Uncertainty Quantification of Data-Driven Output Predictors in the Output Error Setting

We revisit the problem of predicting the output of an LTI system using offline input-output data in the behavioral setting, without relying on parametric models. Existing works calculate the output predictions by projecting the recent data samples onto the column space of a Hankel matrix constructed from offline data, but the noise in this data renders these predictions inexact. While low-rank approximations like truncated singular value decomposition have been proposed to mitigate noise, the ensuing prediction accuracy remains unquantified. This paper introduces two novel upper bounds on prediction error under small-noise assumptions, one using raw data and one using the low-rank approximation. These bounds do not require access to the true system output and depend only on noisy data, a known noise level, and system order. Simulations show that both bounds decrease linearly with noise. Interestingly, applying the de-noising heuristic does not generally improve prediction accuracy or result in a tighter error bound compared to raw data, but it allows for a more general upper bound, as the first upper bound requires a specific condition on the construction of the Hankel matrix.