Received: 21 July 2017 Revised: 28 November 2017 Accepted: 28 November 2017
Distribution and time-series modelling of ordinal
J. Jha A. Biswas
Applied Statistics Unit, Indian Statistical
Institute, Kolkata 203, B.T. Road, Kolkata
J. Jha, Applied Statistics Unit, Indian
Statistical Institute, Kolkata 203, B.T.
Road, Kolkata 700108, India.
In this paper, we discuss the distribution of a circular random variable having
positive probabilities only on uniformly spaced directions. We used the wrap-
ping of geometric distribution to construct two discrete distributions on the
circle. The distributional properties and estimation procedures for the underly-
ing model parameters are discussed. Some tests for the distribution are proposed.
Simulation studies are done to illustrate the distribution. Then, using the distri-
bution, we carry out the time-series analysis of wind direction data where the
sample space also contains the no wind state.
circular dispersion, discrete circular distribution, geometric distribution, wrapping
Distribution and time-series analysis of wind direction are very different from the time-series analysis of linear random
variables because wind direction is measured in angles. The random variables having the sample space as the angles are
called circular or directional random variables. Due to the difference in topological structure, the study of circular random
variables is very different from the linear random variables. The usual definitions for different statistics such as mean
and variance in the linear setup does not hold for the circular setup. Mardia and Jupp (2000) contains a comprehensive
review of the different techniques applied for the analysis of circular random variables. For continuous circular random
variables, such techniques work very well. However, generally, the measurement of wind direction contains additional
constraints. One of the constraints is that, in many cases, in place of the exact wind direction angle, we observe only one
of the finite number of equally spaced directions, for example, north, east, west, or south. Further, in the time-series data
for wind direction, we observe no wind on many of the days. We shall call such observations as null observations. Thus, in
the analysis of the time series of wind directions, we have a discrete circular random variable that is not exactly nominal
in nature. Moreover, the sample space also contains time points in which no wind is observed.
Some distributions have been proposed to analyse the continuous circular random variables. Mardia and Jupp (2000)
mentioned some distributions for the continuous circular random variables. Wrapping is a very common approach for
the distribution modelling in which the probability density of a linear random variable is wrapped to get the probability
density for a circular random variable. If the pdf for a linear random variable X is g
(x), then the wrapping makes the
probability density function (pdf) of a circular random variable Θ as
( + 2k). The examples are
wrapped normal distribution, wrapped Cauchy distribution, etc. Due to the continuity of the probability density function,
the wrapping process in the continuous setup maintains the continuity at the boundaries of the circular random variables,
that is, f
( + 2)∀ ∈[0, 2).
The categorical random variables are of two kinds: nominal and ordinal. In nominal data, there is no particular order-
ing among the categories, and hence, such kind of data sets can be analysed by multinomial distributions, and this
procedure is exactly the same for both the linear and circular random variables. The ordinal random variables have
Environmetrics. 2018;29:e2490. wileyonlinelibrary.com/journal/env Copyright © 2018 John Wiley & Sons, Ltd. 1of13