Granularity refined by knowledge: contingency tables and rough sets as tools of discovery

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 signiicance 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 reenement, 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 o f k n o wledge. 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. 1. ROUGH SETS REPRESENTATION BY CONTINGENCY TABLES Both rough sets and contingency tables are founded on a similar idea of granular empirical data. Both approaches use the representation of empirical objects by n-tuples or vector of attribute values. This representation is common in statistics, databases, machine learning, pattern recognition and many other areas. Speciic to rough sets and contingency tables approaches are indiscernibility classes. Objects are indiscernible if their property tuples are the same. That can happen in practice when domains of attributes contain small numbers of values. That happens in many databases, especially when attribute values are determined with limited accuracy. But if that is not the case, binning also known as discretization, can be used. Rough sets were introduced in 1980's by Zdzislaw P awlak 1991, while the history of contingency tables is several decades older. They play a major role in the theory and practice of statistics Contingency tables have been used as a general tool for expressing knowledge and for knowledge reenement in 49er Zembowicz and Zytkow, 1993, 1996. In this paper we place rough sets in the framework of contingency tables. We argue that contingency tables provide an added …