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.

本文通过引入两个上界来解决使用离线输入 - 输出数据直接预测 LTI 系统输出的问题，同时考虑到噪声对预测精度的影响，通过数值模拟证明噪声水平的增加会线性降低上界，并指出使用去噪启发式方法不一定提高预测准确度或减小上界。