Abstract
Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere, and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper, we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (Wiley 1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, space situational awareness, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments are discussed.
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We are most grateful to three anonymous referees and, in alphabetical order, to Davide Buttarazzi, Marco Di Marzio, Miguel Fernández, Stephan Huckemann, Peter Jupp, Shogo Kato, Christophe Ley, Kanti Mardia, and Louis-Paul Rivest, for their enthusiastic feedback on our original submission and helpful suggestions as to how it might be further improved. This work was supported by Grants PGC2018-097284-B-100, IJCI-2017-32005 and MTM2016-76969-P from the Spanish Ministry of Economy and Competitiveness, and GR18016 from the Junta de Extremadura. All four grants were co-funded with FEDER funds from the European Union.
This invited paper is discussed in the comments available at: https://doi.org/10.1007/s11749-021-00760-4, https://doi.org/10.1007/s11749-021-00761-3, https://doi.org/10.1007/s11749-021-00763-1, https://doi.org/10.1007/s11749-021-00764-0.
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Pewsey, A., García-Portugués, E. Recent advances in directional statistics. TEST 30, 1–58 (2021). https://doi.org/10.1007/s11749-021-00759-x
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DOI: https://doi.org/10.1007/s11749-021-00759-x
Keywords
- Classification
- Clustering
- Dimension reduction
- Distributional models
- Exploratory data analysis
- Hypothesis tests
- Nonparametric methods
- Regression
- Serial dependence
- Software
- Spatial statistics