Triadic Distance Models: Axiomatization and Least Squares Representation

Area of Science:

• Multivariate statistics
• Psychometrics
• Data analysis

Background:

• Traditional distance models focus on pairwise data.
• Analyzing three-way proximity data requires generalized distance concepts.
• Existing methods lack robust frameworks for triadic relationships.

Purpose of the Study:

• To develop an axiomatic framework for triadic dissimilarity, similarity, and distance.
• To introduce and analyze novel distance models for three-way data.
• To provide algorithms for fitting these models and evaluating data representation quality.

Main Methods:

• Axiomatic characterization of triadic measures.
• Detailed study of Minkowski-p (Mp) and presence-absence variable models.
• Development of monotonically convergent algorithms for weighted least squares fitting.
• Derivation of an additive decomposition for scale-free fit evaluation.

Main Results:

• Established an axiomatic framework satisfying the tetrahedral inequality for triadic measures.
• Demonstrated the utility of Mp models and presence-absence models.
• Presented algorithms for fitting Euclidean M1 and M2 models.
• Developed a method for scale-free assessment of model fit.

Conclusions:

• The proposed framework and models effectively generalize distance concepts to three-way data.
• The developed algorithms provide efficient tools for analyzing complex proximity data.
• The methods enable robust and interpretable analysis of three-way, three-mode tables.