Multidimensional scaling of two-mode three-way asymmetric dissimilarities: finding archetypal profiles and clustering

TL;DR

This paper extends multidimensional scaling techniques to analyze three-way asymmetric dissimilarities, enabling the identification of archetypal profiles and clustering in complex relational data with improved interpretability and computational efficiency.

Contribution

It introduces a novel extension of the h-plot method for three-way asymmetric proximity data, allowing extraction of archetypal profiles and clustering with an explicit eigenvector solution.

Findings

Method effectively visualizes three-way asymmetric dissimilarities.

Enables identification of archetypal profiles and clusters.

Demonstrated on a financial dataset with reproducible results.

Abstract

Multidimensional scaling visualizes dissimilarities among objects and reduces data dimensionality. While many methods address symmetric proximity data, asymmetric and especially three-way proximity data (capturing relationships across multiple occasions) remain underexplored. Recent developments, such as the h-plot, enable the analysis of asymmetric and non-reflexive relationships by embedding dissimilarities in a Euclidean space, allowing further techniques like archetypoid analysis to identify representative extreme profiles. However, no existing methods extract archetypal profiles from three-way asymmetric proximity data. This work extends the h-plot methodology to three-way proximity data under both symmetric and asymmetric, conditional and unconditional frameworks. The proposed approach offers several advantages: intuitive interpretability through a unified Euclidean representation; an explicit, eigenvector-based analytical solution free from local minima; scale invariance under linear transformations; computational efficiency for large matrices; and a straightforward goodness-of-fit evaluation. Furthermore, it enables the identification of archetypal profiles and clustering structures for three-way asymmetric proximities. Its performance is compared with existing models for multidimensional scaling and clustering, and illustrated through a financial application. All data and code are provided to facilitate reproducibility.