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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">upper bounds</i> 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.