A Role of (Not) Crisp Discernibility in Rough Set Approach to Numeric Feature Selection

Abstract

We investigate the rough-set-based framework for feature selection in decision tables with numeric attributes. We compare functions evaluating subsets of attributes with respect to their potential in determining the distinguished decision attribute by means of two alternative methods: discernibility-based functions over discretized numeric data, as well as distance-based functions often used in the fuzzy-rough approaches to feature selection. In both cases, the idea is to compare objects belonging to different decision classes, by verifying whether they can be distinguished from each other by using discretized attributes or measuring distances between their values over original numeric attributes. We draw a correspondence between functions evaluating subsets of numeric attributes according to both methodologies. For a subset of numeric attributes, we consider a function measuring the amount of pairs of objects belonging to different decision classes that are not discerned by discretized attributes, averaged over all possible choices of binary discretization cuts over the attribute ranges. We prove that such a function can be rewritten by means of distances between the original numeric attributes. Namely, it is equal to the average fuzzy indiscernibility function computed by using the product t-norm combining indiscernibility degrees obtained over particular attributes. © Springer-Verlag Berlin Heidelberg 2012.