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

TL;DR

This paper develops theoretical bounds for the prediction error of data-driven output predictors in noisy settings, showing how noise affects accuracy and comparing direct data use versus low-rank approximation.

Contribution

It introduces two upper bounds on prediction error in noisy data scenarios, applicable without knowing the true system output, and compares the effectiveness of de-noising heuristics.

Findings

Prediction bounds decrease linearly with noise level.

Using raw data can be as effective as de-noising heuristics.

Bounds do not require ground truth output, only noisy measurements and system order.

Abstract

We revisit the problem of predicting the output of an LTI system directly using offline input-output data (and without the use of a parametric model) in the behavioral setting. Existing works calculate the output predictions by projecting the recent samples of the input and output signals onto the column span of a Hankel matrix consisting of the offline input-output data. However, if the offline data is corrupted by noise, the output prediction is no longer exact. While some prior works propose mitigating noisy data through matrix low-ranking approximation heuristics, such as truncated singular value decomposition, the ensuing prediction accuracy remains unquantified. This paper fills these gaps by introducing two upper bounds on the prediction error under the condition that the noise is sufficiently small relative to the offline data's magnitude. The first bound pertains to prediction using the raw offline data directly, while the second one applies to the case of low-ranking approximation heuristic. Notably, the bounds do not require the ground truth about the system output, relying solely on noisy measurements with a known noise level and system order. Extensive numerical simulations show that both bounds decrease monotonically (and linearly) as a function of the noise level. Furthermore, our results demonstrate that applying the de-noising heuristic in the output error setup does not generally lead to a better prediction accuracy as compared to using raw data directly, nor a smaller upper bound on the prediction error. However, it allows for a more general upper bound, as the first upper bound requires a specific condition on the partitioning of the Hankel matrix.