The Role of Vector AutoRegressive Modeling in Subspace Identification

Some subspace procedures make use, directly or indirectly, of Vector AutoRegressive with eXogenous inputs (VARX) models in a preliminary step. This was first noticed for the CCA method; more recently it has also been proved that the first oblique projection step of a subspace algorithm based on predictor identification (PBSID) is asymptotically equivalent to the SSARX algorithm by Jansson which performs a preliminary VARX modeling step. For the purpose of comparison with more classical methods like CCA a recent work have introduced also an "optimized" version of PBSID. In this paper we shall show that indeed also this latter "optimized" PBSID is equivalent to estimating a long VARX model followed by the "classical" steps of subspace identification. This latter step can be seen as a sort of model reduction. Besides the theoretical interest, we shall argue that this may have also important implications regarding computational complexity