Singular value decomposition of noisy data: mode corruption

Although the singular value decomposition (SVD) and proper orthogonal decomposition have been widely used in fluid mechanics, Venturi (J Fluid Mech 559:215–254, 2006) and Epps and Techet (Exp Fluids 48:355–367, 2010) were among the first to consider how noise in the data affects the results of these decompositions. Herein, we extend those studies using perturbation theory to derive formulae for the 95% confidence intervals of the singular values and vectors, as well as formulae for the root mean square error (rmse) of each noisy SVD mode. Moreover, we show that the rmse is well approximated by $$\epsilon /\tilde{s}_k$$ (where $$\epsilon$$ is the rms noise and $$\tilde{s}_k$$ is the singular value), which provides a useful estimate of the overall uncertainty in each mode.