<title>Granularity refined by knowledge: contingency tables and rough sets as tools of discovery</title>

Abstract

Contingency tables represent data in a granular way and are a well-established tool for inductive generalization of knowledge from data. We show that the basic concepts of rough sets, such as concept approximation, indiscernibility, and reduct can be expressed in the language of contingency tables. We further demonstrate the relevance to rough sets theory of additional probabilistic information available in contingency tables and in particular of statistical tests of significance and predictive strength applied to contingency tables. Tests of both type can help the evaluation mechanisms used in inductive generalization based on rough sets. Granularity of attributes can be improved in feedback with knowledge discovered in data. We demonstrate how 49er's facilities for (1) contingency table refinement, for (2) column and row grouping based on correspondence analysis, and (3) the search for equivalence relations between attributes improve both granularization of attributes and the quality of knowledge. Finally we demonstrate the limitations of knowledge viewed as concept approximation, which is the focus of rough sets. Transcending that focus and reorienting towards the predictive knowledge and towards the related distinction between possible and impossible (or statistically improbable) situations will be very useful in expanding the rough sets approach to more expressive forms of knowledge.