A behavioural approach to exact modelling of finite time series

In this paper, we consider the exact modelling of finite time series in the behavioural framework. Since only a finite portion of the time series is given, the central issue is to postulate a completion of the time series so that the resulting most powerful unfalsified model (MPUM) minimizes certain measures of model complexity. We consider two distinct problems. The first one is the modelling of multiple scalar (q=1) finite time series, where the model degree, or equivalently the dimension of the postulated behaviour, is chosen to be the complexity measure. We give an algorithm for generating the parameters of the minimal complexity polynomial model from the data. The second problem is the modelling of vector (p>1) finite time series, where we consider the complexity measures of model degree and number of free variables (inputs), and show that these are competing objectives. We characterize the Pareto optimal solutions and give an algorithm for computing the model parameters from the data. We prove the optimality of this algorithm using a dynamic programming argument