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86

R. M. Crujeiras

1. Introduction

A simple definition of circular data would state that a circular

observation can be expressed as a point on a unit circle or as a

unit vector, once an initial direction and a sense of rotation have

been chosen. Specific examples of circular data are encountered

in a variety of disciplines such as biology (animal orientation;

Batschelet (1981) devotes a whole volume to the description and

analysis of examples of circular data in biology), environmetrics and

oceanography (wind and waves direction; Oliveira

et al.

(2014) and

Jona–Lasinio

et al.

(2012)), medicine (bone fractures; Mann

et al.

(2003)) or geology (cross–beds; Mardia (1972)). In addition, circular

data can be also generated by wrapping time around, when the goal

is to describe patterns on a certain (daily, monthly, yearly) scale.

Under an assumption of

stationarity

(in the sense that the events

occurrence pattern remains stable along time), sample observations

can be written and analyzed as circular data. For instance, times

when frosting and thawing cycles occur along a day (Oliveira

et

al.

, 2013), or also on a daily basis, circular methods have been

applied for analyzing the chronography of domestic terrorism (Gill

and Hangartner, 2010).

This idea of viewing temporal patterns as circular data is

somehow reflected in the polar area diagram, introduced by Florence

Nightingale (see Figure 1) when reporting the causes of mortality in

the British army in the mid of the nineteenth century. But despite

circular data appear quite frequently in many applied sciences

(and they have been there for a long time...), they are frequently

overlooked and analyzed without accounting for their particular

nature.

In the statistical literature, there are also some (but not many)

classical books dealing with the analysis of circular data, such as the

ones by Fisher (1993), Jammalamadaka and SenGupta (2001) and