On the relation between CCA and predictor-based subspace identification.

There is experimental evidence that a recently proposed subspace algorithm based on predictor identification has a behavior which is very close to prediction error methods in certain simple examples; this observation raises a question concerning its optimality. It is known that time series identification using the Canonical Correlation Analysis (CCA) approach is asymptotically efficient. Asymptotic optimality of CCA has also been proved when measured inputs are white. In this paper we study the relation between the standard CCA approach and the recently proposed subspace procedure based on predictor identification (PBSID¹from now on). In this paper we work under the assumption that there is no feedback; it is shown that CCA and PBSID are asymptotically equivalent precisely in the situations when CCA is optimal. The equivalence holds only asymptotically in the number of data and in the limit as the past horizon goes to infinity. Using some recent results on the asymptotic variance we report counter-examples showing that PBSID is not efficiency in general when measured inputs are not white.