Reconstruction and prediction of nonlinear dynamical systems: a hierarchical Bayes approach with neural nets

When nonlinearity is involved, time series prediction becomes a rather difficult task where the conventional linear methods have limited successes for various reasons. One of the greatest challenges stems from the fact that typical observation data is a scalar time series so that dimension of the nonlinear dynamical system (embedding dimension) is unknown. This paper proposes a hierarchical Bayesian approach to nonlinear time series prediction problems. This class of schemes considers a family of prior distributions parameterized by hyperparameters instead of a single prior so that it enables algorithms less dependent on a particular prior. One can estimate posterior of weight parameters, hyperparameters and embedding dimension by marginalization with respect to the weight parameters and hyperparameters. The proposed scheme is tested against two examples: (i) chaotic time series, and (ii) building air-conditioning load prediction