and Figure 1c shows the ones with process noise of σ p = 0.5.…”

“…Several researchers have addressed this issue; Duke et al [20] and Pan et al [21] conducted error analyses on the DMD algorithms, and Dawson et al [22] and Hemati et al [23] proposed reformulating DMD as a total-least-squares problem to treat the observation noise explicitly. Moreover, there is a line of research on the low-rank approximation of dynamics, including optimized DMD [24], optimal mode decomposition [25], sparsity-promoting DMD [26], and the closed-form solution for a low-rank constrained problem [27].…”

Dynamic reconstruction and data reconstruction for subsampled or irregularly sampled data

“…In particular, a number of generalizations of the DMD algorithm have recently been proposed with each seeking to address various deciencies of the algorithm. Examples of the recent developments include: establishing an appropriate way of selecting the most important DMD modes (Chen et al 2012;Jovanovic et al 2014), nding an optimal low-order projection basis (Wynn et al 2013), improving the robustness of the method to observation noise (Hemati et al 2015;Wynn et al 2013), establishing a stronger link to Koopman operator (Williams et al 2015) or accounting for input/output systems (Proctor et al 2014).…”