  Environmental time series data observed at high frequencies can be studied
with approaches such as hidden Markov and semi-Markov models (HMM and HSMM).
HSMMs extend the HMM by explicitly modeling the time spent in each state. In a
discrete-time HSMM, the duration in each state can be modeled with a
zero-truncated Poisson distribution, where the duration parameter may be
state-specific but constant in time. We extend the HSMM by allowing the
state-specific duration parameters to vary in time and model them as a function
of known covariates observed over a period of time leading up to a state
transition. In addition, we propose a data subsampling approach given that
high-frequency data can violate the conditional independence assumption of the
HSMM. We apply the model to high-frequency data collected by an instrumented
buoy in Lake Mendota. We model the phycocyanin concentration, which is used in
aquatic systems to estimate the relative abundance of blue-green algae, and
identify important time-varying effects associated with the duration in each
state.
