Hidden hybrid Markov/semi-Markov chains

Models that combine Markovian states with implicit geometric state occupancy distributions and semi-Markovian states with explicit state occupancy distributions, are investigated. This type of model retains the flexibility of hidden semi-Markov chains for the modeling of short or medium size homogeneous zones along sequences but also enables the modeling of long zones with Markovian states. The forward-backward algorithm, which in particular enables to implement efficiently the E-step of the EM algorithm, and the Viterbi algorithm for the restoration of the most likely state sequence are derived. It is also shown that macro-states, i.e. series-parallel networks of states with common observation distribution, are not a valid alternative to semi-Markovian states but may be useful at a more macroscopic level to combine Markovian states with semi-Markovian states. This statistical modeling approach is illustrated by the analysis of branching and flowering patterns in plants.