Abstract
When directional data fall on the positive orthant of the hypersphere, they can be modeled using a folded directional distribution. In this paper, we introduce the folded von Mises-Fisher distribution and propose the Bayes classification rule for this distribution. Then we evaluate the performance of this rule and we compare it with the classification rule for the von Mises-Fisher distribution. Finally, we present examples using spherical data from the literature, determining the error rate using the folded von Mises-Fisher rule and comparing it with the error rate obtained using the von Mises-Fisher rule.

Abstract
Data sets that contain an excessive number of zeros appear in several fields of applications. This chapter considers a zero-inflated Lomax distribution as a possible model for these types of data, and presents and analyzes the performance of a Shewhart control chart for process monitoring. Several approaches allow for frequent zero observations, and among them, the most common are zero-inflated models and hurdle models in case of count data, and the use of zero-inflated distributions to model semi-continuous data, that is, data from a continuous distribution with one or more than one point of mass. The chapter presents some motivation for the use of the zero-inflated Lomax distribution together with some properties of this distribution. It proposes a Shewhart-type control chart for monitoring zero-inflated Lomax data, and analyzes its performance under some scenarios.

Abstract
This data-based study aims to understand the progress made by EU countries in recent years on key areas of the circular economy by analyzing some indicators. The data have been collected from the Eurostat database over the period 2013-2021. After a preliminary analysis of the data set, a double principal component analysis has been used. This approach provides insights into the evolution of the countries and correlations between the indicators, highlighting which EU countries are the most (or least) similar to each other. The findings of this study indicate that EU countries collectively have advanced towards a circular economy in various indicators, with certain countries showing more notable progress than others. Some countries are even considered outliers, for positive or negative reasons, in some of the indicators. Overall, the Western EU countries perform better than the Eastern countries on most of the indicators analyzed, especially for resource productivity, municipal waste management, circularity rate, private investment in the circular economy sectors, and gross added value in circular economy sectors. The exceptions are for the generation of municipal waste, percentage of persons employed in circular economy sectors, and greenhouse gas emissions, the ones where the Eastern countries in general perform better. © 2025 Elsevier B.V., All rights reserved.

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
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.

Abstract
Introduction: School lunches are essential in promoting a sustainable, healthy diet for children. However, students’ adherence in developed countries is modest. Objectives: A cross-sectional study with the objective of evaluating school lunch participation and exploring the factors influencing participation and its associations with sociodemographic characteristics, socioeconomic status, body weight, and adherence to the Mediterranean Dietary Pattern. Methodology: Baseline results from the R23 project, collected during the first five weeks of the 2022/23 school year, were used. In total, 806 students, aged 10–16 years, from the 2nd and 3rd cycles of basic education in Portugal, Benavente, participated in this study. Using a face-to-face structured questionnaire, students’ sociodemographic data, socioeconomic status, adherence to the Mediterranean Dietary Pattern, and school lunch participation were assessed. Daily school lunch participation was obtained via the school lunch booking service. Anthropometric measurements (weight, height and waist circumference) were performed on all students with signed consent forms. Results: The school lunch participation rate was 25.3%. Overall, 39.1% of students reported never attending the school canteen. Only 27% of the students eligible for Free or Reduced-Price Meals had an effective adherence to school lunch, and the non-participation rate in this group was 31%. Key barriers included living near to school and dissatisfaction with meal quality. Students' school lunch participation was significantly associated with age, grade, eligibility for Free or Reduced-Price Meals, and adherence to Mediterranean Dietary Pattern. Younger students and students eligible for Free or Reduced-Price Meals went to the canteen more often. While school lunch participation was positively and significantly associated with higher adherence to Mediterranean Dietary Pattern, no significant relationship was observed with body mass index or waist-to-height ratio. Students with greater adherence to Mediterranean Dietary Pattern displayed healthier body mass index and waist-to-height ratio. Conclusions: School lunch participation among adolescents was low, but increased participation was associated with higher adherence to the Mediterranean Dietary Pattern. These findings highlight the importance of school lunches in promoting healthier eating habits.