Article Robust Fitting of a Wrapped Normal Model to Multivariate Circular Data and Outlier Detection Luca Greco 1 * , Giovanni Saraceno 2 and Claudio Agostinelli 2   Citation: Greco, L.; Saraceno, G.; Agostinelli, C. Robust Fitting of a Wrapped Normal Model to Multivariate Circular Data and Outlier Detection. Stats 2021, 4, 454–471. https://doi.org/10.3390/ stats4020028 Academic Editor: Marco Riani Received: 30 April 2021 Accepted: 28 May 2021 Published: 1 June 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 University Giustino Fortunato, 82100 Benevento, Italy 2 Department of Mathematics, University of Trento, 38122 Trento, Italy; giovanni.saraceno@unitn.it (G.S.); claudio.agostinelli@unitn.it (C.A.) * Correspondence: luca.greco@unisannio.it Abstract: In this work, we deal with a robust fitting of a wrapped normal model to multivariate circular data. Robust estimation is supposed to mitigate the adverse effects of outliers on infer- ence. Furthermore, the use of a proper robust method leads to the definition of effective outlier detection rules. Robust fitting is achieved by a suitable modification of a classification-expectation- maximization algorithm that has been developed to perform a maximum likelihood estimation of the parameters of a multivariate wrapped normal distribution. The modification concerns the use of complete-data estimating equations that involve a set of data dependent weights aimed to downweight the effect of possible outliers. Several robust techniques are considered to define weights. The finite sample behavior of the resulting proposed methods is investigated by some numerical studies and real data examples. Keywords: classification; EM; mahalanobis distance; MCD; MM-estimation; weighted likelihood 1. Introduction Circular data arise commonly in many different fields such as earth sciences, meteorol- ogy, biology, physics, and protein bioinformatics. Examples are represented by theanalysis of wind directions [1,2], animal movements [3], handwriting recognition [4], and people orientation [5]. The reader is advised to read the work in [6,7] to become familiar with the topic of circular data and find several stimulating examples and areas of application. In this paper, we deal with the robust fitting of multivariate circular data, according to a wrapped normal model. Robust estimation is supposed to mitigate the adverse effects of outliers on estimation and inference. Outliers are unexpected anomalous values that exhibit a different pattern with respect to the rest of the data, as in the case of data orientated towards certain rare directions [3,8,9]. In circular data modeling, in the univariate case, the data can be represented as points on the circumference of the unit circle. The idea can be extended to the multivariate setting, where observations are supposed to lie on a p−dimensional torus, by revolving the unit circle in a p−dimensional manifold, with p ≥ 2. Therefore, the main aspect of circular data is periodicity, that reflects in the boundedness of the sample space and often of the parametric space. The purpose of robust estimation is twofold: On the one hand we aim to fit a model for the circular data at hand and on the other hand, an effective outlier detection rule can be derived from the robust estimation technique. The latter often gives very important insight into the data generation scheme and statistical analysis. Looking for outliers and investigating their source and nature could unveil unknown random mechanisms that are worth studying and may not have been considered otherwise. It is also important to keep in mind that outliers are model dependent, since they are defined with respect to the specified model. Then, an effective detection of outliers could be a strategy to improve the model [10]. We further remark that an outlier detection rule cannot be derived by Stats 2021, 4, 454–471. https://doi.org/10.3390/stats4020028 https://www.mdpi.com/journal/stats