  Autoregressive Markov switching (ARMS) time series models are used to
represent real-world signals whose dynamics may change over time. They have
found application in many areas of the natural and social sciences, as well as
in engineering. In general, inference in this kind of systems involves two
problems: (a) detecting the number of distinct dynamical models that the signal
may adopt and (b) estimating any unknown parameters in these models. In this
paper, we introduce a class of ARMS time series models that includes many
systems resulting from the discretisation of stochastic delay differential
equations (DDEs). Remarkably, this class includes cases in which the
discretisation time grid is not necessarily aligned with the delays of the DDE,
resulting in discrete-time ARMS models with real (non-integer) delays. We
describe methods for the maximum likelihood detection of the number of
dynamical modes and the estimation of unknown parameters (including the
possibly non-integer delays) and illustrate their application with an ARMS
model of El Ni\~no--southern oscillation (ENSO) phenomenon.
