Monotone and interval approximation of qualitative data (Q579809)

We discuss some issues of constructing quantitative models based on qualitative data. The principle of monotone approximation is proposed for ordinal data, and the principle of interval approximation for nominal data. These approaches are implemented with the aid of certain functionals characterizing the quality of approximation of the original data by the model. The functionals define monotone or nominal regression on qualitative data. The specific feature of the regression problem is that the model fits are without observable counterparts. Only relations on the model data are observable. For instance, we may use pairwise better-worse comparisons, information about the degree of association between objects, information about membership in the same class or in different classes, etc. The same approach is applicable when the model fits have observable counterparts, e.g., the problem of automatic diagnosis from qualitative data.