On Some Parameter Estimation Algorithms for the Nonlinear Exponential Autoregressive Model

Modeling an exponential autoregressive (ExpAR) time series is the basis of solving the corresponding prediction and control problems. This paper investigates the hierarchical parameter estimation methods for the ExpAR model. By the hierarchical identification principle, the original nonlinear optimization problem is transformed into the combination of a linear and nonlinear optimization problem, and then, we derive a hierarchical least squares and stochastic gradient (LS-SG) algorithm. Given the difficulty of determining the step-size in the hierarchical LS-SG algorithm, an approach is proposed to obtain the optimal step-size. To improve the parameter estimation accuracy, the multi-innovation identification theory is employed to develop a hierarchical least squares and multi-innovation stochastic gradient algorithm for the ExpAR model. Two simulation examples are provided to test the effectiveness of the proposed algorithms