Estimation of finite mixture models of skew-symmetric circular distributions
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Estimation of finite mixture models of skew-symmetric circular distributions Yoichi Miyata1
· Takayuki Shiohama2 · Toshihiro Abe3
Received: 20 February 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Analysis of circular data is challenging, since the usual statistical methods are unsuitable and it is necessary to use circular periodic probabilistic models. Because some actual circular datasets exhibit asymmetry and/or multimodality, finite mixtures of symmetric circular distributions to model and fit these data have been investigated. However, it is necessary to question the predominant assumption that each component in the finite mixture model is symmetric. In this study, we consider a finite mixture model of possibly skewed circular distributions and discuss the expectationmaximization (EM) algorithm for the maximum likelihood estimate. It is shown that the maximum likelihood estimator is strongly consistent under some suitable conditions in a finite mixture of skew-symmetric circular distributions. A modified M-step in the EM algorithm is proposed in order to estimate the unknown parameter vectors effectively. To investigate the performance of our proposed model with its estimation procedure, we provide a numerical example as well as data analysis using the records of the time of day of fatal traffic accidents. Keywords Circular statistics · Consistency · EM algorithm · Finite mixtures · Skew-symmetric distributions
1 Introduction Circular or directional data arise in several research fields in the natural sciences, including wave directions in oceanography and wind directions in meteorology. These data also appear in the study of animal movement in biology and protein conformal mapping in bioinformatics. In addition, the records of event times data wrapped to
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00184019-00756-z) contains supplementary material, which is available to authorized users.
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Yoichi Miyata [email protected]
Extended author information available on the last page of the article
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Y. Miyata et al.
certain time periods are considered as circular data, and many applications exist to analyze these data in various fields, such as finance, marketing science, and information science. Analysis of such circular data is challenging, since the usual statistical methods are unsuitable and it is necessary to use circular periodic probabilistic models. The von Mises distribution and wrapped Cauchy (WC) distribution are the most popular symmetric models on the unit circle, and have been applied in many fields. See, for example, Mardia and Jupp (2009). By contrast, because some actual circular datasets exhibit asymmetry and/or multimodality, Fraser et al. (1981), Holzmann et al. (2004), and Banerjee et al. (2005), among others, have investigated finite mixtures of symmetric circular distributions to model and fit these data. Although several approaches exist to model finite mixtures of symmetric circular distributio
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