Triplet Markov models and fast smoothing in switching systems

The aim of the paper is twofold. The first aim is to present a mini tutorial on « pairwise Markov models » (PMMs) and " triplet Markov models " (TMMs) which extend the popular " hidden Markov models " (HMMs). The originality of these extensions is due to the fact that the hidden data does not need to be Markov. More precisely, for X hidden data and Y observed ones, the originality of PMMs is that X does not need to be Markov, and the originality of TMMs is that even (X, Y) does not need to be Markov. In spite of these lacks of Markovianity fast processing methods, similar to those applied in HMMs or their other extensions, remain workable. The second goal is to present an original switching model approximation allowing fast smoothing. The method we propose, called " double filtering based smoothing " (DFBS), uses a particular TMM in which the pair (X, R), where R models switches, is not Markov. It is based on two filters, and uses a class of models, known as conditionally Gaussian observed Markov switching models (CGOMSMs), where exact fast filtering is feasible. The original model is approximated by two CGOMSMs in order to process the past data and the future data in direct and reverse order, respectively. Then state estimates produced by these two models are fused to provide a smoothing estimate. The DFBS is insensitive to the dimensions of the hidden and observation space and appears as an alternative to the classic particle smoothing in the situations where the latter cannot be applied due to its high processing cost