Approximation Quality for Sorting Rules

A method is presented for the approximation quality for sorting rules, which is based on Rough Set Data Analysis. Several types of rules are discussed: The “nominal nominal” case (NN) (the classical Rough Set approach), “nominal ordinal” (NO) rules, and “ordinal ordinal” (OO) rules, and its generalisation to “(nominal, ordinal) ordinal” rules (NO-O). We provide a significance test for the overall approximation quality, and a test for partial influence of attributes based on the bootstrap technology. For the bivariate case, the relationship of – statistics and the proposed approximation quality measure is discussed. It can be shown that in this case a simple linear transformation of the Kendall tau correlation forms an upper bound for the approximation quality of sorting rules. A competing model is also studied. Whereas this method is a promising tool in case of searching for global consistency, we demonstrate that in case of local perturbations in the data set the method may offer questionable results, and that it is dissociated from the theory it claims to support. In the final Section, an example illustrates the introduced concepts and procedures.