Estimation of jump Box–Jenkins models

Jump Box–Jenkins (BJ) models are a collection of a finite set of linear dynamical submodels in BJ form that switch over time, according to a Markov chain. This paper addresses the problem of maximum-a-posteriori estimation of jump BJ models from a given training input/output dataset. The proposed solution method estimates the coefficients of the BJ submodels, the state transition probabilities of the Markov chain regulating the switching of operating modes, and the corresponding mode sequence hidden in the dataset. In particular, the posterior distribution of all the unknown variables characterizing the jump BJ model is derived and then maximized using a coordinate ascent algorithm. The resulting estimation algorithm alternates between Gauss–Newton optimization of the coefficients of the BJ submodels, a method derived based on an instance of prediction error methods tailored to BJ models with switching coefficients, and approximated dynamic programming for optimization of the sequence of active modes. The quality of the proposed estimation approach is evaluated on a numerical example based on synthetic data and in a case study related to segmentation of honeybee dances.