  While subspace identification methods (SIMs) are appealing due to their
simple parameterization for MIMO systems and robust numerical realizations, a
comprehensive statistical analysis of SIMs remains an open problem, especially
in the non-asymptotic regime. In this work, we provide a finite sample analysis
for a class of SIMs, which reveals that the convergence rates for estimating
Markov parameters and system matrices are $\mathcal{O}(1/\sqrt{N})$, in line
with classical asymptotic results. Based on the observation that the model
format in classical SIMs becomes non-causal because of a projection step, we
choose a parsimonious SIM that bypasses the projection step and strictly
enforces a causal model to facilitate the analysis, where a bank of ARX models
are estimated in parallel. Leveraging recent results from finite sample
analysis of an individual ARX model, we obtain an overall error bound of an
array of ARX models and proceed to derive error bounds for system matrices via
robustness results for the singular value decomposition.
