# Semi-Markov Arnason-Schwarz Models

### Ruth King and Roland Langrock

### Universities of Edinburgh and Bielefeld

## Summary

We focus on multi-state capture-recapture-recovery data where observed individuals are recorded in a
set of possible discrete states. Traditionally, the Arnason-Schwarz model has been fitted to such data where the
state process is modeled as a First-order Markov chain, though second-order models have also been proposed and
fitted to data. However, low-order Markov models may not accurately represent the underlying biology. For example,
specifying a (time-independent) first-order Markov process assumes that the dwell-time distribution of each state
(i.e., the duration of a stay in a given state) has a geometric distribution, and hence that the modal dwell-time
is one. Specifying time-dependent or higher-order processes provides additional exibility, but at the expense of a
potentially significant number of additional model parameters. We extend the Arnason-Schwarz model by specifying
a semi-Markov model for the state process, where the dwell-time distribution is specified more generally, using for
example a shifted Poisson or negative binomial distribution. A state expansion technique is applied in order to
represent the resulting semi-Markov Arnason-Schwarz model in terms of a simpler and computationally tractable
hidden Markov model. Semi-Markov Arnason-Schwarz models come with only a very modest increase in the number
of parameters, yet permit a significantly more exible state process. Model selection can be performed using standard
procedures, and in particular via the use of information criteria. The semi-Markov approach allows for important
biological inference to be drawn on the underlying state process, for example on the times spent in the different states.
The feasibility of the approach is demonstrated in a simulation study, before being applied to real data corresponding
to house finches where the states correspond to the presence or absence of conjunctivitis.

### Keywords:

capture-recapture-recovery; dwell-time distribution; hidden Markov model; multi-state model.