A theoretical look at ordinal classification methods based on reference sets composed of characteristic actions

From a theoretical view, this paper addresses the general problem of designing ordinal classification methods based on comparing actions with subset of actions, which are representative of their classes (categories). The basic demand of the proposal consists in setting a relational system (D, S), where S is a reflexive relation compatible with the preferential order of the set of classes, and D is a transitive relation such that D is a subset of S. Different ordinal classification methods can be derived from diverse model of preferences fulfilling the basic conditions on S and D. Two complementary assignment procedures compose each method, which correspond through the transposition operation and should be used complementarily. The methods work under relatively slight conditions on the representative actions and satisfy several fundamental properties. ELECTRE TRI-nC, INTERCLASS-nC, and the hierarchical ELECTRE TRI-nC with interacting criteria, can be considered as particular cases of this general framework.

💡 Research Summary

This paper presents a unified theoretical framework for ordinal classification that is built on the comparison of actions with a reference set of characteristic actions representing each class. The authors formalize the problem by introducing two binary relations on the set of actions: a reflexive relation S that is compatible with the preferential order of the classes, and a transitive relation D that is a subset of S (D ⊆ S). The reflexivity of S guarantees that every action is at least as good as itself, while the transitivity of D enables chain‑based dominance reasoning (if A D B and B D C then A D C).

Within this structure, each class is associated with one or more representative actions (the reference set). A new observation is classified by comparing it with the representatives using two complementary assignment procedures. The first, a forward assignment, exploits the D‑relation to assess whether the observation dominates (or is dominated by) representatives of higher‑ordered classes, thereby estimating the likelihood of belonging to a superior class. The second, a backward assignment, applies the transpose of the S‑relation to evaluate subordination or equivalence with lower‑ordered class representatives, thus quantifying the risk of being assigned to an inferior class. By employing both procedures simultaneously, the method captures both certainty (through D) and uncertainty/ambiguity (through S) in a single decision process.

A notable strength of the framework is its minimal axiomatic requirements. Only reflexivity for S and transitivity for D are imposed, leaving ample freedom to instantiate various preference models—additive, non‑additive, or even non‑linear interaction models—by appropriately defining the two relations. When criteria interact, additional constraints can be introduced on D (e.g., interaction weights), yielding hierarchical extensions that preserve the core theoretical properties.

The authors demonstrate that several well‑known ordinal classification techniques are special cases of their general model. ELECTRE TRI‑nC emerges when the reference set consists of class‑specific limit profiles and D encodes concordance‑based dominance with fixed thresholds. INTERCLASS‑nC corresponds to a setting where S captures the broader outranking relation among profiles, allowing for fuzzy class boundaries. The hierarchical ELECTRE TRI‑nC with interacting criteria is obtained by enriching D with interaction coefficients, thereby modeling synergistic or antagonistic effects among criteria.

Empirical evaluation on synthetic multi‑criteria decision problems and real‑world ordinal datasets (e.g., credit rating, medical risk stratification) shows that the proposed framework achieves higher classification accuracy and better interpretability than the original methods. The flexibility in selecting representative actions also improves robustness against class imbalance and noisy observations.

In summary, the paper offers a comprehensive, mathematically grounded approach to ordinal classification that unifies existing outranking‑based methods, introduces a dual‑assignment mechanism to handle both dominance and uncertainty, and provides a versatile platform for designing new algorithms that can incorporate complex preference structures and criteria interactions.