  Motivated by some cutting edge circular data such as from Smart Home
technologies and roulette spins from online and casino, we construct some new
rich classes of discrete distributions on the circle. We give four new general
methods of construction, namely (i) maximum entropy, (ii) centered wrapping,
(iii) marginalized and (iv) conditionalized methods. We motivate these methods
on the line and then work on the circular case and provide some properties to
gain insight into these constructions. We mainly focus on the last two methods
(iii) and (iv) in the context of circular location families, as they are
amenable to general methodology. We show that the marginalized and
conditionalized discrete circular location families inherit important
properties from their parent continuous families. In particular, for the von
Mises and wrapped Cauchy as the parent distribution, we examine their
properties including the maximum likelihood estimators, the hypothesis test for
uniformity and give a test of serial independence. Using our discrete circular
distributions, we demonstrate how to determine changepoint when the data arise
in a sequence and how to fit mixtures of this distribution. Illustrative
examples are given which triggered the work. For example, for roulette data, we
test for uniformity (unbiasedness) , test for serial correlation, detect
changepoint in streaming roulette-spins data, and fit mixtures. We analyse a
smart home data using our mixtures. We examine the effect of ignoring
discreteness of the underlying population, and discuss marginalized versus
conditionalized approaches. We give various extensions of the families with
skewness and kurtosis, to those supported on an irregular lattice, and discuss
potential extension to general manifolds by showing a construction on the torus
