Anna Emilie J. Wedenborg

Anna Emilie J. Wedenborg, Michael Alexander Harborg, Andreas Bigom et al.

This paper introduces a novel framework for Archetypal Analysis (AA) tailored to ordinal data, particularly from questionnaires. Unlike existing methods, the proposed method, Ordinal Archetypal Analysis (OAA), bypasses the two-step process of transforming ordinal data into continuous scales and operates directly on the ordinal data. We extend traditional AA methods to handle the subjective nature of questionnaire-based data, acknowledging individual differences in scale perception. We introduce the Response Bias Ordinal Archetypal Analysis (RBOAA), which learns individualized scales for each subject during optimization. The effectiveness of these methods is demonstrated on synthetic data and the European Social Survey dataset, highlighting their potential to provide deeper insights into human behavior and perception. The study underscores the importance of considering response bias in cross-national research and offers a principled approach to analyzing ordinal data through Archetypal Analysis.

A. Emilie J. Wedenborg, Morten Mørup

Archetypal analysis (AA) is a matrix decomposition method that identifies distinct patterns using convex combinations of the data points denoted archetypes with each data point in turn reconstructed as convex combinations of the archetypes. AA thereby forms a polytope representing trade-offs of the distinct aspects in the data. Most existing methods for AA are designed for continuous data and do not exploit the structure of the data distribution. In this paper, we propose two new optimization frameworks for archetypal analysis for binary data. i) A second order approximation of the AA likelihood based on the Bernoulli distribution with efficient closed-form updates using an active set procedure for learning the convex combinations defining the archetypes, and a sequential minimal optimization strategy for learning the observation specific reconstructions. ii) A Bernoulli likelihood based version of the principal convex hull analysis (PCHA) algorithm originally developed for least squares optimization. We compare these approaches with the only existing binary AA procedure relying on multiplicative updates and demonstrate their superiority on both synthetic and real binary data. Notably, the proposed optimization frameworks for AA can easily be extended to other data distributions providing generic efficient optimization frameworks for AA based on tailored likelihood functions reflecting the underlying data distribution.