On the relation between CCA and predictor-based subspace identification

In this paper, we investigate the relation between a recently proposed subspace method based on predictor identification (PBSID), known also as "whitening filter algorithm," and the classical CCA algorithm. The comparison is motivated by i) the fact that CCA is known to be asymptotically efficient for time series identification and optimal for white measured inputs and ii) some recent results showing that a number of recently developed algorithms are very closely related to PBSID. We show that PBSID is asymptotically equivalent to CCA precisely in the situations in which CCA is optimal while an "optimized" version of PBSID behaves no worse than CCA also for nonwhite inputs. Even though PBSID (and its optimized version) are consistent regardless of the presence of feedback, in this paper we work under the assumption that there is no feedback to make the comparison with CCA meaningful. The results of this paper imply that the "optimized" PBSID, besides being able to handle feedback, is to be preferred to CCA also when there is no feedback; only in very specific cases (white or no inputs) are the two algorithms (asymptotically) equivalent.