Emergence of the wrapped Cauchy distribution in mixed directional data
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Emergence of the wrapped Cauchy distribution in mixed directional data Joseph D. Bailey1 · Edward A. Codling1 Received: 20 December 2019 / Accepted: 28 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Inferring the most appropriate distribution (or distributions) to describe observed directional data is important in many applications of circular statistics. In particular, animal movement paths are typically analysed and modelled by considering the distribution of step lengths and turning (or absolute) angles. Here we demonstrate that a single-wrapped Cauchy distribution can appear to fit directional data mixed from two different underlying wrapped normal distributions. We derive mathematical expressions to calculate the parameter space for which this occurs and illustrate the result by analysing an example data set of the movements of African bull elephants (Loxodonta Africana). We conclude that the presence of a wrapped Cauchy distribution in observed directional data can, in certain cases, be explained by data coming from two distinct underlying distributions. We discuss how this may relate to the presence of multiple movement modes within an observed path when analysing animal movement data. Keywords Circular distributions · Wrapped normal · Wrapped Cauchy · Animal movement · Directional data Mathematics Subject Classification MSC 62P10 · MSC 92B05
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s1018 2-020-00380-7) contains supplementary material, which is available to authorized users. * Joseph D. Bailey [email protected] 1
Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK
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J. D. Bailey, E. A. Codling
1 Introduction The analysis and applications of circular statistics to directional data play a significant role in the study of many environmental processes from plant phenology (Morellato et al. 2010) and tree growth (Aradottir et al. 1997) to wind direction (Masseran et al. 2013) and the general movement patterns of animals and cells (Rivest et al. 2016; Landler et al. 2018). Ascertaining the distribution which most closely describes circular data is important as characteristics of circular distributions, such as sharper peaks and slow-decaying tails, have significant effects on the qualitative and quantitative results of descriptive and predictive models. The most common distributions used to describe angular data are the wrapped normal (WN), von Mises (vM) (or circular normal) and the wrapped Cauchy (WC) (Jammalamadaka and SenGupta 2001; Mardia and Jupp 2000; McClintock et al. 2012; McClintock and Michelot 2018). These are defined by a probability density function (PDF) on the unit circle, and in the case of the WN and WC distributions, can be formed by ‘wrapping’ the equivalent one-dimensional distributions on the real line around the unit circle (Stephens 1963; Jammalamadaka and SenGupta 2001; Mardia and Jupp 2000; Abe and Shimatani 2018)
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