[Paper Review] Depth and depth-based classification with R-package ddalpha

Oleksii Pokotylo, Pavlo Mozharovskyi+1 more|Kölner Universitäts PublikationsServer (Universität zu Köln)|Aug 14, 2016

Advanced Statistical Methods and Models36 references20 citations

TL;DR

This paper introduces the R package ddalpha, which implements data depth and depth-based classification methods for multivariate and functional data. It proposes the $DD\alpha$-procedure, a robust, nonparametric classifier that transforms data into depth space using multiple depth functions and applies an $\alpha$-procedure separator, achieving superior classification accuracy compared to $k$-NN and polynomial classifiers in empirical tests.

ABSTRACT

Following the seminal idea of Tukey, data depth is a function that measures how close an arbitrary point of the space is located to an implicitly defined center of a data cloud. Having undergone theoretical and computational developments, it is now employed in numerous applications with classification being the most popular one. The R-package ddalpha is a software directed to fuse experience of the applicant with recent achievements in the area of data depth and depth-based classification. ddalpha provides an implementation for exact and approximate computation of most reasonable and widely applied notions of data depth. These can be further used in the depth-based multivariate and functional classifiers implemented in the package, where the $DDα$-procedure is in the main focus. The package is expandable with user-defined custom depth methods and separators. The implemented functions for depth visualization and the built-in benchmark procedures may also serve to provide insights into the geometry of the data and the quality of pattern recognition.

Motivation & Objective

• To develop a comprehensive, user-extensible R package for data depth and depth-based classification.
• To address limitations in existing depth-based classifiers, such as instability in polynomial fitting and poor handling of points beyond the convex hull (outsiders).
• To provide efficient, robust, and computationally feasible depth computation methods for high-dimensional and functional data.
• To enable flexible, nonparametric classification through depth transformation and the $\alpha$-procedure separator.
• To support empirical parameter tuning via built-in benchmarking and cross-validation procedures.

Proposed method

• Transform each training point into a q-dimensional vector of depth values relative to each of q classes, forming a depth space.
• Apply the $\alpha$-procedure, a nonparametric, robust, and computationally efficient separator, in the depth space for classification.
• Handle 'outsiders'—points with zero depth beyond the convex hull—via a dedicated treatment strategy to maintain classification capability.
• Support functional data classification through a location-slope (LS-) transform prior to depth computation.
• Implement both exact and approximate algorithms for various depth functions (e.g., halfspace, simplicial, projection, Mahalanobis) with configurable parameters.
• Enable user-defined depth methods and separators, enhancing extensibility and adaptability to specific data geometries.

Experimental results

Research questions

• RQ1 How can data depth be effectively leveraged for multivariate and functional classification in a computationally efficient and robust manner?
• RQ2 What depth functions best preserve data topology while minimizing computational burden and instability in classification?
• RQ3 How can points lying outside the convex hull of training data (outsiders) be meaningfully classified without compromising performance?
• RQ4 What is the relative performance of the $DD\alpha$-procedure compared to existing depth-based and non-depth-based classifiers like $k$-NN and polynomial classifiers?
• RQ5 How can parameters for depth computation and classification be automatically tuned using cross-validation and benchmarking procedures?

Key findings

• The $DD\alpha$-procedure achieves lower error rates than both the polynomial classifier and $k$-NN in empirical evaluations on real data.
• The $DD\alpha$ classifier demonstrates superior robustness and stability compared to polynomial-based methods, which can suffer from numerical instability.
• Approximate depth computation (e.g., random halfspace, projection, and simplicial approximations) enables scalable analysis with controlled precision loss.
• Outsider treatment provides a computationally efficient solution for points beyond the convex hull, preserving classification capability without requiring complex reweighting.
• Cross-validation of depth functions, separators, and parameters (e.g., $L$, $S$, $k$, num.directions) significantly improves classification accuracy.
• The package's benchmarking framework allows systematic parameter tuning, with error rates estimated via iterative cross-validation across multiple parameter sets.