Getulio Jose Amorim Amaral

The goal of this paper is to propose supervised learning methods for three-dimensional landmark data. We introduce novel adaptations of Linear Discriminant Analysis (SLDA), Quadratic Discriminant Analysis (SQDA), and K-Nearest Neighbors (Shape-KNN), specifically modified to operate within the non-Euclidean geometry of Kendall’s pre-shape sphere and its corresponding tangent space. To systematically evaluate the performance of the proposed methods, we conducted Monte Carlo simulations, sampling data directly from the manifold using the von Mises-Fisher distribution. The numerical results reveal that SLDA demonstrates superior robustness and stability in high-dimensional, noisy scenarios, whereas SQDA is highly susceptible to the dimensionality when sample sizes are limited. Moreover, we evaluate the models on a real-world 3D dataset. In this application, SQDA achieved the highest accuracy, showcasing its ability to effectively model class-specific covariance structures. Ultimately, this study provides a comprehensive framework and practical guidelines for classifying 3D morphological data in statistical shape analysis.

The development of unsupervised learning methods is crucial for the validation of scientific hypotheses in various fields, such as in statistical shape analysis. The concept of shape has been successfully applied in medicine and evolutionary biology in particular. Such a concept results from the extraction of location, scale, and rotation from multivariate data and paves the way for the comparison of important latent features within data sets. When working with shapes, however, it is necessary to work with data in a non-Euclidean space, the complex unit hypersphere CSk−1. In this paper, we focus on planar shapes and would like to propose clustering tools that work well on data with low concentration, which is the most critical condition. To this end, we propose an adaptation of the k-means algorithm by integrating the nearest neighbors method with stochastic distances derived from the complex Bingham distribution, a well-defined assumption on CSk−1. Using Monte Carlo experiments, we quantify the performance of our method and compare it with other current clustering tools. We also use our proposal to analyze the geometric aspects of the facial skull in children. The results from both artificial and real-world datasets favor our proposed method over existing techniques.

The Cardioid (C) distribution is one of the most important models for modeling circular data. Although some of its structural properties have been derived, this distribution is not appropriate for asymmetry and multimodal phenomena in the circle, and then extensions are required. There are various general methods that can be used to produce circular distributions. This paper proposes four extensions of the C distribution based on the beta, Kumaraswamy, gamma, and Marshall–Olkin generators. We obtain a unique linear representation of their densities and some mathematical properties. Inference procedures for the parameters are also investigated. We perform two applications on real data, where the new models are compared to the C distribution and one of its extensions.

Spatial cluster is a spatial analysis and mapping technique for cluster identification in spatial phenomena. This work proposes a nonparametric scan method for cluster detection based on the empirical likelihood, and it was compared with the Kulldorff test. The main contribution of this method is that no family of distributions has to be assumed in the analysis. The method is able to deal with the presence of overdispersion, zero inflated and other characteristics that are common in real data. The proposal was evaluated via simulations studies considering data following a zero inflated Poisson distribution. The results show that the proposed method can substantially reduce the probabilities of the error type I for zero inflated data, with low power probabilities for cluster with size less than eight observations. In a case study of measle in São Paulo, Brazil, only the Kulldorff test identified a cluster. It is recommended that a cluster detected by the spatial scan statistic of Kulldorff should be interpreted with caution when it is not confirmed by the empirical likelihood scan statistic.

Directional data are related to vectors on the unit sphere. When these vectors are not signed, this kind of data is called axial data. The Watson distribution is one of the main models for axial data. This model is equipped by two parameters: dominant axis and concentration. Based on the Renyi divergence and the Bhattacharya and Hellinger distances, we propose three hypothesis tests to check if two samples come from populations having the same concentration parameter. Results from synthetic and real data indicate that the proposed tests can yield good performance on Watson data.