Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Definition 1 (Classification on Ordinal Data Problems[16, 18, 91, 93, 146]). <s> The original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation. It operates on a data table composed of a set U of objects (actions) described by a set Q of attributes. Its basic notions are: indiscernibility relation on U, lower and upper approximation of either a subset or a partition of U, dependence and reduction of attributes from Q, and decision rules derived from lower approximations and boundaries of subsets identified with decision classes. The original rough set idea is failing, however, when preference-orders of attribute domains (criteria) are to be taken into account. Precisely, it cannot handle inconsistencies following from violation of the dominance principle. This inconsistency is characteristic for preferential information used in multicriteria decision analysis (MCDA) problems, like sorting, choice or ranking. In order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary. The main change is the substitution of the indiscernibility relation by a dominance relation, which permits approximation of ordered sets in multicriteria sorting. To approximate preference relations in multicriteria choice and ranking problems, another change is necessary: substitution of the data table by a pairwise comparison table, where each row corresponds to a pair of objects described by binary relations on particular criteria. In all those MCDA problems, the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model. It is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax. In order to workout a recommendation in one of the MCDA problems, we propose exploitation procedures of the set of decision rules. Finally, some other recently obtained results are given: rough approximations by means of similarity relations, rough set handling of missing data, comparison of the rough set model with Sugeno and Choquet integrals, and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive. <s> BIB001 </s> Multicriteria Models for Learning Ordinal Data: A Literature Review <s> Definition 1 (Classification on Ordinal Data Problems[16, 18, 91, 93, 146]). <s> The mathematical representation of human preferences has been a subject of study for researchers in different fields. In multi-criteria decision making (MCDM) and fuzzy modeling, preference models are typically constructed by interacting with the human decision maker (DM). However, it is known that a DM often has difficulties to specify precise values for certain parameters of the model. He/she instead feels more comfortable to give holistic judgements for some of the alternatives. Inference and elicitation procedures then assist the DM to find a satisfactory model and to assess unjudged alternatives. In a related but more statistical way, machine learning algorithms can also infer preference models with similar setups and purposes, but here less interaction with the DM is required/allowed. In this article we discuss the main differences between both types of inference and, in particular, we present a hybrid approach that combines the best of both worlds. This approach consists of a very general kernel-based framework for constructing and inferring preference models. Additive models, for which interpretability is preserved, and utility models can be considered as special cases. Besides generality, important benefits of this approach are its robustness to noise and good scalability. We show in detail how this framework can be utilized to aggregate single-criterion outranking relations, resulting in a flexible class of preference models for which domain knowledge can be specified by a DM. <s> BIB002
Classifying on ordinal data problems consists on finding the best mapping f : Pattern x x x is also referred as instance, example or alternative. Moreover, x x x can be represented in a vector fashion where each entry is identified as a feature, attribute or criterion, i.e., x x x = {x 1 ,... ,x d }. A dataset is a tuple consisted of N patterns and its target classes (or outcomes), and , the order relation on the set of classes. Literature usually differentiates attribute and criterion BIB001 . Consequently, the problem in analysis can be substantially different. In an ordinal data problem as the credit scoring, an alternative (to which client a loan should be granted) is characterised by several criteria, each one representing a level of importance to the decision maker (DM) (the bank). Here, criteria is used instead of attribute being the former more adequate for the ordinal problem BIB001 BIB002 . The usage of the term ranking is also common in the MC field. However, such term is usually mentioned to other subjects aside classification.