Contributions to depth-based classification and computation of the Tukey depth

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Classification has become increasingly valuable across various scientific and everyday contexts, necessitating ongoing enhancements to its methodological foundations. This book explores recent advancements in data classification through the innovative concept of the statistical depth function. The initial chapter offers a brief overview of depth and classification. Chapter 2 presents a depth-based methodology for supervised learning known as the DDalpha-classifier, which employs a two-step process: first, it maps the learning sample into a depth space relative to training classes (DD-plot), and then optimally separates it using the projective-invariant alpha-procedure. This technique is nonparametric, robust, and efficient. Chapter 3 delves into the issue of "outsiders"—points that cannot be classified based solely on depth values—offering various treatments and discussing classifier configuration. Chapter 4 extends these concepts to functional data, demonstrating the Bayes optimality of the procedure under standard conditions and conducting extensive experiments with both synthetic and 50 real-world classification problems. Finally, Chapter 5 introduces two algorithms for the exact computation of Tukey depth: one utilizes breadth-first spreading over cone segmentation via linear programming, while the other is combinatorial, facilitating straightforward implementation.