Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition

Abstract

Dynamic mode decomposition (DMD) provides a practical means of extracting insightful dynamical information from fluids datasets. Like any data processing technique, DMD’s usefulness is limited by its ability to extract real and accurate dynamical features from noise-corrupted data. Here, we show analytically that DMD is biased to sensor noise, and quantify how this bias depends on the size and noise level of the data. We present three modifications to DMD that can be used to remove this bias: (1) a direct correction of the identified bias using known noise properties, (2) combining the results of performing DMD forwards and backwards in time, and (3) a total least-squares-inspired algorithm. We discuss the relative merits of each algorithm and demonstrate the performance of these modifications on a range of synthetic, numerical, and experimental datasets. We further compare our modified DMD algorithms with other variants proposed in the recent literature.

Appendix: Quantifying the size of the bias in DMD

We seek to quantify the magnitude of this bias present in DMD that was derived in Sect. 2.2, subject to certain simplifying assumptions on the nature of the data and noise. If the noise is uniform, and spatially and temporally independent, then $\mathbb {E}( \tilde{N}_X\tilde{N}_X^* ) = \mathbb {E}(U^*N_X N_X^*U) = U^*m \sigma _N^2U = m \sigma _N^2 I$, where $\sigma _N^2$ is the variance of each independent component of the noise matrix. Furthermore, if we assure that we are projecting onto the POD modes of the clean data, then $(\tilde{X} \tilde{X}^*) = \Sigma ^2$, where $U \Sigma V^*$ is the singular value decomposition of X. Thus, with these assumptions, Eq. (6) can be simplified to give

$$\begin{aligned} \mathbb {E}(\tilde{A}_m) \approx \tilde{A}(I - m\sigma _N^2\Sigma ^{-2}). \end{aligned}$$

(24)

The (diagonal) entries $\Sigma _i^2$ of $\Sigma ^2$ have the interpretation of being the energy content of the ith POD mode. We then should expect that $\Sigma _i^2 \sim mnq_i \sigma _X^2$, where $\sigma _X^2$ is the RMS value of the elements in the data matrix X, and $q_i = \frac{\Sigma _i^2}{\text {Trace}(\Sigma ^2)}$ is the proportion of the total energy of the system contained in the ith POD mode. For this scaling, we make the assumption that adding/removing rows and columns of data (i.e., varying m and n) does not affect either $\sigma _X$ or $q_i$. The bias term $m\sigma _N^2\Sigma ^{-2}$ is a diagonal matrix whose ith entry has a size $(e_b)_i$ proportional to

$$\begin{aligned} (e_b)_i \sim \frac{1}{n q_i SNR^2}, \end{aligned}$$

(25)

where SNR is the signal-to-noise ratio. Thus, sensor noise has the effect of reducing the diagonal entries of the computed $\tilde{A}_m$ matrix by a multiplicative factor of $1-\frac{\sigma _N^2}{n q_i \sigma _X^2}$, which means that POD coefficients are predicted to decay more rapidly than they actually do. This effect is most pronounced for lower-energy modes, for which the $q_i$ is smaller. We thus expect to identify with DMD (continuous-time) eigenvalues that are further into the left half plane than they should be (or would be if we applied DMD to noise-free data). Duke et al. (2012) argue in the case of periodic data that the growth rate of the eigenvalues should typically be the most challenging to identify, since there are a range of preexisting methods that can identify frequencies. Here, we have argued that it is precisely this growth rate that is most affected by noise. Importantly, the amount of bias is independent of m, which suggests that the bias component of the error will be particularly dominant when we have a large number of low-dimensional snapshots. Importantly, this suggests that one cannot always effectively reduce the effect of noise by simply using more snapshots of data, since the bias error will eventually become the dominant error.

While we can now quantify the magnitude of the bias in DMD, we do not as yet know how it compares to the random component of the error that would arise from a given realization of noise. To do this, we will estimate the typical size of the variance of individual entries of $\tilde{A}$, using the standard definition

$$\begin{aligned} var\left[ \tilde{A}_{ij} \right]&= \mathbb {E}\left\{ \left( (\tilde{Y}_m \tilde{X}_m^+)_{ij} - \mathbb {E}\left[ (\tilde{Y}_m \tilde{X}_m^+)_{ij}\right] \right) \right. \\&\quad \quad \left. \times \left( (\tilde{Y}_m\tilde{X}_m^+)_{ij} - \mathbb {E}\left[ (\tilde{Y}_m \tilde{X}_m^+)_{ij}\right] \right) \right\} . \end{aligned}$$

(26)

Referring back to Eq. (3), if we exclude terms that are quadratic or higher in noise and assume that the noise covariance matrix is sufficiently close to its expected value, we find that

$$\begin{aligned} (\tilde{Y}_m \tilde{X}_m^+)&- \mathbb {E}\left[ (\tilde{Y}_m \tilde{X}_m^+)\right] \\&= (\tilde{Y} + \tilde{N}_Y)(\tilde{X} + \tilde{N}_X) (\tilde{X} \tilde{X}^* + \tilde{X}\tilde{N}_X^* + \tilde{N}_X \tilde{X}^*+\tilde{N}_X\tilde{N}_X^*)^{-1} \\&\quad \quad - \tilde{Y}\tilde{X}^+ - \mathbb {E}(\tilde{N}_X\tilde{N}_X^*)\Sigma ^{-2}\\&= \left[ \tilde{Y}\tilde{X}^+(\tilde{X} \tilde{N}_X^* + \tilde{N}_X\tilde{X}^*)+ \tilde{N}_Y\tilde{X}^* + \tilde{Y}\tilde{N}_X^*\right] \Sigma ^{-2}. \end{aligned}$$

Elements of the terms $\tilde{X} \tilde{N}_X^*,\ \tilde{N}_X\tilde{Y}^*, \ \tilde{N}_Y\tilde{X}^*$, and $\tilde{Y}\tilde{N}_X^*$ are uncorrelated sums over m random terms, with each term in the sum having variance $n q_i\sigma _X^2\sigma _N^2$ where as before $q_i$ is the energy fraction in the $i^{\text{ t }h}$ POD mode. This means that the sum will have variance $mn q_i\sigma _X^2\sigma _N^2$. Assuming that $\tilde{Y}\tilde{X}^+ (=\tilde{A})$ does not greatly change the magnitude of quantities that it multiplies and assuming that $q_i$ remains constant when varying m and n, this means that we find that

$$\begin{aligned} var\left[ \tilde{A}_{ij} \right] \sim \frac{\sigma _N^2}{mn\sigma _X^2}. \end{aligned}$$

(27)

Thus, the expected size of the random error in applying DMD to noisy data is

$$\begin{aligned} e_r \sim \frac{1}{m^{1/2}n^{1/2}SNR}. \end{aligned}$$

(28)

Comparing Eq. (28) with Eq. (25), we propose that the bias in DMD will be the dominant source of error whenever

$$\begin{aligned} m^{1/2}SNR > n^{1/2}. \end{aligned}$$