A Hybrid Initialization Method for the Maximum Likelihood Estimation of Box-Jenkins Models

Local minimum basin on the search route drags gradient-related optimization towards the erroneous optimal in maximum likelihood (ML) estimation. Under such circumstance, wrong system information will be delivered. To deal with this problem, a reliable starting point that is close enough to the true parameters is preferable for ML estimation. Here we present a two-step method to provide such point under periodic excitation based on hierarchy identification of Box-Jenkins (BJ) models. In this approach, the estimates for the noise and system model are derived in turn through fast converging gradient-related and least squares minimization of global convergent cost functions. It can also be proven that the estimates asymptotically converge to the true values in the square root of $N$ (data number). Simulation results show that the method output can help the further ML process handle local convergence when the number of periods is moderate. If the number of periods gets bigger, applying the method alone is a good choice for simplicity since the covariance matrix of estimate tends to the Cramer-Rae lower bound.