Abstract
When directional data fall in the positive orthant of the unit hypersphere, a folded directional distribution is preferred over a simple directional distribution for modeling the data. Since directional data, especially axial data, can be modeled using a Watson distribution, this paper considers a folded Watson distribution for such cases. We first address the parameter estimation of this distribution using maximum likelihood, which requires a numerical algorithm to solve the likelihood equations. We use the Expectation-Maximization (EM) algorithm to obtain these estimates and to analyze the properties of the concentration estimator through simulation. Next, we propose the Bayes rule for a folded Watson distribution and evaluate its performance through simulation in various scenarios, comparing it with the Bayes rule for the Watson distribution. Finally, we present examples using both simulated and real data available in the literature.

Abstract
Os artigos incluídos neste livro, após processo de revisão por pares, são desenvolvimentos de trabalhos apresentados nas JOCLAD 2014-17 e mostram, mais uma vez, a interdisciplinariedade e a diversidade das áreas que integram esta Jornadas.

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We propose an approach to cluster and classify compositional data. We transform the compositional data into directional data using the square root transformation. To cluster the compositional data, we apply the identification of a mixture of Watson distributions on the hypersphere and to classify the compositional data into predefined groups, we apply Bayes rules based on the Watson distribution to the directional data. We then compare our clustering results with those obtained in hierarchical clustering and in the K-means clustering using the log-ratio transformations of the data and compare our classification results with those obtained in linear discriminant analysis using log-ratio transformations of the data. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

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