Multiscale time series modelling with an application to the relativistic electron intensity at the geosynchronous orbit

In this paper, a Bayesian system identification approach to multiscale time series modelling is proposed, where multiscale means that the output of the system is observed at one(coarse) resolution while the input of the system is observed at another (One) resolution. The proposed method identifies linear models at different levels of resolution where the link between the two resolutions is realised via non-overlapping averaging process. This averaged time series at the coarse level of resolution is assumed to be a set of observations from an implied process so that the implied process and the output of the system result in an errors-in-variables ARMAX model at the coarse level of resolution. By using a Bayesian inference and Markov Chain Monte Carlo (MCMC) method, such a modelling framework results in different dynamical models at different levels of resolution at the same time. The new method is also shown to have the ability to combine information across different levels of resolution. An application to the analysis of the relativistic electron intensity at the geosynchronous orbit is used to illustrate the new method.