  Time series subject to change in regime have attracted much interest in
domains such as econometry, finance or meteorology. For discrete-valued
regimes, some models such as the popular Hidden Markov Chain (HMC) describe
time series whose state process is unknown at all time-steps. Sometimes, time
series are firstly labelled thanks to some annotation function. Thus, another
category of models handles the case with regimes observed at all time-steps. We
present a novel model which addresses the intermediate case: (i) state
processes associated to such time series are modelled by Partially Hidden
Markov Chains (PHMCs); (ii) a linear autoregressive (LAR) model drives the
dynamics of the time series, within each regime. We describe a variant of the
expection maximization (EM) algorithm devoted to PHMC-LAR model learning. We
propose a hidden state inference procedure and a forecasting function that take
into account the observed states when existing. We assess inference and
prediction performances, and analyze EM convergence times for the new model,
using simulated data. We show the benefits of using partially observed states
to decrease EM convergence times. A fully labelled scheme with unreliable
labels also speeds up EM. This offers promising prospects to enhance PHMC-LAR
model selection. We also point out the robustness of PHMC-LAR to labelling
errors in inference task, when large training datasets and moderate labelling
error rates are considered. Finally, we highlight the remarkable robustness to
error labelling in the prediction task, over the whole range of error rates.
