Choosing the future horizon in closed-loop CCA-type Subspace Algorithms

The length of the future horizon is one of the principal “user choices” in subspace identification. It is well known that for the CCA algorithm the asymptotic variance decreases as the future horizon increases when input signals are white (or absent). There are also examples, when the input is not white, in which the asymptotic variance is minimized when the future horizon is equal to the system order. In this paper we shall discuss how (and why) the choice of the future horizon influences the accuracy. In particular we shall show that the asymptotic variance is a monotonically non-increasing function of the future horizon under more general assumptions than previously known. For the situations not covered by the above result, we shall also describe a computational procedure, based on some recently derived asymptotic variance formulas, which allows to optimize the choice of this parameter. This is, to the best of the author’s knowledge, the first theoretically sound attempt to automatically select the length of the future horizon. Even though we strongly believe much remains to be done, we hope the considerations contained in this paper will provide a starting point towards a completely automated choice of user parameters in subspace identification. As an intermediate result we show the a version of the CCA algorithm introduced by Larimore is asymptotically equivalent to a number of recently studied methods, complementing recent results appeared in the literature. The setup we consider includes closed loop identification.