Reconstructions and predictions of nonlinear dynamical systems: a hierarchical Bayesian approach

An attempt is made to reconstruct model nonlinear dynamical systems from scalar time series data via a hierarchical Bayesian framework. Reconstruction is performed by fitting given training data with a parameterized family of functions without overfitting. The reconstructed model dynamical systems are compared with respect to (approximated) model marginal likelihood, which is a natural Bayesian information criterion. The best model is selected with respect to this criterion and is utilized to make predictions. The results are applied to two problems: (i) chaotic time series prediction and (ii) building air-conditioning load prediction. The former is a very good class of problems for checking the abilities of prediction algorithms for at least two reasons. First, since no linear dynamical systems can admit chaotic behavior, an algorithm must capture the nonlinearities behind the time series. Second, chaotic dynamical systems are sensitive to initial conditions. More precisely, the error grows exponentially with respect to time so that crispness of capturing nonlinearities is also important. Experimental results appear to indicate that the proposed scheme can capture difficult nonlinearities behind the chaotic time series data. The latter class of problems (air conditioning load prediction) is motivated by a great amount of demand for reducing CO₂ emissions associated with electric power generation. The authors won a prediction competition using the proposed algorithm; therefore, it appears to be reasonably sound